Search: id:A002623 Results 1-1 of 1 results found. %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, Table of n, a(n) for n=0..1000 %H A002623 Index entries for sequences related to linear recurrences with constant coefficients %H A002623 P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5. %H A002623 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 203 %H A002623 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 413 %H A002623 Vladeta Jovovic, Binary matrices up to row and column permutations %H A002623 G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006. %H A002623 S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992. %H A002623 S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992. %H A002623 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A002623 Index entries for Molien series %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 Search completed in 0.002 seconds