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%I M1581 N0616 #730 Apr 17 2024 11:10:04
%S 0,2,6,12,20,30,42,56,72,90,110,132,156,182,210,240,272,306,342,380,
%T 420,462,506,552,600,650,702,756,812,870,930,992,1056,1122,1190,1260,
%U 1332,1406,1482,1560,1640,1722,1806,1892,1980,2070,2162,2256,2352,2450,2550
%N Oblong (or promic, pronic, or heteromecic) numbers: a(n) = n*(n+1).
%C 4*a(n) + 1 are the odd squares A016754(n).
%C The word "pronic" (used by Dickson) is incorrect. - _Michael Somos_
%C According to the 2nd edition of Webster, the correct word is "promic". - _R. K. Guy_
%C a(n) is the number of minimal vectors in the root lattice A_n (see Conway and Sloane, p. 109).
%C Let M_n denote the n X n matrix M_n(i, j) = (i + j); then the characteristic polynomial of M_n is x^(n-2) * (x^2 - a(n)*x - A002415(n)). - _Benoit Cloitre_, Nov 09 2002
%C The greatest LCM of all pairs (j, k) for j < k <= n for n > 1. - _Robert G. Wilson v_, Jun 19 2004
%C First differences are a(n+1) - a(n) = 2*n + 2 = 2, 4, 6, ... (while first differences of the squares are (n+1)^2 - n^2 = 2*n + 1 = 1, 3, 5, ...). - _Alexandre Wajnberg_, Dec 29 2005
%C 25 appended to these numbers corresponds to squares of numbers ending in 5 (i.e., to squares of A017329). - _Lekraj Beedassy_, Mar 24 2006
%C A rapid (mental) multiplication/factorization technique -- a generalization of Lekraj Beedassy's comment: For all bases b >= 2 and positive integers n, c, d, k with c + d = b^k, we have (n*b^k + c)*(n*b^k + d) = a(n)*b^(2*k) + c*d. Thus the last 2*k base-b digits of the product are exactly those of c*d -- including leading 0(s) as necessary -- with the preceding base-b digit(s) the same as a(n)'s. Examples: In decimal, 113*117 = 13221 (as n = 11, b = 10 = 3 + 7, k = 1, 3*7 = 21, and a(11) = 132); in octal, 61*67 = 5207 (52 is a(6) in octal). In particular, for even b = 2*m (m > 0) and c = d = m, such a product is a square of this type. Decimal factoring: 5609 is immediately seen to be 71*79. Likewise, 120099 = 301*399 (k = 2 here) and 99990000001996 = 9999002*9999998 (k = 3). - _Rick L. Shepherd_, Jul 24 2021
%C Number of circular binary words of length n + 1 having exactly one occurrence of 01. Example: a(2) = 6 because we have 001, 010, 011, 100, 101 and 110. Column 1 of A119462. - _Emeric Deutsch_, May 21 2006
%C The sequence of iterated square roots sqrt(N + sqrt(N + ...)) has for N = 1, 2, ... the limit (1 + sqrt(1 + 4*N))/2. For N = a(n) this limit is n + 1, n = 1, 2, .... For all other numbers N, N >= 1, this limit is not a natural number. Examples: n = 1, a(1) = 2: sqrt(2 + sqrt(2 + ...)) = 1 + 1 = 2; n = 2, a(2) = 6: sqrt(6 + sqrt(6 + ...)) = 1 + 2 = 3. - _Wolfdieter Lang_, May 05 2006
%C Nonsquare integers m divisible by ceiling(sqrt(m)), except m = 0. - _Max Alekseyev_, Nov 27 2006
%C The number of off-diagonal elements of an (n + 1) X (n + 1) matrix. - _Artur Jasinski_, Jan 11 2007
%C a(n) is equal to the number of functions f:{1, 2} -> {1, 2, ..., n + 1} such that for a fixed x in {1, 2} and a fixed y in {1, 2, ..., n + 1} we have f(x) <> y. - Aleksandar M. Janjic and _Milan Janjic_, Mar 13 2007
%C Numbers m >= 0 such that round(sqrt(m+1)) - round(sqrt(m)) = 1. - _Hieronymus Fischer_, Aug 06 2007
%C Numbers m >= 0 such that ceiling(2*sqrt(m+1)) - 1 = 1 + floor(2*sqrt(m)). - _Hieronymus Fischer_, Aug 06 2007
%C Numbers m >= 0 such that fract(sqrt(m+1)) > 1/2 and fract(sqrt(m)) < 1/2 where fract(x) is the fractional part (fract(x) = x - floor(x), x >= 0). - _Hieronymus Fischer_, Aug 06 2007
%C X values of solutions to the equation 4*X^3 + X^2 = Y^2. To find Y values: b(n) = n(n+1)(2n+1). - _Mohamed Bouhamida_, Nov 06 2007
%C Nonvanishing diagonal of A132792, the infinitesimal Lah matrix, so "generalized factorials" composed of a(n) are given by the elements of the Lah matrix, unsigned A111596, e.g., a(1)*a(2)*a(3) / 3! = -A111596(4,1) = 24. - _Tom Copeland_, Nov 20 2007
%C If Y is a 2-subset of an n-set X then, for n >= 2, a(n-2) is the number of 2-subsets and 3-subsets of X having exactly one element in common with Y. - _Milan Janjic_, Dec 28 2007
%C a(n) coincides with the vertex of a parabola of even width in the Redheffer matrix, directed toward zero. An integer p is prime if and only if for all integer k, the parabola y = kx - x^2 has no integer solution with 1 < x < k when y = p; a(n) corresponds to odd k. - _Reikku Kulon_, Nov 30 2008
%C The third differences of certain values of the hypergeometric function 3F2 lead to the squares of the oblong numbers i.e., 3F2([1, n + 1, n + 1], [n + 2, n + 2], z = 1) - 3*3F2([1, n + 2, n + 2], [n + 3, n + 3], z = 1) + 3*3F2([1, n + 3, n + 3], [n + 4, n + 4], z = 1) - 3F2([1, n + 4, n + 4], [n + 5, n + 5], z = 1) = (1/((n+2)*(n+3)))^2 for n = -1, 0, 1, 2, ... . See also A162990. - _Johannes W. Meijer_, Jul 21 2009
%C a(A007018(n)) = A007018(n+1), see sequence A007018 (1, 2, 6, 42, 1806, ...), i.e., A007018(n+1) = A007018(n)-th oblong numbers. - _Jaroslav Krizek_, Sep 13 2009
%C Generalized factorials, [a.(n!)] = a(n)*a(n-1)*...*a(0) = A010790(n), with a(0) = 1 are related to A001263. - _Tom Copeland_, Sep 21 2011
%C For n > 1, a(n) is the number of functions f:{1, 2} -> {1, ..., n + 2} where f(1) > 1 and f(2) > 2. Note that there are n + 1 possible values for f(1) and n possible values for f(2). For example, a(3) = 12 since there are 12 functions f from {1, 2} to {1, 2, 3, 4, 5} with f(1) > 1 and f(2) > 2. - _Dennis P. Walsh_, Dec 24 2011
%C a(n) gives the number of (n + 1) X (n + 1) symmetric (0, 1)-matrices containing two ones (see [Cameron]). - _L. Edson Jeffery_, Feb 18 2012
%C a(n) is the number of positions of a domino in a rectangled triangular board with both legs equal to n + 1. - _César Eliud Lozada_, Sep 26 2012
%C a(n) is the number of ordered pairs (x, y) in [n+2] x [n+2] with |x-y| > 1. - _Dennis P. Walsh_, Nov 27 2012
%C a(n) is the number of injective functions from {1, 2} into {1, 2, ..., n + 1}. - _Dennis P. Walsh_, Nov 27 2012
%C a(n) is the sum of the positive differences of the partition parts of 2n + 2 into exactly two parts (see example). - _Wesley Ivan Hurt_, Jun 02 2013
%C a(n)/a(n-1) is asymptotic to e^(2/n). - _Richard R. Forberg_, Jun 22 2013
%C Number of positive roots in the root system of type D_{n + 1} (for n > 2). - _Tom Edgar_, Nov 05 2013
%C Number of roots in the root system of type A_n (for n > 0). - _Tom Edgar_, Nov 05 2013
%C From _Felix P. Muga II_, Mar 18 2014: (Start)
%C a(m), for m >= 1, are the only positive integer values t for which the Binet-de Moivre formula for the recurrence b(n) = b(n-1) + t*b(n-2) with b(0) = 0 and b(1) = 1 has a root of a square. PROOF (as suggested by _Wolfdieter Lang_, Mar 26 2014): The sqrt(1 + 4t) appearing in the zeros r1 and r2 of the characteristic equation is (a positive) integer for positive integer t precisely if 4t + 1 = (2m + 1)^2, that is t = a(m), m >= 1. Thus, the characteristic roots are integers: r1 = m + 1 and r2 = -m.
%C Let m > 1 be an integer. If b(n) = b(n-1) + a(m)*b(n-2), n >= 2, b(0) = 0, b(1) = 1, then lim b(n+1)/b(n) = m + 1 as n approaches infinity. (End)
%C Cf. A130534 for relations to colored forests, disposition of flags on flagpoles, and colorings of the vertices (chromatic polynomial) of the complete graphs (here simply K_2). - _Tom Copeland_, Apr 05 2014
%C The set of integers n for which n + sqrt(n + sqrt(n + sqrt(n + sqrt(n + ...)... is an integer. - _Leslie Koller_, Apr 11 2014
%C a(n-1) is the largest number k such that (n*k)/(n+k) is an integer. - _Derek Orr_, May 22 2014
%C Number of ways to place a domino and a singleton on a strip of length n - 2. - _Ralf Stephan_, Jun 09 2014
%C With offset 1, this appears to give the maximal number of crossings between n nonconcentric circles of equal radius. - _Felix Fröhlich_, Jul 14 2014
%C For n > 1, the harmonic mean of the n values a(1) to a(n) is n + 1. The lowest infinite sequence of increasing positive integers whose cumulative harmonic mean is integral. - _Ian Duff_, Feb 01 2015
%C a(n) is the maximum number of queens of one color that can coexist without attacking one queen of the opponent's color on an [n+2] X [n+2] chessboard. The lone queen can be placed in any position on the perimeter of the board. - _Bob Selcoe_, Feb 07 2015
%C With a(0) = 1, a(n-1) is the smallest positive number not in the sequence such that Sum_{i = 1..n} 1/a(i-1) has a denominator equal to n. - _Derek Orr_, Jun 17 2015
%C The positive members of this sequence are a proper subsequence of the so-called 1-happy couple products A007969. See the W. Lang link there, eq. (4), with Y_0 = 1, with a table at the end. - _Wolfdieter Lang_, Sep 19 2015
%C For n > 0, a(n) is the reciprocal of the area bounded above by y = x^(n-1) and below by y = x^n for x in the interval [0, 1]. Summing all such areas visually demonstrates the formula below giving Sum_{n >= 1} 1/a(n) = 1. - _Rick L. Shepherd_, Oct 26 2015
%C It appears that, except for a(0) = 0, this is the set of positive integers n such that x*floor(x) = n has no solution. (For example, to get 3, take x = -3/2.) - _Melvin Peralta_, Apr 14 2016
%C If two independent real random variables, x and y, are distributed according to the same exponential distribution: pdf(x) = lambda * exp(-lambda * x), lambda > 0, then the probability that n - 1 <= x/y < n is given by 1/a(n). - _Andres Cicuttin_, Dec 03 2016
%C a(n) is equal to the sum of all possible differences between n different pairs of consecutive odd numbers (see example). - _Miquel Cerda_, Dec 04 2016
%C a(n+1) is the dimension of the space of vector fields in the plane with polynomial coefficients up to order n. - _Martin Licht_, Dec 04 2016
%C It appears that a(n) + 3 is the area of the largest possible pond in a square (A268311). - _Craig Knecht_, May 04 2017
%C Also the number of 3-cycles in the (n+3)-triangular honeycomb acute knight graph. - _Eric W. Weisstein_, Jul 27 2017
%C Also the Wiener index of the (n+2)-wheel graph. - _Eric W. Weisstein_, Sep 08 2017
%C The left edge of a Floyd's triangle that consists of even numbers: 0; 2, 4; 6, 8, 10; 12, 14, 16, 18; 20, 22, 24, 26, 28; ... giving 0, 2, 6, 12, 20, ... The right edge generates A028552. - _Waldemar Puszkarz_, Feb 02 2018
%C a(n+1) is the order of rowmotion on a poset obtained by adjoining a unique minimal (or maximal) element to a disjoint union of at least two chains of n elements. - _Nick Mayers_, Jun 01 2018
%C From _Juhani Heino_, Feb 05 2019: (Start)
%C For n > 0, 1/a(n) = n/(n+1) - (n-1)/n.
%C For example, 1/6 = 2/3 - 1/2; 1/12 = 3/4 - 2/3.
%C Corollary of this:
%C Take 1/2 pill.
%C Next day, take 1/6 pill. 1/2 + 1/6 = 2/3, so your daily average is 1/3.
%C Next day, take 1/12 pill. 2/3 + 1/12 = 3/4, so your daily average is 1/4.
%C And so on. (End)
%C From _Bernard Schott_, May 22 2020: (Start)
%C For an oblong number m >= 6 there exists a Euclidean division m = d*q + r with q < r < d which are in geometric progression, in this order, with a common integer ratio b. For b >= 2 and q >= 1, the Euclidean division is m = qb*(qb+1) = qb^2 * q + qb where (q, qb, qb^2) are in geometric progression.
%C Some examples with distinct ratios and quotients:
%C 6 | 4 30 | 25 42 | 18
%C ----- ----- -----
%C 2 | 1 , 5 | 1 , 6 | 2 ,
%C and also:
%C 42 | 12 420 | 100
%C ----- -----
%C 6 | 3 , 20 | 4 .
%C Some oblong numbers also satisfy a Euclidean division m = d*q + r with q < r < d that are in geometric progression in this order but with a common noninteger ratio b > 1 (see A335064). (End)
%C For n >= 1, the continued fraction expansion of sqrt(a(n)) is [n; {2, 2n}]. For n=1, this collapses to [1; {2}]. - _Magus K. Chu_, Sep 09 2022
%C a(n-2) is the maximum irregularity over all trees with n vertices. The extremal graphs are stars. (The irregularity of a graph is the sum of the differences between the degrees over all edges of the graph.) - _Allan Bickle_, May 29 2023
%C For n > 0, number of diagonals in a regular 2*(n+1)-gon that are not parallel to any edge (cf. A367204). - _Paolo Xausa_, Mar 30 2024
%C a(n-1) is the maximum Zagreb index over all trees with n vertices. The extremal graphs are stars. (The Zagreb index of a graph is the sum of the squares of the degrees over all vertices of the graph.) - _Allan Bickle_, Apr 11 2024
%D W. W. Berman and D. E. Smith, A Brief History of Mathematics, 1910, Open Court, page 67.
%D J. H. Conway and R. K. Guy, The Book of Numbers, 1996, p. 34.
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag.
%D L. E. Dickson, History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, p. 357, 1952.
%D L. E. Dickson, History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, pp. 6, 232-233, 350 and 407, 1952.
%D H. Eves, An Introduction to the History of Mathematics, revised, Holt, Rinehart and Winston, 1964, page 72.
%D Nicomachus of Gerasa, Introduction to Arithmetic, translation by Martin Luther D'Ooge, Ann Arbor, University of Michigan Press, 1938, p. 254.
%D Granino A. Korn and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968), pp. 980-981.
%D C. S. Ogilvy and J. T. Anderson, Excursions in Number Theory, Oxford University Press, 1966, pp. 61-62.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D F. J. Swetz, From Five Fingers to Infinity, Open Court, 1994, p. 219.
%H T. D. Noe, <a href="/A002378/b002378.txt">Table of n, a(n) for n = 0..1000</a>
%H D. Applegate, M. LeBrun and N. J. A. Sloane, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Sloane/carry2.html">Dismal Arithmetic</a>, J. Int. Seq. 14 (2011) # 11.9.8.
%H Allan Bickle and Zhongyuan Che, <a href="https://doi.org/10.1016/j.dam.2023.01.020">Irregularities of Maximal k-degenerate Graphs</a>, Discrete Applied Math. 331 (2023) 70-87.
%H Allan Bickle, <a href="https://doi.org/10.20429/tag.2024.000105">A Survey of Maximal k-degenerate Graphs and k-Trees</a>, Theory and Applications of Graphs 0 1 (2024) Article 5.
%H Allan Bickle, <a href="https://ajc.maths.uq.edu.au/pdf/89/ajc_v89_p167.pdf">Zagreb Indices of Maximal k-degenerate Graphs</a>, Australas. J. Combin. 89 1 (2024) 167-178.
%H Alin Bostan, Frédéric Chyzak, and Vincent Pilaud, <a href="https://arxiv.org/abs/2303.10986">Refined product formulas for Tamari intervals</a>, arXiv:2303.10986 [math.CO], 2023.
%H H. Bottomley, <a href="/A002378/a002378.gif">Illustration of initial terms of A000217, A002378</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=410">Encyclopedia of Combinatorial Structures 410</a>
%H P. Cameron, T. Prellberg and D. Stark, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v13i1r85">Asymptotics for incidence matrix classes</a>, Electron. J. Combin. 13 (2006), #R85, p. 11.
%H S. Crowley, <a href="http://vixra.org/abs/1202.0066">Two new zeta constants: fractal string and hypergeometric aspects of the Riemann zeta function</a>, viXra:1202.0066, 2012.
%H J. Estes and B. Wei, <a href="https://doi.org/10.1007/s10878-012-9515-6">Sharp bounds of the Zagreb indices of k-trees</a>, J Comb Optim 27 (2014), 271-291.
%H Ricardo Gómez Aíza, <a href="https://arxiv.org/abs/2402.16111">Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis</a>, arXiv:2402.16111 [math.CO], 2024. See p. 23.
%H I. Gutman and K. Das, <a href="https://match.pmf.kg.ac.rs/electronic_versions/Match50/match50_83-92.pdf">The first Zagreb index 30 years after</a>, MATCH Commun. Math. Comput. Chem. 50 (2004), 83-92.
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>
%H L. B. W. Jolley, <a href="https://archive.org/details/summationofserie00joll">Summation of Series</a>, Dover, 1961
%H Refik Keskin and Olcay Karaatli, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Karaatli/karaatli5.html">Some New Properties of Balancing Numbers and Square Triangular Numbers</a>, Journal of Integer Sequences, Vol. 15 (2012), Article #12.1.4.
%H Craig Knecht, <a href="/A002378/a002378.png">Largest pond by area in a square</a>
%H Enrique Navarrete and Daniel Orellana, <a href="https://arxiv.org/abs/1907.10023">Finding Prime Numbers as Fixed Points of Sequences</a>, arXiv:1907.10023 [math.NT], 2019.
%H László Németh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Nemeth/nemeth6.html">The trinomial transform triangle</a>, J. Int. Seqs., Vol. 21 (2018), Article #18.7.3. Also <a href="https://arxiv.org/abs/1807.07109">arXiv:1807.07109</a> [math.NT], 2018.
%H Lee Melvin Peralta, <a href="https://doi.org/10.5951/mathteacher.111.2.0150">Solutions to the Equation [x]x = n</a>, The Mathematics Teacher, Vol. 111, No. 2 (October 2017), pp. 150-154.
%H Aleksandar Petojević, <a href="http://dx.doi.org/10.5937/MatMor0801037P">A Note about the Pochhammer Symbol</a>, Mathematica Moravica, Vol. 12-1 (2008), 37-42.
%H A. Petojevic and N. Dapic, <a href="http://www.mi.sanu.ac.rs/~gvm/radovi/AP-Budva.pdf">The vAm(a,b,c;z) function</a>, Preprint 2013.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H John D. Roth, David A. Garren, and R. Clark Robertson, <a href="https://doi.org/10.1109/ICASSP39728.2021.9413937">Integer Carrier Frequency Offset Estimation in OFDM with Zadoff-Chu Sequences</a>, IEEE Int'l Conference on Acoustics, Speech and Signal Processing (ICASSP 2021) 4850-4854.
%H Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
%H Michelle Rudolph-Lilith, <a href="http://arxiv.org/abs/1508.07894">On the Product Representation of Number Sequences, with Application to the Fibonacci Family</a>, arXiv preprint arXiv:1508.07894 [math.NT], 2015.
%H Amelia Carolina Sparavigna, <a href="https://www.researchgate.net/publication/333844014">Groupoids of OEIS A002378 and A016754 Numbers (oblong and odd square numbers)</a>, Politecnico di Torino (Italy, 2019).
%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.3471358">The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences</a>, Politecnico di Torino, Italy (2019), [math.NT].
%H J. Striker and N. Williams, <a href="https://arxiv.org/abs/1108.1172">Promotion and Rowmotion </a>, arXiv preprint arXiv:1108.1172 [math.CO], 2011-2012.
%H D. Suprijanto and Rusliansyah, <a href="http://dx.doi.org/10.12988/ams.2014.4140">Observation on Sums of Powers of Integers Divisible by Four</a>, Applied Mathematical Sciences, Vol. 8, 2014, no. 45, 2219 - 2226.
%H R. Tijdeman, <a href="http://www.math.leidenuniv.nl/~tijdeman/tij1.ps">Some applications of Diophantine approximation</a>, pp. 261-284 of Surveys in Number Theory (Urbana, May 21, 2000), ed. M. A. Bennett et al., Peters, 2003.
%H G. Villemin's Almanach of Numbers, <a href="http://villemin.gerard.free.fr/Wwwgvmm/Geometri/Pronique.htm">Nombres Proniques</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CrownGraph.html">Crown Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PronicNumber.html">Pronic Number</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LeibnizHarmonicTriangle.html">Leibniz Harmonic Triangle</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WheelGraph.html">Wheel Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WienerIndex.html">Wiener Index</a>
%H Wolfram Research, <a href="http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=Hypergeometric3F2">Hypergeometric Function 3F2</a>, The Wolfram Functions site.
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F G.f.: 2*x/(1-x)^3. - _Simon Plouffe_ in his 1992 dissertation.
%F a(n) = a(n-1) + 2*n, a(0) = 0.
%F Sum_{n >= 1} a(n) = n*(n+1)*(n+2)/3 (cf. A007290, partial sums).
%F Sum_{n >= 1} 1/a(n) = 1. (Cf. Tijdeman)
%F Sum_{n >= 1} (-1)^(n+1)/a(n) = log(4) - 1 = A016627 - 1 [Jolley eq (235)].
%F 1 = 1/2 + Sum_{n >= 1} 1/[2*a(n)] = 1/2 + 1/4 + 1/12 + 1/24 + 1/40 + 1/60 + ... with partial sums: 1/2, 3/4, 5/6, 7/8, 9/10, 11/12, 13/14, ... - _Gary W. Adamson_, Jun 16 2003
%F a(n)*a(n+1) = a(n*(n+2)); e.g., a(3)*a(4) = 12*20 = 240 = a(3*5). - _Charlie Marion_, Dec 29 2003
%F Sum_{k = 1..n} 1/a(k) = n/(n+1). - _Robert G. Wilson v_, Feb 04 2005
%F a(n) = A046092(n)/2. - _Zerinvary Lajos_, Jan 08 2006
%F Log 2 = Sum_{n >= 0} 1/a(2n+1) = 1/2 + 1/12 + 1/30 + 1/56 + 1/90 + ... = (1 - 1/2) + (1/3 - 1/4) + (1/5 - 1/6) + (1/7 - 1/8) + ... = Sum_{n >= 0} (-1)^n/(n+1) = A002162. - _Gary W. Adamson_, Jun 22 2003
%F a(n) = A110660(2*n). - _N. J. A. Sloane_, Sep 21 2005
%F a(n-1) = n^2 - n = A000290(n) - A000027(n) for n >= 1. a(n) is the inverse (frequency distribution) sequence of A000194(n). - _Mohammad K. Azarian_, Jul 26 2007
%F (2, 6, 12, 20, 30, ...) = binomial transform of (2, 4, 2). - _Gary W. Adamson_, Nov 28 2007
%F a(n) = 2*Sum_{i=0..n} i = 2*A000217(n). - _Artur Jasinski_, Jan 09 2007, and _Omar E. Pol_, May 14 2008
%F a(n) = A006503(n) - A000292(n). - _Reinhard Zumkeller_, Sep 24 2008
%F a(n) = A061037(4*n) = (n+1/2)^2 - 1/4 = ((2n+1)^2 - 1)/4 = (A005408(n)^2 - 1)/4. - _Paul Curtz_, Oct 03 2008 and _Klaus Purath_, Jan 13 2022
%F a(0) = 0, a(n) = a(n-1) + 1 + floor(x), where x is the minimal positive solution to fract(sqrt(a(n-1) + 1 + x)) = 1/2. - _Hieronymus Fischer_, Dec 31 2008
%F E.g.f.:(x+2)*x*exp(x). - _Geoffrey Critzer_, Feb 06 2009
%F Product_{i >= 2} (1-1/a(i)) = -2*sin(Pi*A001622)/Pi = -2*sin(A094886)/A000796 = 2*A146481. - _R. J. Mathar_, Mar 12 2009, Mar 15 2009
%F E.g.f.: ((-x+1)*log(-x+1)+x)/x^2 also Integral_{x = 0..1} ((-x+1)*log(-x+1) + x)/x^2 = zeta(2) - 1. - _Stephen Crowley_, Jul 11 2009
%F a(n) = floor((n + 1/2)^2). a(n) = A035608(n) + A004526(n+1). - _Reinhard Zumkeller_, Jan 27 2010
%F a(n) = 2*(2*A006578(n) - A035608(n)). - _Reinhard Zumkeller_, Feb 07 2010
%F a(n-1) = floor(n^5/(n^3 + n^2 + 1)). - _Gary Detlefs_, Feb 11 2010
%F For n > 1: a(n) = A173333(n+1, n-1). - _Reinhard Zumkeller_, Feb 19 2010
%F a(n) = A004202(A000217(n)). - _Reinhard Zumkeller_, Feb 12 2011
%F a(n) = A188652(2*n+1) + 1. - _Reinhard Zumkeller_, Apr 13 2011
%F For n > 0 a(n) = 1/(Integral_{x=0..Pi/2} 2*(sin(x))^(2*n-1)*(cos(x))^3). - _Francesco Daddi_, Aug 02 2011
%F a(n) = A002061(n+1) - 1. - _Omar E. Pol_, Oct 03 2011
%F a(0) = 0, a(n) = A005408(A034856(n)) - A005408(n-1). - _Ivan N. Ianakiev_, Dec 06 2012
%F a(n) = A005408(A000096(n)) - A005408(n). - _Ivan N. Ianakiev_, Dec 07 2012
%F a(n) = A001318(n) + A085787(n). - _Omar E. Pol_, Jan 11 2013
%F Sum_{n >= 1} 1/(a(n))^(2s) = Sum_{t = 1..2*s} binomial(4*s - t - 1, 2*s - 1) * ( (1 + (-1)^t)*zeta(t) - 1). See Arxiv:1301.6293. - _R. J. Mathar_, Feb 03 2013
%F a(n)^2 + a(n+1)^2 = 2 * a((n+1)^2), for n > 0. - _Ivan N. Ianakiev_, Apr 08 2013
%F a(n) = floor(n^2 * e^(1/n)) and a(n-1) = floor(n^2 / e^(1/n)). - _Richard R. Forberg_, Jun 22 2013
%F a(n) = 2*C(n+1, 2), for n >= 0. - _Felix P. Muga II_, Mar 11 2014
%F A005369(a(n)) = 1. - _Reinhard Zumkeller_, Jul 05 2014
%F Binomial transform of [0, 2, 2, 0, 0, 0, ...]. - _Alois P. Heinz_, Mar 10 2015
%F a(2n) = A002943(n) for n >= 0, a(2n-1) = A002939(n) for n >= 1. - _M. F. Hasler_, Oct 11 2015
%F For n > 0, a(n) = 1/(Integral_{x=0..1} (x^(n-1) - x^n) dx). - _Rick L. Shepherd_, Oct 26 2015
%F a(n) = A005902(n) - A007588(n). - _Peter M. Chema_, Jan 09 2016
%F For n > 0, a(n) = Lim_{m -> infinity} (1/m)*1/(Sum_{i, m*n..m*(n+1)} 1/i^2), with error of ~1/m. - _Richard R. Forberg_, Jul 27 2016
%F From _Ilya Gutkovskiy_, Jul 28 2016: (Start)
%F Dirichlet g.f.: zeta(s-2) + zeta(s-1).
%F Convolution of nonnegative integers (A001477) and constant sequence (A007395).
%F Sum_{n >= 0} a(n)/n! = 3*exp(1). (End)
%F From _Charlie Marion_, Mar 06 2020: (Start)
%F a(n)*a(n+2k-1) + (n+k)^2 = ((2n+1)*k + n^2)^2.
%F a(n)*a(n+2k) + k^2 = ((2n+1)*k + a(n))^2. (End)
%F Product_{n>=1} (1 + 1/a(n)) = cosh(sqrt(3)*Pi/2)/Pi. - _Amiram Eldar_, Jan 20 2021
%F A generalization of the Dec 29 2003 formula, a(n)*a(n+1) = a(n*(n+2)), follows. a(n)*a(n+k) = a(n*(n+k+1)) + (k-1)*n*(n+k+1). - _Charlie Marion_, Jan 02 2023
%e a(3) = 12, Since 2(3)+2 = 8 has 4 partitions with exactly two parts: (7,1), (6,2), (5,3), (4,4). Taking the positive differences of the parts in each partition and adding, we get: 6 + 4 + 2 + 0 = 12. - _Wesley Ivan Hurt_, Jun 02 2013
%e G.f. = 2*x + 6*x^2 + 12*x^3 + 20*x^4 + 30*x^5 + 42*x^6 + 56*x^7 + ...
%e a(1) = 2, since 45-43 = 2
%e a(2) = 6, since 47-45 = 2 and 47-43 = 4, then 2+4 = 6
%e a(3) = 12, since 49-47 = 2, 49-45 = 4, and 49-43 = 6, then 2+4+6 = 12. - _Miquel Cerda_, Dec 04 2016
%p A002378 := proc(n)
%p n*(n+1) ;
%p end proc:
%p seq(A002378(n),n=0..100) ;
%t Table[n(n + 1), {n, 0, 50}] (* _Robert G. Wilson v_, Jun 19 2004 *)
%t oblongQ[n_] := IntegerQ @ Sqrt[4 n + 1]; Select[Range[0, 2600], oblongQ] (* _Robert G. Wilson v_, Sep 29 2011 *)
%t 2 Accumulate[Range[0, 50]] (* _Harvey P. Dale_, Nov 11 2011 *)
%t LinearRecurrence[{3, -3, 1}, {2, 6, 12}, {0, 20}] (* _Eric W. Weisstein_, Jul 27 2017 *)
%o (PARI) {a(n) = n*(n+1)};
%o (PARI) concat(0, Vec(2*x/(1-x)^3 + O(x^100))) \\ _Altug Alkan_, Oct 26 2015
%o (PARI) is(n)=my(m=sqrtint(n)); m*(m+1)==n \\ _Charles R Greathouse IV_, Nov 01 2018
%o (PARI) is(n)=issquare(4*n+1) \\ _Charles R Greathouse IV_, Mar 16 2022
%o (Haskell)
%o a002378 n = n * (n + 1)
%o a002378_list = zipWith (*) [0..] [1..]
%o -- _Reinhard Zumkeller_, Aug 27 2012, Oct 12 2011
%o (Magma) [n*(n+1) : n in [0..100]]; // _Wesley Ivan Hurt_, Oct 26 2015
%o (Scala) (2 to 100 by 2).scanLeft(0)(_ + _) // _Alonso del Arte_, Sep 12 2019
%o (Python)
%o def a(n): return n*(n+1)
%o print([a(n) for n in range(51)]) # _Michael S. Branicky_, Jan 13 2022
%Y Partial sums of A005843 (even numbers). Twice triangular numbers (A000217).
%Y 1/beta(n, 2) in A061928.
%Y Cf. A035106, A087811, A119462, A127235, A049598, A124080, A033996, A028896, A046092, A000217, A005563, A046092, A001082, A059300, A059297, A059298, A166373, A002943 (bisection), A002939 (bisection), A078358 (complement).
%Y A036689 is a subsequence. Cf. numbers of the form n*(n*k-k+4)/2 listed in A226488. - _Bruno Berselli_, Jun 10 2013
%Y Row n=2 of A185651.
%Y Cf. A007745, A169810, A213541, A005369.
%Y Cf. A281026. - _Bruno Berselli_, Jan 16 2017
%Y Cf. A045943 (4-cycles in triangular honeycomb acute knight graph), A028896 (5-cycles), A152773 (6-cycles).
%Y Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
%Y Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
%Y Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
%Y A335064 is a subsequence.
%Y Second column of A003506.
%Y Cf. A002378, A046092, A028896 (irregularities of maximal k-degenerate graphs).
%Y Cf. A347213 (Dgf at s=4).
%Y Cf. A002378, A152811, A371912 (Zagreb indices of maximal k-degenerate graphs).
%K nonn,easy,core,nice
%O 0,2
%A _N. J. A. Sloane_
%E Additional comments from _Michael Somos_
%E Comment and cross-reference added by _Christopher Hunt Gribble_, Oct 13 2009
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