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%I M3356 N1350 #775 Apr 17 2024 10:57:41
%S 0,1,4,9,16,25,36,49,64,81,100,121,144,169,196,225,256,289,324,361,
%T 400,441,484,529,576,625,676,729,784,841,900,961,1024,1089,1156,1225,
%U 1296,1369,1444,1521,1600,1681,1764,1849,1936,2025,2116,2209,2304,2401,2500
%N The squares: a(n) = n^2.
%C To test if a number is a square, see Cohen, p. 40. - _N. J. A. Sloane_, Jun 19 2011
%C Zero followed by partial sums of A005408 (odd numbers). - _Jeremy Gardiner_, Aug 13 2002
%C Begin with n, add the next number, subtract the previous number and so on ending with subtracting a 1: a(n) = n + (n+1) - (n-1) + (n+2) - (n-2) + (n+3) - (n-3) + ... + (2n-1) - 1 = n^2. - _Amarnath Murthy_, Mar 24 2004
%C Sum of two consecutive triangular numbers A000217. - _Lekraj Beedassy_, May 14 2004
%C Numbers with an odd number of divisors: {d(n^2) = A048691(n); for the first occurrence of 2n + 1 divisors, see A071571(n)}. - _Lekraj Beedassy_, Jun 30 2004
%C See also A000037.
%C First sequence ever computed by electronic computer, on EDSAC, May 06 1949 (see Renwick link). - _Russ Cox_, Apr 20 2006
%C Numbers k such that the imaginary quadratic field Q(sqrt(-k)) has four units. - _Marc LeBrun_, Apr 12 2006
%C For n > 0: number of divisors of (n-1)th power of any squarefree semiprime: a(n) = A000005(A006881(k)^(n-1)); a(n) = A000005(A000400(n-1)) = A000005(A011557(n-1)) = A000005(A001023(n-1)) = A000005(A001024(n-1)). - _Reinhard Zumkeller_, Mar 04 2007
%C If a 2-set Y and an (n-2)-set Z are disjoint subsets of an n-set X then a(n-2) is the number of 3-subsets of X intersecting both Y and Z. - _Milan Janjic_, Sep 19 2007
%C Numbers a such that a^1/2 + b^1/2 = c^1/2 and a^2 + b = c. - _Cino Hilliard_, Feb 07 2008 (this comment needs clarification, _Joerg Arndt_, Sep 12 2013)
%C Numbers k such that the geometric mean of the divisors of k is an integer. - _Ctibor O. Zizka_, Jun 26 2008
%C Equals row sums of triangle A143470. Example: 36 = sum of row 6 terms: (23 + 7 + 3 + 1 + 1 + 1). - _Gary W. Adamson_, Aug 17 2008
%C Equals row sums of triangles A143595 and A056944. - _Gary W. Adamson_, Aug 26 2008
%C Number of divisors of 6^(n-1) for n > 0. - _J. Lowell_, Aug 30 2008
%C Denominators of Lyman spectrum of hydrogen atom. Numerators are A005563. A000290-A005563 = A000012. - _Paul Curtz_, Nov 06 2008
%C a(n) is the number of all partitions of the sum 2^2 + 2^2 + ... + 2^2, (n-1) times, into powers of 2. - _Valentin Bakoev_, Mar 03 2009
%C a(n) is the maximal number of squares that can be 'on' in an n X n board so that all the squares turn 'off' after applying the operation: in any 2 X 2 sub-board, a square turns from 'on' to 'off' if the other three are off. - _Srikanth K S_, Jun 25 2009
%C Zero together with the numbers k such that 2 is the number of perfect partitions of k. - _Juri-Stepan Gerasimov_, Sep 26 2009
%C Totally multiplicative sequence with a(p) = p^2 for prime p. - _Jaroslav Krizek_, Nov 01 2009
%C Satisfies A(x)/A(x^2), A(x) = A173277: (1, 4, 13, 32, 74, ...). - _Gary W. Adamson_, Feb 14 2010
%C Positive members are the integers with an odd number of odd divisors and an even number of even divisors. See also A120349, A120359, A181792, A181793, A181795. - _Matthew Vandermast_, Nov 14 2010
%C Besides the first term, this sequence is the denominator of Pi^2/6 = 1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36 + ... . - _Mohammad K. Azarian_, Nov 01 2011
%C Partial sums give A000330. - _Omar E. Pol_, Jan 12 2013
%C Drmota, Mauduit, and Rivat proved that the Thue-Morse sequence along the squares is normal; see A228039. - _Jonathan Sondow_, Sep 03 2013
%C a(n) can be decomposed into the sum of the four numbers [binomial(n, 1) + binomial(n, 2) + binomial(n-1, 1) + binomial(n-1, 2)] which form a "square" in Pascal's Triangle A007318, or the sum of the two numbers [binomial(n, 2) + binomial(n+1, 2)], or the difference of the two numbers [binomial(n+2, 3) - binomial(n, 3)]. - _John Molokach_, Sep 26 2013
%C In terms of triangular tiling, the number of equilateral triangles with side length 1 inside an equilateral triangle with side length n. - _K. G. Stier_, Oct 30 2013
%C Number of positive roots in the root systems of type B_n and C_n (when n > 1). - _Tom Edgar_, Nov 05 2013
%C Squares of squares (fourth powers) are also called biquadratic numbers: A000583. - _M. F. Hasler_, Dec 29 2013
%C For n > 0, a(n) is the largest integer k such that k^2 + n is a multiple of k + n. More generally, for m > 0 and n > 0, the largest integer k such that k^(2*m) + n is a multiple of k + n is given by k = n^(2*m). - _Derek Orr_, Sep 03 2014
%C For n > 0, a(n) is the number of compositions of n + 5 into n parts avoiding the part 2. - _Milan Janjic_, Jan 07 2016
%C a(n), for n >= 3, is also the number of all connected subtrees of a cycle graph, having n vertices. - _Viktar Karatchenia_, Mar 02 2016
%C On every sequence of natural continuous numbers with an even number of elements, the summatory of the second half of the sequence minus the summatory of the first half of the sequence is always a square. Example: Sequence from 61 to 70 has an even number of elements (10). Then 61 + 62 + 63 + 64 + 65 = 315; 66 + 67 + 68 + 69 + 70 = 340; 340 - 315 = 25. (n/2)^2 for n = number of elements. - _César Aguilera_, Jun 20 2016
%C On every sequence of natural continuous numbers from n^2 to (n+1)^2, the sum of the differences of pairs of elements of the two halves in every combination possible is always (n+1)^2. - _César Aguilera_, Jun 24 2016
%C Suppose two circles with radius 1 are tangent to each other as well as to a line not passing through the point of tangency. Create a third circle tangent to both circles as well as the line. If this process is continued, a(n) for n > 0 is the reciprocals of the radii of the circles, beginning with the largest circle. - _Melvin Peralta_, Aug 18 2016
%C Does not satisfy Benford's law [Ross, 2012]. - _N. J. A. Sloane_, Feb 08 2017
%C Numerators of the solution to the generalization of the Feynman triangle problem, with an offset of 2. If each vertex of a triangle is joined to the point (1/p) along the opposite side (measured say clockwise), then the area of the inner triangle formed by these lines is equal to (p - 2)^2/(p^2 - p + 1) times the area of the original triangle, p > 2. For example, when p = 3, the ratio of the areas is 1/7. The denominators of the ratio of the areas is given by A002061. [Cook & Wood, 2004] - _Joe Marasco_, Feb 20 2017
%C Equals row sums of triangle A004737, n >= 1. - _Martin Michael Musatov_, Nov 07 2017
%C Right-hand side of the binomial coefficient identity Sum_{k = 0..n} (-1)^(n+k+1)*binomial(n,k)*binomial(n + k,k)*(n - k) = n^2. - _Peter Bala_, Jan 12 2022
%C Conjecture: For n>0, min{k such that there exist subsets A,B of {0,1,2,...,a(n)-1} such that |A|=|B|=k and A+B contains {0,1,2,...,a(n)-1}} = n. - _Michael Chu_, Mar 09 2022
%C Number of 3-permutations of n elements avoiding the patterns 132, 213, 321. See Bonichon and Sun. - _Michel Marcus_, Aug 20 2022
%C Number of intercalates in cyclic Latin squares of order 2n (cyclic Latin squares of odd order do not have intercalates). - _Eduard I. Vatutin_, Feb 15 2024
%D G. L. Alexanderson et al., The William Lowell Putnam Mathematical Competition, Problems and Solutions: 1965-1984, "December 1967 Problem B4(a)", pp. 8(157) MAA Washington DC 1985.
%D T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 2.
%D R. P. Burn & A. Chetwynd, A Cascade Of Numbers, "The prison door problem" Problem 4 pp. 5-7; 79-80 Arnold London 1996.
%D H. Cohen, A Course in Computational Algebraic Number Theory, Springer, 1996, p. 40.
%D E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), p. 6.
%D M. Gardner, Time Travel and Other Mathematical Bewilderments, Chapter 6 pp. 71-2, W. H. Freeman NY 1988.
%D Granino A. Korn and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968), p. 982.
%D Alfred S. Posamentier, The Art of Problem Solving, Section 2.4 "The Long Cell Block" pp. 10-1; 12; 156-7 Corwin Press Thousand Oaks CA 1996.
%D Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.
%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 J. K. Strayer, Elementary Number Theory, Exercise Set 3.3 Problems 32, 33, p. 88, PWS Publishing Co. Boston MA 1996.
%D C. W. Trigg, Mathematical Quickies, "The Lucky Prisoners" Problem 141 pp. 40, 141, Dover NY 1985.
%D R. Vakil, A Mathematical Mosaic, "The Painted Lockers" pp. 127;134 Brendan Kelly Burlington Ontario 1996.
%H Franklin T. Adams-Watters, <a href="/A000290/b000290.txt">The first 10000 squares: Table of n, n^2 for n = 0..10000</a>
%H Valentin P. Bakoev, <a href="http://dx.doi.org/10.1016/S0012-365X(03)00096-7">Algorithmic approach to counting of certain types m-ary partitions</a>, Discrete Mathematics, 275 (2004) pp. 17-41.
%H Stefano Barbero, Umberto Cerruti, and Nadir Murru, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barbero/barbero5.html">Transforming Recurrent Sequences by Using the Binomial and Invert Operators</a>, J. Int. Seq. 13 (2010) # 10.7.7, section 4.4.
%H Anicius Manlius Severinus Boethius, <a href="https://archive.org/stream/aniciimanliitor01friegoog#page/n112/mode/2up">De institutione arithmetica libri duo</a>, Book 2, sections 10-12.
%H Nicolas Bonichon and Pierre-Jean Morel, <a href="https://arxiv.org/abs/2202.12677">Baxter d-permutations and other pattern avoiding classes</a>, arXiv:2202.12677 [math.CO], 2022.
%H Henry Bottomley, <a href="http://fs.gallup.unm.edu/Bottomley-Sm-Mult-Functions.htm">Some Smarandache-type multiplicative sequences</a>
%H R. J. Cook and G. V. Wood, <a href="http://www.jstor.org/stable/3620856"> Feynman's Triangle</a>, Mathematical Gazette, 88:299-302 (2004).
%H John Derbyshire, <a href="http://www.johnderbyshire.com/Opinions/Diaries/Puzzles/2002-11.html">Monkeys and Doors</a>
%H Michael Drmota, Christian Mauduit, and Joël Rivat, <a href="https://www.dmg.tuwien.ac.at/drmota/alongsquares.pdf">The Thue-Morse Sequence Along The Squares is Normal</a>, Abstract, ÖMG-DMV Congress, 2013.
%H Ralph Greenberg, <a href="http://www.math.washington.edu/~greenber/MathPoet.html">Math for Poets</a>
%H Guo-Niu Han, <a href="http://www-irma.u-strasbg.fr/~guoniu/papers/p77puzzle.pdf">Enumeration of Standard Puzzles</a>
%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a> [Cached copy]
%H Vi Hart, <a href="https://www.youtube.com/watch?v=v-pyuaThp-c">Doodling in Math Class: Connecting Dots</a> (2012) [Video].
%H Nick Hobson, <a href="/A000290/a000290.py.txt">Python program to determine whether an integer is in sequence A000290</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=338">Encyclopedia of Combinatorial Structures 338</a>
%H Milan Janjic, <a href="http://pmf.unibl.org/wp-content/uploads/2017/10/enumfun.pdf">Enumerative Formulas for Some Functions on Finite Sets</a>
%H Milan Janjic, <a href="https://www.semanticscholar.org/paper/801b6b226bfe1d6b002fb4946c3957d7052132bd">Two Enumerative Functions</a>
%H Milan Janjić, <a href="https://arxiv.org/abs/1905.04465">On Restricted Ternary Words and Insets</a>, arXiv:1905.04465 [math.CO], 2019.
%H L. B. W. Jolley, <a href="https://archive.org/details/summationofserie00joll">Summation of Series</a>, Dover, 1961
%H Sameen Ahmed Khan, <a href="https://doi.org/10.12732/ijam.v33i2.6">Sums of the powers of reciprocals of polygonal numbers</a>, Int'l J. of Appl. Math. (2020) Vol. 33, No. 2, 265-282.
%H Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.html">Complementary Equations</a>, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
%H Hyun Kwang Kim, <a href="http://dx.doi.org/10.1090/S0002-9939-02-06710-2">On Regular Polytope Numbers</a>, Proc. Amer. Math. Soc., 131 (2002), 65-75.
%H J. H. McKay, <a href="https://www.jstor.org/stable/2315185">The William Lowell Putnam Mathematical Competition, Problem B4(a)</a>, The American Mathematical Monthly, vol. 75, no. 7, 1968, pp. 732-739.
%H Matthew Parker, <a href="https://oeis.org/A000290/a000290_1M.7z">The first million squares (7-Zip compressed file)</a>
%H Ed Pegg, Jr., <a href="http://www.mathpuzzle.com/MAA/07-Sequence%20Pictures/mathgames_12_08_03.html">Sequence Pictures</a>, Math Games column, Dec 08 2003.
%H Ed Pegg, Jr., <a href="/A000043/a000043_2.pdf">Sequence Pictures</a>, Math Games column, Dec 08 2003 [Cached copy, with permission (pdf only)]
%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; arXiv:0911.4975 [math.NT], 2009.
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polnum01.jpg">Illustration of initial terms of A000217, A000290, A000326, A000384, A000566, A000567</a>
%H Yash Puri and Thomas Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%H Franck Ramaharo, <a href="https://arxiv.org/abs/1805.10680">A generating polynomial for the pretzel knot</a>, arXiv:1805.10680 [math.CO], 2018.
%H William S. Renwick, <a href="http://www.cl.cam.ac.uk/Relics/elog.html">EDSAC log</a>.
%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.
%H Kenneth A. Ross, <a href="https://www.jstor.org/stable/10.4169/math.mag.85.1.36">First Digits of Squares and Cubes</a>, Math. Mag. 85 (2012) 36-42.
%H John Scholes, <a href="https://mks.mff.cuni.cz/kalva/putnam/putn67.html">28th Putnam 1967 Prob.B4(a)</a>
%H James A. Sellers, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL7/Sellers/sellers58.html">Partitions Excluding Specific Polygonal Numbers As Parts</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.4.
%H N. J. A. Sloane, <a href="/A000217/a000217.gif">Illustration of initial terms of A000217, A000290, A000326</a>
%H Michael Somos, <a href="http://cis.csuohio.edu/~somos/rfmc.txt">Rational Function Multiplicative Coefficients</a>
%H Nathan Sun, <a href="https://arxiv.org/abs/2208.08506">On d-permutations and Pattern Avoidance Classes</a>, arXiv:2208.08506 [math.CO], 2022.
%H Dinoj Surendran, <a href="http://web.archive.org/web/20101206041638/http://www.uz.ac.zw/science/maths/zimaths/jailer.htm">Chimbumu and Chickwama get out of jail</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SquareNumber.html">Square Number</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Unit.html">Unit</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WienerIndex.html">Wiener Index</a>
%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).
%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>
%H <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>
%H <a href="/index/Be#Benford">Index entries for sequences related to Benford's law</a>
%F G.f.: x*(1 + x) / (1 - x)^3.
%F E.g.f.: exp(x)*(x + x^2).
%F Dirichlet g.f.: zeta(s-2).
%F a(n) = a(-n).
%F Multiplicative with a(p^e) = p^(2*e). - _David W. Wilson_, Aug 01 2001
%F Sum of all matrix elements M(i, j) = 2*i/(i+j) (i, j = 1..n). a(n) = Sum_{i = 1..n} Sum_{j = 1..n} 2*i/(i + j). - _Alexander Adamchuk_, Oct 24 2004
%F a(0) = 0, a(1) = 1, a(n) = 2*a(n-1) - a(n-2) + 2. - _Miklos Kristof_, Mar 09 2005
%F From _Pierre CAMI_, Oct 22 2006: (Start)
%F a(n) is the sum of the odd numbers from 1 to 2*n - 1.
%F a(0) = 0, a(1) = 1, then a(n) = a(n-1) + 2*n - 1. (End)
%F For n > 0: a(n) = A130064(n)*A130065(n). - _Reinhard Zumkeller_, May 05 2007
%F a(n) = Sum_{k = 1..n} A002024(n, k). - _Reinhard Zumkeller_, Jun 24 2007
%F Left edge of the triangle in A132111: a(n) = A132111(n, 0). - _Reinhard Zumkeller_, Aug 10 2007
%F Binomial transform of [1, 3, 2, 0, 0, 0, ...]. - _Gary W. Adamson_, Nov 21 2007
%F a(n) = binomial(n+1, 2) + binomial(n, 2).
%F This sequence could be derived from the following general formula (cf. A001286, A000330): n*(n+1)*...*(n+k)*(n + (n+1) + ... + (n+k))/((k+2)!*(k+1)/2) at k = 0. Indeed, using the formula for the sum of the arithmetic progression (n + (n+1) + ... + (n+k)) = (2*n + k)*(k + 1)/2 the general formula could be rewritten as: n*(n+1)*...*(n+k)*(2*n+k)/(k+2)! so for k = 0 above general formula degenerates to n*(2*n + 0)/(0 + 2) = n^2. - _Alexander R. Povolotsky_, May 18 2008
%F From a(4) recurrence formula a(n+3) = 3*a(n+2) - 3*a(n+1) + a(n) and a(1) = 1, a(2) = 4, a(3) = 9. - _Artur Jasinski_, Oct 21 2008
%F The recurrence a(n+3) = 3*a(n+2) - 3*a(n+1) + a(n) is satisfied by all k-gonal sequences from a(3), with a(0) = 0, a(1) = 1, a(2) = k. - _Jaume Oliver Lafont_, Nov 18 2008
%F a(n) = floor(n*(n+1)*(Sum_{i = 1..n} 1/(n*(n+1)))). - _Ctibor O. Zizka_, Mar 07 2009
%F Product_{i >= 2} 1 - 2/a(i) = -sin(A063448)/A063448. - _R. J. Mathar_, Mar 12 2009
%F a(n) = A002378(n-1) + n. - _Jaroslav Krizek_, Jun 14 2009
%F a(n) = n*A005408(n-1) - (Sum_{i = 1..n-2} A005408(i)) - (n-1) = n*A005408(n-1) - a(n-1) - (n-1). - _Bruno Berselli_, May 04 2010
%F a(n) == 1 (mod n+1). - _Bruno Berselli_, Jun 03 2010
%F a(n) = a(n-1) + a(n-2) - a(n-3) + 4, n > 2. - _Gary Detlefs_, Sep 07 2010
%F a(n+1) = Integral_{x >= 0} exp(-x)/( (Pn(x)*exp(-x)*Ei(x) - Qn(x))^2 +(Pi*exp(-x)*Pn(x))^2 ), with Pn the Laguerre polynomial of order n and Qn the secondary Laguerre polynomial defined by Qn(x) = Integral_{t >= 0} (Pn(x) - Pn(t))*exp(-t)/(x-t). - _Groux Roland_, Dec 08 2010
%F Euler transform of length-2 sequence [4, -1]. - _Michael Somos_, Feb 12 2011
%F A162395(n) = -(-1)^n * a(n). - _Michael Somos_, Mar 19 2011
%F a(n) = A004201(A000217(n)); A007606(a(n)) = A000384(n); A007607(a(n)) = A001105(n). - _Reinhard Zumkeller_, Feb 12 2011
%F Sum_{n >= 1} 1/a(n)^k = (2*Pi)^k*B_k/(2*k!) = zeta(2*k) with Bernoulli numbers B_k = -1, 1/6, 1/30, 1/42, ... for k >= 0. See A019673, A195055/10 etc. [Jolley eq 319].
%F Sum_{n>=1} (-1)^(n+1)/a(n)^k = 2^(k-1)*Pi^k*(1-1/2^(k-1))*B_k/k! [Jolley eq 320] with B_k as above.
%F A007968(a(n)) = 0. - _Reinhard Zumkeller_, Jun 18 2011
%F A071974(a(n)) = n; A071975(a(n)) = 1. - _Reinhard Zumkeller_, Jul 10 2011
%F a(n) = A199332(2*n - 1, n). - _Reinhard Zumkeller_, Nov 23 2011
%F For n >= 1, a(n) = Sum_{d|n} phi(d)*psi(d), where phi is A000010 and psi is A001615. - _Enrique Pérez Herrero_, Feb 29 2012
%F a(n) = A000217(n^2) - A000217(n^2 - 1), for n > 0. - _Ivan N. Ianakiev_, May 30 2012
%F a(n) = (A000217(n) + A000326(n))/2. - _Omar E. Pol_, Jan 11 2013
%F a(n) = A162610(n, n) = A209297(n, n) for n > 0. - _Reinhard Zumkeller_, Jan 19 2013
%F a(A000217(n)) = Sum_{i = 1..n} Sum_{j = 1..n} i*j, for n > 0. - _Ivan N. Ianakiev_, Apr 20 2013
%F a(n) = A133280(A000217(n)). - _Ivan N. Ianakiev_, Aug 13 2013
%F a(2*a(n)+2*n+1) = a(2*a(n)+2*n) + a(2*n+1). - _Vladimir Shevelev_, Jan 24 2014
%F a(n+1) = Sum_{t1+2*t2+...+n*tn = n} (-1)^(n+t1+t2+...+tn)*multinomial(t1+t2 +...+tn,t1,t2,...,tn)*4^(t1)*7^(t2)*8^(t3+...+tn). - _Mircea Merca_, Feb 27 2014
%F a(n) = floor(1/(1-cos(1/n)))/2 = floor(1/(1-n*sin(1/n)))/6, n > 0. - _Clark Kimberling_, Oct 08 2014
%F a(n) = ceiling(Sum_{k >= 1} log(k)/k^(1+1/n)) = -Zeta'[1+1/n]. Thus any exponent greater than 1 applied to k yields convergence. The fractional portion declines from A073002 = 0.93754... at n = 1 and converges slowly to 0.9271841545163232... for large n. - _Richard R. Forberg_, Dec 24 2014
%F a(n) = Sum_{j = 1..n} Sum_{i = 1..n} ceiling((i + j - n + 1)/3). - _Wesley Ivan Hurt_, Mar 12 2015
%F a(n) = Product_{j = 1..n-1} 2 - 2*cos(2*j*Pi/n). - _Michel Marcus_, Jul 24 2015
%F From _Ilya Gutkovskiy_, Jun 21 2016: (Start)
%F Product_{n >= 1} (1 + 1/a(n)) = sinh(Pi)/Pi = A156648.
%F Sum_{n >= 0} 1/a(n!) = BesselI(0, 2) = A070910. (End)
%F a(n) = A028338(n, n-1), n >= 1 (second diagonal). - _Wolfdieter Lang_, Jul 21 2017
%F For n >= 1, a(n) = Sum_{d|n} sigma_2(d)*mu(n/d) = Sum_{d|n} A001157(d)*A008683(n/d). - _Ridouane Oudra_, Apr 15 2021
%F a(n) = Sum_{i = 1..2*n-1} ceiling(n - i/2). - _Stefano Spezia_, Apr 16 2021
%F From _Richard L. Ollerton_, May 09 2021: (Start) For n >= 1,
%F a(n) = Sum_{k=1..n} psi(n/gcd(n,k)).
%F a(n) = Sum_{k=1..n} psi(gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)).
%F a(n) = Sum_{k=1..n} sigma_2(n/gcd(n,k))*mu(gcd(n,k))/phi(n/gcd(n,k)).
%F a(n) = Sum_{k=1..n} sigma_2(gcd(n,k))*mu(n/gcd(n,k))/phi(n/gcd(n,k)). (End)
%F a(n) = (A005449(n) + A000326(n))/3. - _Klaus Purath_, May 13 2021
%F Let T(n) = A000217(n), then a(T(n)) + a(T(n+1)) = T(a(n+1)). - _Charlie Marion_, Jun 27 2022
%F a(n) = Sum_{k=1..n} sigma_1(k) + Sum_{i=1..n} (n mod i). - _Vadim Kataev_, Dec 07 2022
%e For n = 8, a(8) = 8 * 15 - (1 + 3 + 5 + 7 + 9 + 11 + 13) - 7 = 8 * 15 - 49 - 7 = 64. - _Bruno Berselli_, May 04 2010
%e G.f. = x + 4*x^2 + 9*x^3 + 16*x^4 + 25*x^5 + 36*x^6 + 49*x^7 + 64*x^8 + 81*x^9 + ...
%e a(4) = 16. For n = 4 vertices, the cycle graph C4 is A-B-C-D-A. The subtrees are: 4 singles: A, B, C, D; 4 pairs: A-B, BC, C-D, A-D; 4 triples: A-B-C, B-C-D, C-D-A, D-A-B; 4 quads: A-B-C-D, B-C-D-A, C-D-A-B, D-A-B-C; 4 + 4 + 4 + 4 = 16. - _Viktar Karatchenia_, Mar 02 2016
%p A000290 := n->n^2; seq(A000290(n), n=0..50);
%p A000290 := -(1+z)/(z-1)^3; # _Simon Plouffe_, in his 1992 dissertation, for sequence starting at a(1)
%t Array[#^2 &, 51, 0] (* _Robert G. Wilson v_, Aug 01 2014 *)
%t LinearRecurrence[{3, -3, 1}, {0, 1, 4}, 60] (* _Vincenzo Librandi_, Jul 24 2015 *)
%t CoefficientList[Series[-(x^2 + x)/(x - 1)^3, {x, 0, 50}], x] (* _Robert G. Wilson v_, Jul 23 2018 *)
%t Range[0, 99]^2 (* _Alonso del Arte_, Nov 21 2019 *)
%o (Magma) [ n^2 : n in [0..1000]];
%o (PARI) {a(n) = n^2};
%o (PARI) b000290(maxn)=for(n=0,maxn,print(n," ",n^2);) \\ _Anatoly E. Voevudko_, Nov 11 2015
%o (Haskell)
%o a000290 = (^ 2)
%o a000290_list = scanl (+) 0 [1,3..] -- _Reinhard Zumkeller_, Apr 06 2012
%o (Maxima) A000290(n):=n^2$ makelist(A000290(n),n,0,30); /* _Martin Ettl_, Oct 25 2012 */
%o (Scheme) (define (A000290 n) (* n n)) ;; _Antti Karttunen_, Oct 06 2017
%o (Scala) (0 to 59).map(n => n * n) // _Alonso del Arte_, Oct 07 2019
%o (Python) # See Hobson link
%o (Python)
%o def A000290(n): return n**2 # _Chai Wah Wu_, Nov 13 2022
%Y Cf. A092205, A128200, A005408, A128201, A002522, A005563, A008865, A059100, A143051, A143470, A143595, A056944, A001157 (inverse Möbius transform), A001788 (binomial transform), A228039, A001105, A004159, A159918, A173277, A095794, A162395, A186646 (Pisano periods), A028338 (2nd diagonal).
%Y A row or column of A132191.
%Y This sequence is related to partitions of 2^n into powers of 2, as it is shown in A002577. So A002577 connects the squares and A000447. - _Valentin Bakoev_, Mar 03 2009
%Y Boustrophedon transforms: A000697, A000745.
%Y Cf. A342819.
%K nonn,core,easy,nice,mult
%O 0,3
%A _N. J. A. Sloane_
%E Incorrect comment and example removed by _Joerg Arndt_, Mar 11 2010
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