%I A002623 M2640 N1050
%S A002623 1,3,7,13,22,34,50,70,95,125,161,203,252,308,372,444,525,615,715,825,
%T A002623 946,1078,1222,1378,1547,1729,1925,2135,2360,2600,2856,3128,3417,3723,
%U A002623 4047,4389,4750,5130,5530,5950,6391,6853,7337,7843,8372,8924,9500
%N A002623 G.f.: 1/((1-x)^3*(1-x^2)).
%C A002623 Also number of nondegenerate triangles that can be made from rods of 
               length 1,2,3,4,...,n (Alfred Bruckstein, freddy(AT)cs.technion.ac.il).
%C A002623 Also number of circumscribable (or escrible) quadrilaterals that can 
               be made from rods of length 1,2,3,4,....,n (xpolakis(AT)otenet.gr, 
               Antreas P. Hatzipolakis)
%C A002623 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 (vladeta(AT)eunet.rs).
%C A002623 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 
               (se16(AT)btinternet.com), Aug 08 2000
%C A002623 Also Molien series for certain 4-D representation of cyclic group of 
               order 2.
%C A002623 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 A002623 a(n) = A108561(n+4,n) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Jun 10 2005
%C A002623 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 A002623 a(n-2) is the number of ways to pick two non-overlapping subwords of 
               equal nonzero length from a word of length n. - Michael Somos Oct 
               22 2006
%C A002623 Partial sums of A002620. - G.H.J. van Rees (vanrees(AT)cs.umanitoba.ca), 
               Feb 16 2007
%C A002623 Comment from Philippe LALLOUET (philip.lallouet(AT)orange.fr), Oct 19 
               2007: (Start) Also number of squares of any size in a staircase of 
               n steps built with unit squares:
%C A002623 .__
%C A002623 |__|__
%C A002623 |__|__|__
%C A002623 |__|__|__|
%C A002623 For a staircase of 3 steps 6 squares of size 1 and 1 square of size 2, 
               hence c(3)=7. (End)
%D A002623 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002623 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002623 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 74, Problem 7.
%D A002623 P. Diaconis, R. L. Graham and B. Sturmfels, Primitive partition identities, 
               in Combinatorics: Paul Erdos is Eighty, Vol. 2, Bolyai Soc. Math. 
               Stud., 2, 1996, pp. 173-192.
%D A002623 E. Fix and J. L. Hodges, Jr., Significance probabilities of the Wilcoxon 
               test, Annals Math. Stat., 26 (1955), 301-312.
%D A002623 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 A002623 M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. 
               Computers, 22 (1973), 1048-1051.
%D A002623 M. E. Larsen, The eternal triangle - a history of a counting problem, 
               College Math. J., 20 (1989), 370-392.
%D A002623 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 A002623 J. Silverman, V. E. Vickers and J. M. Mooney, On the number of Costas 
               arrays as a function of array size, Proc. IEEE, 76 (1988), 851-853.
%D A002623 Brian OSullivan and Thomas Busch, Spontaneous emission in ultra-cold 
               spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 
               2008. [Eq 10a, lambda=2]
%H A002623 T. D. Noe, <a href="b002623.txt">Table of n, a(n) for n=0..1000</a>
%H A002623 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A002623 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Sequences realized by oligomorphic permutation groups</a>, J. Integ. 
               Seqs. Vol. 3 (2000), #00.1.5.
%H A002623 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=203">
               Encyclopedia of Combinatorial Structures 203</a>
%H A002623 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=413">
               Encyclopedia of Combinatorial Structures 413</a>
%H A002623 Vladeta Jovovic, <a href="a005748.PDF">Binary matrices up to row and 
               column permutations</a>
%H A002623 G. Nebe, E. M. Rains and N. J. A. Sloane, <a href="http://www.research.att.com/
               ~njas/doc/cliff2.html">Self-Dual Codes and Invariant Theory</a>, 
               Springer, Berlin, 2006.
%H A002623 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A002623 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A002623 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               TriangleCounting.html">Link to a section of The World of Mathematics.</
               a>
%H A002623 <a href="Sindx_Mo.html#Molien">Index entries for Molien series</a>
%F A002623 a(n+1) = a(n) + {(k-1)*k if n=2*k} or {k*k if n=2*k+1}.
%F A002623 a(n) = a(n-2)+A000217(n+1) = A002717(n+2)-A000292(n+1)
%F A002623 Also: a(n)=C(n, 3)-a(n-1) with a(0)=0 and A002623(0)=a(3); a(n)=A002623(n-3). 
               - Labos E. (labos(AT)ana.hu), Apr 26 2003
%F A002623 Sum{k=0..n, (-1)^(n-k)C(k+3, 3) } - Paul Barry (pbarry(AT)wit.ie), Jul 
               01 2003
%F A002623 The signed version 1, -3, 7, .... has a(n)=(4n^3+30n^2+68n+45)(-1)^n/
               48+1/16. This is the partial sums of the signed version of A000292. 
               - Paul Barry (pbarry(AT)wit.ie), Jul 01 2003
%F A002623 a(n)=sum{k=0..n, floor((k+2)^2/4)}; a(n)=sum{k=0..n, sum{j=0..k, sum{i=0..j, 
               (1+(-1)^i)/2 }}}; - Paul Barry (pbarry(AT)wit.ie), Jul 21 2003
%F A002623 Ordered union of A002412(n+1) and A016061(n+1). - Lekraj Beedassy (blekraj(AT)yahoo.com), 
               Oct 13 2003
%F A002623 Comment from Radu Grigore (radugrigore(AT)gmail.com), Jun 19 2004: 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:
%F A002623 .../\
%F A002623 ../\/\
%F A002623 ./\/\/\
%F A002623 /\/\/\/\
%F A002623 a[n] == a[n - 2] + (n*(n - 1))/2, a[1] == 0, a[2] == 1; (3*(-1)^n - 3*(-1)^(2*n) 
               + 8*n - 12*(-1)^(2*n)*n + 12* n^2 - 6*(-1)^(2*n)*n^2 + 4*n^3)/48 
               - Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 14 2004
%F A002623 a(n) = ((2*n+3)*(n+2)*(n+1)/6-floor((n+2)/2))/4; a(n) = sum(floor(i^2/
               4), i=2..n+1) - Jerry W. Lewis (JLewis(AT)wyeth.com), Mar 23 2005
%F A002623 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 A002623 a(-5-n)=-a(n). - Michael Somos Sep 04 2006
%F A002623 Euler transform of length 2 sequence [ 3, 1]. - Michael Somos Sep 04 
               2006
%F A002623 a(n) = ceiling( (n+3)(n+1)(2n+7) ) - G.H.J. van Rees (vanrees(AT)cs.umanitoba.ca), 
               Feb 16 2007
%F A002623 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 (thomas.wieder(AT)t-online.de), 
               Feb 18 2007
%F A002623 a(n) = (n+(1+(-1)^n)/2)*(n + (5+(-1)^n)/2)*(2*n+3+2*(-1)^n)/24. - Philippe 
               LALLOUET (philip.lallouet(AT)orange.fr), Oct 19 2007
%F A002623 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 (qntmpkt(AT)yahoo.com), 
               Oct 25 2007
%e A002623 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.
%p A002623 A002623 := n->(1/16)*(1+(-1)^n)+(n+1)/8+binomial(n+2,2)/4+binomial(n+3,
               3)/2;
%p A002623 seq( ((2*n+3)*(n+2)*(n+1)/6-floor((n+2)/2))/4,n=1..47); (Lewis)
%p A002623 A002623:=1/(z+1)/(z-1)**4; [S. Plouffe in his 1992 dissertation.]
%o A002623 (PARI) a(n)=(8+34/3*n+5*n^2+2/3*n^3)\8
%Y A002623 A002620(n+3)=a(n+1)-a(n).
%Y A002623 Cf. A002717 (a companion sequence), A002727, A006148.
%Y A002623 Partial sums of A002620. Sums of 2 consecutive terms give A000292.
%Y A002623 Cf. A000217, A057524, A134446.
%Y A002623 Sequence in context: A136219 A078582 A051336 this_sequence A081662 A091652 
               A134197
%Y A002623 Adjacent sequences: A002620 A002621 A002622 this_sequence A002624 A002625 
               A002626
%K A002623 nonn,easy,nice
%O A002623 0,2
%A A002623 N. J. A. Sloane (njas(AT)research.att.com).
%E A002623 PARI formula and more terms from Michael Somos