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%I M0985 #261 Jan 19 2024 16:56:04
%S 0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,
%T 48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,
%U 94,96,98,100,102,104,106,108,110,112,114,116,118,120
%N The nonnegative even numbers: a(n) = 2n.
%C -2, -4, -6, -8, -10, -12, -14, ... are the trivial zeros of the Riemann zeta function. - Vivek Suri (vsuri(AT)jhu.edu), Jan 24 2008
%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 2-subsets of X intersecting both Y and Z. - _Milan Janjic_, Sep 19 2007
%C A134452(a(n)) = 0; A134451(a(n)) = 2 for n > 0. - _Reinhard Zumkeller_, Oct 27 2007
%C Omitting the initial zero gives the number of prime divisors with multiplicity of product of terms of n-th row of A077553. - _Ray Chandler_, Aug 21 2003
%C A059841(a(n))=1, A000035(a(n))=0. - _Reinhard Zumkeller_, Sep 29 2008
%C (APSO) Alternating partial sums of (a-b+c-d+e-f+g...) = (a+b+c+d+e+f+g...) - 2*(b+d+f...), it appears that APSO(A005843) = A052928 = A002378 - 2*(A116471), with A116471=2*A008794. - _Eric Desbiaux_, Oct 28 2008
%C A056753(a(n)) = 1. - _Reinhard Zumkeller_, Aug 23 2009
%C Twice the nonnegative numbers. - _Juri-Stepan Gerasimov_, Dec 12 2009
%C The number of hydrogen atoms in straight-chain (C(n)H(2n+2)), branched (C(n)H(2n+2), n > 3), and cyclic, n-carbon alkanes (C(n)H(2n), n > 2). - _Paul Muljadi_, Feb 18 2010
%C For n >= 1; a(n) = the smallest numbers m with the number of steps n of iterations of {r - (smallest prime divisor of r)} needed to reach 0 starting at r = m. See A175126 and A175127. A175126(a(n)) = A175126(A175127(n)) = n. Example (a(4)=8): 8-2=6, 6-2=4, 4-2=2, 2-2=0; iterations has 4 steps and number 8 is the smallest number with such result. - _Jaroslav Krizek_, Feb 15 2010
%C For n >= 1, a(n) = numbers k such that arithmetic mean of the first k positive integers is not integer. A040001(a(n)) > 1. See A145051 and A040001. - _Jaroslav Krizek_, May 28 2010
%C Union of A179082 and A179083. - _Reinhard Zumkeller_, Jun 28 2010
%C a(k) is the (Moore lower bound on and the) order of the (k,4)-cage: the smallest k-regular graph having girth four: the complete bipartite graph with k vertices in each part. - _Jason Kimberley_, Oct 30 2011
%C For n > 0: A048272(a(n)) <= 0. - _Reinhard Zumkeller_, Jan 21 2012
%C Let n be the number of pancakes that have to be divided equally between n+1 children. a(n) is the minimal number of radial cuts needed to accomplish the task. - _Ivan N. Ianakiev_, Sep 18 2013
%C For n > 0, a(n) is the largest number k such that (k!-n)/(k-n) is an integer. - _Derek Orr_, Jul 02 2014
%C a(n) when n > 2 is also the number of permutations simultaneously avoiding 213, 231 and 321 in the classical sense which can be realized as labels on an increasing strict binary tree with 2n-1 nodes. See A245904 for more information on increasing strict binary trees. - _Manda Riehl_ Aug 07 2014
%C It appears that for n > 2, a(n) = A020482(n) + A002373(n), where all sequences are infinite. This is consistent with Goldbach's conjecture, which states that every even number > 2 can be expressed as the sum of two prime numbers. - _Bob Selcoe_, Mar 08 2015
%C Number of partitions of 4n into exactly 2 parts. - _Colin Barker_, Mar 23 2015
%C Number of neighbors in von Neumann neighborhood. - _Dmitry Zaitsev_, Nov 30 2015
%C Unique solution b( ) of the complementary equation a(n) = a(n-1)^2 - a(n-2)*b(n-1), where a(0) = 1, a(1) = 3, and a( ) and b( ) are increasing complementary sequences. - _Clark Kimberling_, Nov 21 2017
%C Also the maximum number of non-attacking bishops on an (n+1) X (n+1) board (n>0). (Cf. A000027 for rooks and queens (n>3), A008794 for kings or A030978 for knights.) - _Martin Renner_, Jan 26 2020
%C Integer k is even positive iff phi(2k) > phi(k), where phi is Euler's totient (A000010) [see reference De Koninck & Mercier]. - _Bernard Schott_, Dec 10 2020
%C Number of 3-permutations of n elements avoiding the patterns 132, 213, 312 and also number of 3-permutations avoiding the patterns 213, 231, 321. See Bonichon and Sun. - _Michel Marcus_, Aug 20 2022
%D T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 2.
%D J.-M. De Koninck and A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 529a pp. 71 and 257, Ellipses, 2004, Paris.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H N. J. A. Sloane, <a href="/A005843/b005843.txt">Table of n, a(n) for n = 0..10000</a>
%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 David Callan, <a href="https://arxiv.org/abs/1911.02209">On Ascent, Repetition and Descent Sequences</a>, arXiv:1911.02209 [math.CO], 2019.
%H Charles Cratty, Samuel Erickson, Frehiwet Negass, and Lara Pudwell, <a href="http://www.valpo.edu/mathematics-statistics/files/2015/07/Pattern-Avoidance-in-Double-Lists.pdf">Pattern Avoidance in Double Lists</a>, preprint, 2015.
%H Kevin Fagan, <a href="http://chesswanks.com/pot/IntelligenceTest.jpg">Drabble cartoon, Jun 15 1987: Intelligence Test</a>
%H Adam M. Goyt and Lara K. Pudwell, <a href="http://arxiv.org/abs/1203.3786">Avoiding colored partitions of two elements in the pattern sense</a>, arXiv preprint arXiv:1203.3786 [math.CO], 2012, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Goyt/goyt4.html">J. Int. Seq. 15 (2012) # 12.6.2</a>
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%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 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EvenNumber.html">Even Number</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HamiltonianCycle.html">Hamiltonian Cycle</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RiemannZetaFunctionZeros.html">Riemann Zeta Function Zeros</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Alkane">Alkane</a>
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F G.f.: 2*x/(1-x)^2.
%F E.g.f.: 2*x*exp(x). - _Geoffrey Critzer_, Aug 25 2012
%F G.f. with interpolated zeros: 2x^2/((1-x)^2 * (1+x)^2); e.g.f. with interpolated zeros: x*sinh(x). - _Geoffrey Critzer_, Aug 25 2012
%F Inverse binomial transform of A036289, n*2^n. - _Joshua Zucker_, Jan 13 2006
%F a(0) = 0, a(1) = 2, a(n) = 2a(n-1) - a(n-2). - _Jaume Oliver Lafont_, May 07 2008
%F a(n) = Sum_{k=1..n} floor(6n/4^k + 1/2). - _Vladimir Shevelev_, Jun 04 2009
%F a(n) = A034856(n+1) - A000124(n) = A000217(n) + A005408(n) - A000124(n) = A005408(n) - 1. - _Jaroslav Krizek_, Sep 05 2009
%F a(n) = Sum_{k>=0} A030308(n,k)*A000079(k+1). - _Philippe Deléham_, Oct 17 2011
%F Digit sequence 22 read in base n-1. - _Jason Kimberley_, Oct 30 2011
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Vincenzo Librandi_, Dec 23 2011
%F a(n) = 2*n = Product_{k=1..2*n-1} 2*sin(Pi*k/(2*n)), n >= 0 (undefined product := 1). See an Oct 09 2013 formula contribution in A000027 with a reference. - _Wolfdieter Lang_, Oct 10 2013
%F From _Ilya Gutkovskiy_, Aug 19 2016: (Start)
%F Convolution of A007395 and A057427.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/2 = (1/2)*A002162 = (1/10)*A016655. (End)
%F From _Bernard Schott_, Dec 10 2020: (Start)
%F Sum_{n>=1} 1/a(n)^2 = Pi^2/24 = A222171.
%F Sum_{n>=1} (-1)^(n+1)/a(n)^2 = Pi^2/48 = A245058. (End)
%e G.f. = 2*x + 4*x^2 + 6*x^3 + 8*x^4 + 10*x^5 + 12*x^6 + 14*x^7 + 16*x^8 + ...
%p A005843 := n->2*n;
%p A005843:=2/(z-1)**2; # _Simon Plouffe_ in his 1992 dissertation
%t Range[0,120,2] (* _Harvey P. Dale_, Aug 16 2011 *)
%o (Magma) [ 2*n : n in [0..100]];
%o (R) seq(0,200,2)
%o (PARI) A005843(n) = 2*n
%o (Haskell)
%o a005843 = (* 2)
%o a005843_list = [0, 2 ..] -- _Reinhard Zumkeller_, Feb 11 2012
%o (Python) def a(n): return 2*n # _Martin Gergov_, Oct 20 2022
%Y a(n)=2*A001477(n). - _Juri-Stepan Gerasimov_, Dec 12 2009
%Y Cf. A000027, A002061, A005408, A001358, A077553, A077554, A077555, A002024, A087112, A157888, A157889, A140811, A157872, A157909, A157910, A165900.
%Y Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306 (k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), this sequence (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7). - _Jason Kimberley_, Oct 30 2011
%Y Cf. A231200 (boustrophedon transform).
%K nonn,easy,core,nice
%O 0,2
%A _N. J. A. Sloane_
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