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%I A006331 M1963 #200 Aug 02 2024 22:38:12
%S A006331 0,2,10,28,60,110,182,280,408,570,770,1012,1300,1638,2030,2480,2992,
%T A006331 3570,4218,4940,5740,6622,7590,8648,9800,11050,12402,13860,15428,
%U A006331 17110,18910,20832,22880,25058,27370,29820,32412,35150,38038,41080,44280
%N A006331 a(n) = n*(n+1)*(2*n+1)/3.
%C A006331 Triangles in rhombic matchstick arrangement of side n.
%C A006331 Maximum accumulated number of electrons at energy level n. - _Scott A. Brown_, Feb 28 2000
%C A006331 Let M_n denote the n X n matrix M_n(i,j)=i^2+j^2; then the characteristic polynomial of M_n is x^n - a(n)x^(n-1) - .... - _Michael Somos_, Nov 14 2002
%C A006331 Convolution of odds (A005408) and evens (A005843). - _Graeme McRae_, Jun 06 2006
%C A006331 a(n) is the number of non-monotonic functions with domain {0,1,2} and codomain {0,1,...,n}. - _Dennis P. Walsh_, Apr 25 2011
%C A006331 For any odd number 2n+1, find Sum_{aTable of n, a(n) for n = 0..10000
%H A006331 J. L. Bailey, Jr., A table to facilitate the fitting of certain logistic curves, Annals Math. Stat., Vol. 2 (1931), pp. 355-359.
%H A006331 J. L. Bailey, A table to facilitate the fitting of certain logistic curves, Annals Math. Stat., Vol. 2 (1931), pp. 355-359. [Annotated scanned copy]
%H A006331 Rowan Beckworth, Basic atomic information.
%H A006331 Allan Bickle and Zhongyuan Che, Irregularities of Maximal k-degenerate Graphs, Discrete Applied Math. 331 (2023) 70-87.
%H A006331 Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
%H A006331 P. Cameron, T. Prellberg and D. Stark, Asymptotics for incidence matrix classes, Electron. J. Combin. 13 (2006), #R85, p. 11.
%H A006331 Jose Manuel Garcia Calcines, Luis Javier Hernandez Paricio, and Maria Teresa Rivas Rodriguez, Semi-simplicial combinatorics of cyclinders and subdivisions, arXiv:2307.13749 [math.CO], 2023. See p. 25.
%H A006331 N. S. S. Gu, H. Prodinger and S. Wagner, Bijections for a class of labeled plane trees, Eur. J. Combinat., Vol. 31 (2010), pp. 720-732, doi|10.1016/j.ejc.2009.10.007, Theorem 2 at n=3.
%H A006331 Germain Kreweras, Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, Vol. 6 (1965), circa p. 82.
%H A006331 Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A006331 Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
%H A006331 Dennis Walsh, Notes on finite monotonic and non-monotonic functions.
%H A006331 Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
%F A006331 G.f.: 2*x*(1 + x)/(1 - x)^4. - _Simon Plouffe_ (in his 1992 dissertation)
%F A006331 a(n) = 2*binomial(n+1,3) + 2*binomial(n+2,3).
%F A006331 a(n) = 2*A000330(n) = A002492(n)/2.
%F A006331 a(n) = Sum_{i=0..n} T(i,n-i), array T as in A048147. - _N. J. A. Sloane_, Dec 11 1999
%F A006331 From the formula for the sum of squares of positive integers 1^2 + 2^2 + 3^2 + ... + n^2 = n*(n+1)(2*n+1)/6, if we multiply both sides by 2 we get Sum_{k=0..n} 2*k^2 = n*(n+1)*(2*n+1)/3, which is an alternative formula for this sequence. - _Mike Warburton_, Sep 08 2007
%F A006331 10*a(n) = A016755(n) - A001845(n); since A016755 are odd cubes and A001845 centered octahedral numbers, 10*a(n) are the "odd cubes without their octahedral contents." - _Damien Pras_, Mar 19 2011
%F A006331 a(n) = sum(a*b), where the summing is over all unordered partitions 2*n+1=a+b. - _Vladimir Shevelev_, May 11 2012
%F A006331 a(n) = binomial(2*n+2, 3)/2. - _Ronan Flatley_, Dec 13 2012
%F A006331 a(n) = A000292(n) + A002411(n). - _Omar E. Pol_, Jan 11 2013
%F A006331 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3, with a(0)=0, a(1)=2, a(2)=10, a(3)=28. - _Harvey P. Dale_, Apr 12 2013
%F A006331 a(n) = A208532(n+1,2). - _Philippe Deléham_, Dec 05 2013
%F A006331 Sum_{n>0} 1/a(n) = 9 - 12*log(2). - _Enrique Pérez Herrero_, Dec 03 2014
%F A006331 a(n) = A000292(n-1) + (n+1)*A000217(n). - _J. M. Bergot_, Sep 02 2015
%F A006331 a(n) = 2*(A000332(n+3) - A000332(n+1)). - _Antal Pinter_, Sep 20 2015
%F A006331 From _Bruno Berselli_, May 17 2018: (Start)
%F A006331 a(n) = n*A002378(n) - Sum_{k=0..n-1} A002378(k) for n>0, a(0)=0. Also:
%F A006331 A163102(n) = n*a(n) - Sum_{k=0..n-1} a(k) for n>0, A163102(0)=0. (End)
%F A006331 a(n) = A005900(n) - A000290(n) = A096000(n) - A000578(n+1) = A000578(n+1) - A084980(n+1) = A000578(n+1) - A077415(n)-1 = A112524(n) + 1 = A188475(n) - 1 = A061317(n) - A100178(n) = A035597(n+1) - A006331(n+1). - _Bruce J. Nicholson_, Jun 24 2018
%F A006331 E.g.f.: (1/3)*exp(x)*x*(6 + 9*x + 2*x^2). - _Stefano Spezia_, Jan 05 2020
%F A006331 Sum_{n>=1} (-1)^(n+1)/a(n) = 3*Pi - 9. - _Amiram Eldar_, Jan 04 2022
%e A006331 For n=2, a(2)=10 since there are 10 non-monotonic functions f from {0,1,2} to {0,1,2}, namely, functions f =