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%I M2640 N1050 #285 Mar 25 2024 15:26:45
%S 1,3,7,13,22,34,50,70,95,125,161,203,252,308,372,444,525,615,715,825,
%T 946,1078,1222,1378,1547,1729,1925,2135,2360,2600,2856,3128,3417,3723,
%U 4047,4389,4750,5130,5530,5950,6391,6853,7337,7843,8372,8924,9500
%N Expansion of 1/((1-x)^4*(1+x)).
%C Also a(n) is the number of nondegenerate triangles that can be made from rods of lengths 1 to n+1. - _Alfred Bruckstein_; corrected by _Hans Rudolf Widmer_, Nov 02 2023
%C Also number of circumscribable (or escrible) quadrilaterals that can be made from rods of length 1,2,3,4,...,n. - Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr)
%C Also number of 2 X n binary matrices up to row and column permutation (see the link: Binary matrices up to row and column permutations). - _Vladeta Jovovic_
%C Also partial sum of alternate triangular numbers (1, 3, 1+6, 3+10, 1+6+15, 3+10+21, etc.); and also number of triangles pointing in opposite direction to largest triangle in triangular matchstick arrangement of side n+2 (cf. A002717, also the Larsen article). - _Henry Bottomley_, Aug 08 2000
%C Ordered union of A002412(n+1) and A016061(n+1). - _Lekraj Beedassy_, Oct 13 2003
%C Also Molien series for certain 4-D representation of cyclic group of order 2. - _N. J. A. Sloane_, Jun 12 2004
%C From Radu Grigore (radugrigore(AT)gmail.com), Jun 19 2004: (Start)
%C a(n) = floor( (n+2)*(n+4)*(2n+3) / 24 ). E.g., a(2) = floor(4*6*7/24) = 7 because there are 7 upside down triangles (6 of size one and 1 of size two) in the matchstick figure:
%C /\
%C /\/\
%C /\/\/\
%C /\/\/\/\
%C (End)
%C Number of non-congruent non-parallelogram trapezoids with positive integer sides (trapezints) and perimeter 2n+5. Also with perimeter 2n+8. - _Michael Somos_, May 12 2005
%C a(n) = A108561(n+4,n) for n > 0. - _Reinhard Zumkeller_, Jun 10 2005
%C Also number of nonisomorphic planes with n points and 2 lines. E.g., a(0)=1 because with no points, we just have two empty lines. a(1)=3 because the one point may belong to 0, 1 or 2 lines. a(2)=7 because there are 7 ways to determine which of 2 points belong to which of 2 lines, up to isomorphism, i.e., up to a bijection f on the sets of points and a bijection g on the sets of lines, such that A belongs to a iff f(A) belongs to g(a). - Bjorn Kjos-Hanssen (bjorn(AT)math.uconn.edu), Nov 10 2005
%C a(n-2) is the number of ways to pick two non-overlapping subwords of equal nonzero length from a word of length n. E.g., a(5-2)=a(3)=13 since the word 12345 of length 5 has the following subword pairs: 1,2; 1,3; 1,4; 1,5; 2,3; 2,4; 2,5; 3,4; 3,5; 4,5; 12,34; 12,45; 23,45. - _Michael Somos_, Oct 22 2006
%C Partial sums of A002620. - G.H.J. van Rees (vanrees(AT)cs.umanitoba.ca), Feb 16 2007
%C From Philippe LALLOUET (philip.lallouet(AT)orange.fr), Oct 19 2007: (Start)
%C Also number of squares of any size in a staircase of n steps built with unit squares:
%C __
%C |__|__
%C |__|__|__
%C |__|__|__|
%C For a staircase of 3 steps 6 squares of size 1 and 1 square of size 2, hence c(3)=7.
%C Columns sums of:
%C 1 3 6 10 15 21 28 ...
%C 1 3 6 10 15 ...
%C 1 3 6 ...
%C 1 ...
%C ---------------------
%C 1 3 7 13 22 34 50 ...
%C (End)
%C a(n) = sum of row n+1 of triangle A134446. Also, binomial transform of [1, 2, 2, 0, 1, -2, 4, -8, 16, -32, ...]. - _Gary W. Adamson_, Oct 25 2007
%C Let b(n) be the number of 4-tuples (w,x,y,z) having all terms in {1,...,n} and 2w=x+y+z+n; then b(n+3) = a(n) for n >= 0. - _Clark Kimberling_, May 08 2012
%C a(n) is the number of 3-tuples (w,x,y) having all terms in {0,...,n} and w >= x+y and x <= y. - _Clark Kimberling_, Jun 04 2012
%C Also, number of unlabeled bipartite graphs with two left vertices and n right vertices. - _Yavuz Oruc_, Jan 14 2018
%C Also number of triples (x,y,z) with 0 < x <= y <= z <= n + 1, x + y > z. - _Ralf Steiner_, Feb 06 2020
%C Bisections A002412 and A016061: a(2*k) = k*(k+1)*(4*k-1)/3! and a(2*k+1) = (k+1)*(k+2)*(4*k+9)/3!, for k >= 0. See the Woolhouse link, II. Solution by Stephen Watson, p. 65, with index shifts. - _Mo Li_, Apr 02 2020
%C Also, Wiener index of the square of the path graph P_(n+2). - _Allan Bickle_, Aug 01 2020
%C Maximum Wiener index of all maximal 2-degenerate graphs with n+2 vertices. (A maximal 2-degenerate graph can be constructed from a 2-clique by iteratively adding a new 2-leaf (vertex of degree 2) adjacent to two existing vertices.) The extremal graphs are squares of paths, so the bound also applies to 2-trees and maximal outerplanar graphs. - _Allan Bickle_, Sep 15 2022
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 74, Problem 7.
%D P. Diaconis, R. L. Graham and B. Sturmfels, Primitive partition identities, in Combinatorics: Paul Erdős is Eighty, Vol. 2, Bolyai Soc. Math. Stud., 2, 1996, pp. 173-192.
%D H. Gupta, Partitions of j-partite numbers into twelve or a smaller number of parts. Collection of articles dedicated to Professor P. L. Bhatnagar on his sixtieth birthday. Math. Student 40 (1972), 401-441 (1974).
%D I. Siap, Linear codes over F_2 + u*F_2 and their complete weight enumerators, in Codes and Designs (Ohio State, May 18, 2000), pp. 259-271. De Gruyter, 2002.
%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).
%H T. D. Noe, <a href="/A002623/b002623.txt">Table of n, a(n) for n = 0..1000</a>
%H A. Atmaca, and A. Yavuz Oruç, <a href="https://doi.org/10.1016/j.akcej.2017.11.008">On the size of two families of unlabeled bipartite graphs</a>, AKCE International Journal of Graphs and Combinatorics, 2017.
%H M. Benoumhani and M. Kolli, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Benoumhani/benoumhani6.html">Finite topologies and partitions</a>, JIS 13 (2010) # 10.3.5, Lemma 6 5th line.
%H Allan Bickle and Zhongyuan Che, <a href="https://arxiv.org/abs/1908.09202">Wiener indices of maximal k-degenerate graphs</a>, arXiv:1908.09202 [math.CO], 2019.
%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 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H S. B. Ekhad and D. Zeilberger, <a href="http://arxiv.org/abs/1511.06791">Computerizing the Andrews-Fraenkel-Sellers Proofs on the Number of m-ary partitions mod m (and doing MUCH more!)</a>, arXiv preprint arXiv:1511.06791 [math.CO], 2015.
%H E. Fix and J. L. Hodges, <a href="/A000601/a000601.pdf">Significance probabilities of the Wilcoxon test</a>, Annals Math. Stat., 26 (1955), 301-312. [Annotated scanned copy]
%H E. Fix and J. L. Hodges, Jr., <a href="https://doi.org/10.1214/aoms/1177728547">Significance probabilities of the Wilcoxon test</a>, Annals Math. Stat., 26 (1955), 301-312.
%H E. Gonzalez-Jimenez and X. Xarles, <a href="http://arxiv.org/abs/1301.5122">On a conjecture of Rudin on squares in Arithmetic Progressions</a>, arXiv preprint arXiv:1301.5122 [math.NT], 2013.
%H H. Gupta, <a href="/A001840/a001840.pdf">Partitions of j-partite numbers into twelve or a smaller number of parts</a>, Math. Student 40 (1972), 401-441 (1974). [Annotated scanned copy]
%H M. A. Harrison, <a href="http://dx.doi.org/10.1109/T-C.1973.223649">On the number of classes of binary matrices</a>, IEEE Trans. Computers, 22 (1973), 1048-1051. doi:10.1109/T-C.1973.223649.
%H M. A. Harrison, <a href="/A000711/a000711.pdf">On the number of classes of binary matrices</a>, IEEE Transactions on Computers, C-22.12 (1973), 1048-1052. (Annotated scanned copy)
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=203">Encyclopedia of Combinatorial Structures 203</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=413">Encyclopedia of Combinatorial Structures 413</a>
%H Vladeta Jovovic, <a href="/A005748/a005748.pdf">Binary matrices up to row and column permutations</a>
%H A. Kerber, <a href="/A002727/a002727.pdf">Experimentelle Mathematik</a>, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 19 (1988), 77-83. [Annotated scanned copy]. See page 79.
%H W. Lanssens, B. Demoen, and P.-L. Nguyen, <a href="https://lirias.kuleuven.be/1656296">The Diagonal Latin Tableau and the Redundancy of its Disequalities</a>, Report CW 666, July 2014, Department of Computer Science, KU Leuven
%H M. E. Larsen, <a href="http://www.jstor.org/stable/2686923">The eternal triangle - a history of a counting problem</a>, College Math. J., 20 (1989), 370-392.
%H P. Lisonek, <a href="/A005045/a005045_2.pdf">Quasi-polynomials: A case study in experimental combinatorics</a>, RISC-Linz Report Series No. 93-18, 1983. (Annotated scanned copy)
%H Math StackExchange, <a href="https://math.stackexchange.com/questions/4085977">cycle index for S_2 X S_4</a>, Apr. 2021
%H B. Misek, <a href="http://dml.cz/dmlcz/108444">On the number of classes of strongly equivalent incidence matrices</a>, (Czech with English summary) Casopis Pest. Mat. 89 1964 211-218. See page 217.
%H G. Nebe, E. M. Rains and N. J. A. Sloane, <a href="http://neilsloane.com/doc/cliff2.html">Self-Dual Codes and Invariant Theory</a>, Springer, Berlin, 2006.
%H Brian O'Sullivan and Thomas Busch, <a href="http://arxiv.org/abs/0810.0231">Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas</a>, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 10a, lambda=2]
%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 Giovanni Resta, <a href="/A002623/a002623.png">Illustration for a(8)=70.</a>
%H Marko Riedel, <a href="https://math.stackexchange.com/questions/4085977/">The O.g.f. of inequivalent colorings of a 2xN board with C colors in N is (1/2) 1/(1-w)^(C^2) + (1/2) 1/(1+w)^(C(C-1)/2)*1/(1-w)^(C(C+1)/2) by PET</a>
%H Atsuto Seko, <a href="https://doi.org/10.1063/5.0129045">Tutorial: Systematic development of polynomial machine learning potentials for elemental and alloy systems</a>, J. Appl. Phys. (2023) Vol. 133, 011101.
%H J. Silverman, V. E. Vickers and J. M. Mooney, <a href="https://doi.org/10.1109/5.7156">On the number of Costas arrays as a function of array size</a>, Proc. IEEE, 76 (1988), 851-853.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TriangleCounting.html">Triangle Counting.</a>
%H W. S. B. Woolhouse, <a href="https://babel.hathitrust.org/cgi/pt?id=nyp.33433062732734&view=1up&seq=307">Problem 2420. On the probability of the number of triangles</a>, Mathematical questions with their solutions, v. 9 (June 1868), pp. 63-65.
%H <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-2,3,-1).
%F a(n+1) = a(n) + {(k-1)*k if n=2*k} or {k*k if n=2*k+1}.
%F a(n)+a(n+1) = A000292(n+1).
%F a(n) = a(n-2) + A000217(n+1) = A002717(n+2) - A000292(n+1).
%F Also: a(n) = C(n+3, 3) - a(n-1) with a(0)=1. - _Labos Elemer_, Apr 26 2003
%F From _Paul Barry_, Jul 01 2003: (Start)
%F a(n) = Sum_{k=0..n} (-1)^(n-k)*C(k+3,3).
%F The signed version 1, -3, 7, ... has the formula:
%F a(n) = (4*n^3 + 30*n^2 + 68*n + 45)*(-1)^n/48 + 1/16.
%F This is the partial sums of the signed version of A000292. (End)
%F From _Paul Barry_, Jul 21 2003: (Start)
%F a(n) = Sum_{k=0..n} floor((k+2)^2/4).
%F a(n) = Sum_{k=0..n} Sum_{j=0..k} Sum_{i=0..j} (1+(-1)^i)/2. (End)
%F a(n) = a(n - 2) + (n*(n - 1))/2, with n>2, a(1)=0, a(2)=1; a(n) = (4*n^3+6*n^2-4*n+3*(-1)^n-3)/48, with offset 2. - Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 14 2004 (formula simplified by _Bruno Berselli_, Aug 29 2013)
%F a(n) = ((2*n+3)*(n+2)*(n+1)/6-floor((n+2)/2))/4, with offset 1. - Jerry W. Lewis (JLewis(AT)wyeth.com), Mar 23 2005
%F a(n) = 2*a(n-1) - a(n-2) + 1 + floor(n/2). - Bjorn Kjos-Hanssen (bjorn(AT)math.uconn.edu), Nov 10 2005
%F A002620(n+3) = a(n+1) - a(n). - _Michael Somos_, Sep 04 1999
%F Euler transform of length 2 sequence [ 3, 1]. - _Michael Somos_, Sep 04 2006
%F a(n) = -a(-5-n) for all n in Z. - _Michael Somos_, Sep 04 2006
%F Let P(i,k) be the number of integer partitions of n into k parts, then with k=2 we have a(n) = sum_{m=1}^{n} sum_{i=k}^{m} P(i,k). For k=1 we get A000217 = triangular numbers. - _Thomas Wieder_, Feb 18 2007
%F a(n) = (n+(3+(-1)^n)/2)*(n+(7+(-1)^n)/2)*(2*n+5-2*(-1)^n)/24. - Philippe LALLOUET (philip.lallouet(AT)orange.fr), Oct 19 2007 (corrected by _Bruno Berselli_, Aug 30 2013)
%F From _Johannes W. Meijer_, May 20 2011: (Start)
%F a(n) = A006918(n+1) + A006918(n).
%F a(n) = A058187(n-2) + 2*A058187(n-1) + A058187(n). (End)
%F a(0)=1, a(1)=3, a(2)=7, a(3)=13, a(4)=22; for n > 4, a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5). - _Harvey P. Dale_, Jul 19 2011
%F a(n) = Sum_{i=0..n+2} floor(i/2)*ceiling(i/2). - _Bruno Berselli_, Aug 30 2013
%F a(n) = 15/16 + (1/16)*(-1)^n + (17/12)*n + (5/8)*n^2 + (1/12)*n^3. - _Robert Israel_, Jul 07 2014
%F a(n) = Sum_{i=0..n+2} (n+1-i)*floor(i/2+1). - _Bruno Berselli_, Apr 04 2017
%F a(n) = 1 + floor((2*n^3 + 15*n^2 + 34*n) / 24). - _Allan Bickle_, Aug 01 2020
%F E.g.f.: ((24 + 51*x + 21*x^2 + 2*x^3)*cosh(x) + (21 + 51*x + 21*x^2 + 2*x^3)*sinh(x))/24. - _Stefano Spezia_, Jun 02 2021
%e G.f. = 1 + 3*x + 7*x^2 + 13*x^3 + 22*x^4 + 34*x^5 + 50*x^6 + 70*x^7 + 95*x^8 + ...
%p A002623 := n->(1/16)*(1+(-1)^n)+(n+1)/8+binomial(n+2,2)/4+binomial(n+3,3)/2;
%p seq( ((2*n+3)*(n+2)*(n+1)/6-floor((n+2)/2))/4,n=1..47); # Lewis
%p a := n -> ((-1)^n*3 + 45 + 68*n + 30*n^2 + 4*n^3) / 48:
%p seq(a(n), n=0..46); # _Peter Luschny_, Jan 22 2018
%t CoefficientList[Series[1/((1-x)^3(1-x^2)),{x,0,50}],x] (* or *) LinearRecurrence[{3,-2,-2,3,-1},{1,3,7,13,22},50] (* _Harvey P. Dale_, Jul 19 2011 *)
%t Table[((2 n^3 + 15 n^2 + 34 n + 45 / 2 + (3/2) (-1)^n) / 24), {n, 0, 100}] (* _Vincenzo Librandi_, Jan 15 2018 *)
%t a[ n_] := Floor[(n + 2)*(n + 4)*(2*n + 3)/24]; (* _Michael Somos_, Feb 19 2024 *)
%o (PARI) {a(n) = (8 + 34/3*n + 5*n^2 + 2/3*n^3) \ 8}; /* _Michael Somos_, Sep 04 1999 */
%o (PARI) x='x+O('x^50); Vec(1/((1 - x)^3 * (1 - x^2))) \\ _Indranil Ghosh_, Apr 04 2017
%o (Python)
%o def A002623(n): return ((n+2)*(n+4)*((n<<1)+3)>>3)//3 # _Chai Wah Wu_, Mar 25 2024
%Y Cf. A002620 (first differences), A000292, A001752 (partial sums), A062109 (binomial transf.).
%Y Bisections A002412, A016061.
%Y Cf. also A002717 (a companion sequence), A002727, A006148, A057524, A134446, A014125, A122046, A122047.
%Y Cf. A000217, A006918, A058187, A108561.
%Y The maximum Wiener index of all maximal k-degenerate graphs for k=1..6 are given in A000292, A002623 (this sequence), A014125, A122046, A122047, A175724, respectively.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_
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