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A008805 Triangular numbers repeated.

%I A008805 #168 Dec 30 2023 11:02:03
%S A008805 1,1,3,3,6,6,10,10,15,15,21,21,28,28,36,36,45,45,55,55,66,66,78,78,91,
%T A008805 91,105,105,120,120,136,136,153,153,171,171,190,190,210,210,231,231,
%U A008805 253,253,276,276,300,300,325,325,351,351,378,378,406,406,435,435
%N A008805 Triangular numbers repeated.
%C A008805 Number of choices for nonnegative integers x,y,z such that x and y are even and x + y + z = n.
%C A008805 Diagonal sums of A002260, when arranged as a number triangle. - _Paul Barry_, Feb 28 2003
%C A008805 a(n) = number of partitions of n+4 such that the differences between greatest and smallest parts are 2: a(n-4) = A097364(n,2) for n>3. - _Reinhard Zumkeller_, Aug 09 2004
%C A008805 For n >= i, i=4,5, a(n-i) is the number of incongruent two-color bracelets of n beads, i from them are black (cf. A005232, A032279), having a diameter of symmetry. - _Vladimir Shevelev_, May 03 2011
%C A008805 Prefixing A008805 by 0,0,0,0 gives the sequence c(0), c(1), ... defined by c(n)=number of (w,x,y) such that w = 2x+2y, where w,x,y are all in {1,...,n}; see A211422. - _Clark Kimberling_, Apr 15 2012
%C A008805 Partial sums of positive terms of A142150. - _Reinhard Zumkeller_, Jul 07 2012
%C A008805 The sum of the first parts of the nondecreasing partitions of n+2 into exactly two parts, n >= 0. - _Wesley Ivan Hurt_, Jun 08 2013
%C A008805 Number of the distinct symmetric pentagons in a regular n-gon, see illustration for some small n in links. - _Kival Ngaokrajang_, Jun 25 2013
%C A008805 a(n) is the number of nonnegative integer solutions to the equation x + y + z = n such that x + y <= z. For example, a(4) = 6 because we have 0+0+4 = 0+1+3 = 0+2+2 = 1+0+3 = 1+1+2 = 2+0+2. - _Geoffrey Critzer_, Jul 09 2013
%C A008805 a(n) is the number of distinct opening moves in n X n tic-tac-toe. - _I. J. Kennedy_, Sep 04 2013
%C A008805 a(n) is the number of symmetry-allowed, linearly-independent terms at n-th order in the series expansion of the T2 X t2 vibronic perturbation matrix, H(Q) (cf. Opalka & Domcke). - _Bradley Klee_, Jul 20 2015
%C A008805 a(n-1) also gives the number of D_4 (dihedral group of order 4) orbits of an n X n square grid with squares coming in either of two colors and only one square has one of the colors. - _Wolfdieter Lang_, Oct 03 2016
%C A008805 Also, this sequence is the third column in the triangle of the coefficients of the sum of two consecutive Fibonacci polynomials F(n+1, x) and F(n, x) (n>=0) in ascending powers of x. - _Mohammad K. Azarian_, Jul 18 2018
%C A008805 In an n-person symmetric matching pennies game (a zero-sum normal-form game) with n > 2 symmetric and indistinguishable players, each with two strategies (viz. heads or tails), a(n-3) is the number of distinct subsets of players that must play the same strategy to avoid incurring losses (single pure Nash equilibrium in the reduced game). The total number of distinct partitions is A000217(n-1). - _Ambrosio Valencia-Romero_, Apr 17 2022
%C A008805 a(n) is the number of connected bipartite graphs with n+1 edges and a stable set of cardinality 2. - _Christian Barrientos_, Jun 15 2022
%C A008805 a(n) is the number of 132-avoiding odd Grassmannian permutations of size n+2. - _Juan B. Gil_, Mar 10 2023
%C A008805 Consider a regular n-gon with all diagonals drawn. Define a "layer" to be the set of all regions sharing an edge with the exterior. Removing a layer creates another layer. Count the layers, removing them until none remain. The number of layers is a(n-2). See illustration. - _Christopher Scussel_, Nov 07 2023
%D A008805 H. D. Brunk, An Introduction to Mathematical Statistics, Ginn, Boston, 1960; p. 360.
%H A008805 Vincenzo Librandi, <a href="/A008805/b008805.txt">Table of n, a(n) for n = 0..3000</a>
%H A008805 G. E. Andrews, M. Beck, and N. Robbins, <a href="http://arxiv.org/abs/1406.3374">Partitions with fixed differences between largest and smallest parts</a>, arXiv preprint arXiv:1406.3374 [math.NT], 2014.
%H A008805 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 46.
%H A008805 Juan B. Gil and Jessica A. Tomasko, <a href="https://arxiv.org/abs/2207.12617">Pattern-avoiding even and odd Grassmannian permutations</a>, arXiv:2207.12617 [math.CO], 2022.
%H A008805 Jia Huang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Huang/huang8.html">Partially Palindromic Compositions</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 19.
%H A008805 Kival Ngaokrajang, <a href="/A008805/a008805.jpg">The distinct symmetric 5-gons in a regular n-gon for n = 6..13</a>
%H A008805 D. Opalka and W. Domcke, <a href="http://dx.doi.org/10.1063/1.3382912">High-order expansion of T2xt2 Jahn-Teller potential energy surfaces in tetrahedral molecules</a>, J. Chem. Phys., 132, 154108 (2010).
%H A008805 Christopher Scussel, <a href="/A008805/a008805.pdf">Illustration of layers in regular n-gons with all diagonals drawn</a>
%H A008805 Vladimir Shevelev, <a href="https://arxiv.org/abs/0710.1370">A problem of enumeration of two-color bracelets with several variations</a>, arXiv:0710.1370 [math.CO], 2007-2011.
%H A008805 <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>
%H A008805 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%H A008805 <a href="/index/Mo#Molien">Index entries for Molien series</a>
%F A008805 G.f.: 1/((1-x)*(1-x^2)^2) = 1/((1+x)^2*(1-x)^3).
%F A008805 E.g.f.: (exp(x)*(2*x^2 +12*x+ 11) - exp(-x)*(2*x -5))/16.
%F A008805 a(-n) = a(-5+n).
%F A008805 a(n) = binomial(floor(n/2)+2, 2). - _Vladimir Shevelev_, May 03 2011
%F A008805 From _Paul Barry_, May 31 2003: (Start)
%F A008805 a(n) = ((2*n +5)*(-1)^n + (2*n^2 +10*n +11))/16.
%F A008805 a(n) = Sum_{k=0..n} ((k+2)*(1+(-1)^k))/4. (End)
%F A008805 From _Paul Barry_, Apr 16 2005: (Start)
%F A008805 a(n) = Sum_{k=0..n} floor((k+2)/2)*(1-(-1)^(n+k-1))/2.
%F A008805 a(n) = Sum_{k=0..floor(n/2)} floor((n-2k+2)/2). (End)
%F A008805 A signed version is given by Sum_{k=0..n} (-1)^k*floor(k^2/4). - _Paul Barry_, Aug 19 2003
%F A008805 a(n) = A108299(n-2,n)*(-1)^floor((n+1)/2) for n>1. - _Reinhard Zumkeller_, Jun 01 2005
%F A008805 a(n) = A004125(n+3) - A049798(n+2). - _Carl Najafi_, Jan 31 2013
%F A008805 a(n) = Sum_{i=1..floor((n+2)/2)} i. - _Wesley Ivan Hurt_, Jun 08 2013
%F A008805 a(n) = (1/2)*floor((n+2)/2)*(floor((n+2)/2)+1). - _Wesley Ivan Hurt_, Jun 08 2013
%F A008805 From _Wesley Ivan Hurt_, Apr 22 2015: (Start)
%F A008805 a(n) = a(n-1) +2*a(n-2) -2*a(n-3) -a(n-4) +a(n-5).
%F A008805 a(n) = (2*n +3 +(-1)^n)*(2*n +7 +(-1)^n)/32. (End)
%F A008805 a(n-1) = A054252(n,1) = A054252(n^2-1), n >= 1. See a Oct 03 2016 comment above. - _Wolfdieter Lang_, Oct 03 2016
%F A008805 a(n) = A000217(A008619(n)). - _Guenther Schrack_, Sep 12 2018
%F A008805 From _Ambrosio Valencia-Romero_, Apr 17 2022: (Start)
%F A008805 a(n) = a(n-1) if n odd, a(n) = a(n-1) + (n+2)/2 if n is even, for n > 0, a(0) = 1.
%F A008805 a(n) = (n+1)*(n+3)/8 if n odd, a(n) = (n+2)*(n+4)/8 if n is even, for n >= 0.
%F A008805 a(n) = A002620(n+2) - a(n-1), for n > 0, a(0) = 1.
%F A008805 a(n) = A142150(n+2) + a(n-1), for n > 0, a(0) = 1.
%F A008805 a(n) = A000217(n+3)/2 - A135276(n+3)/2. (End)
%e A008805 a(5) = 6, since (5) + 2 = 7 has three nondecreasing partitions with exactly 2 parts: (1,6),(2,5),(3,4). The sum of the first parts of these partitions = 1 + 2 + 3 = 6. - _Wesley Ivan Hurt_, Jun 08 2013
%p A008805 A008805:=n->(2*n+3+(-1)^n)*(2*n+7+(-1)^n)/32: seq(A008805(n), n=0..50); # _Wesley Ivan Hurt_, Apr 22 2015
%t A008805 CoefficientList[Series[1/(1-x^2)^2/(1-x), {x, 0, 50}], x]
%t A008805 Table[Binomial[Floor[n/2] + 2, 2], {n, 0, 57}] (* _Michael De Vlieger_, Oct 03 2016 *)
%o A008805 (PARI) a(n)=(n\2+2)*(n\2+1)/2
%o A008805 (Haskell)
%o A008805 import Data.List (transpose)
%o A008805 a008805 = a000217 . (`div` 2) . (+ 1)
%o A008805 a008805_list = drop 2 $ concat $ transpose [a000217_list, a000217_list]
%o A008805 -- _Reinhard Zumkeller_, Feb 01 2013
%o A008805 (Magma) [(2*n+3+(-1)^n)*(2*n+7+(-1)^n)/32 : n in [0..50]]; // _Wesley Ivan Hurt_, Apr 22 2015
%o A008805 (Sage) [(2*n +3 +(-1)^n)*(2*n +7 +(-1)^n)/32 for n in (0..60)] # _G. C. Greubel_, Sep 12 2019
%o A008805 (GAP) List([0..60], n-> (2*n +3 +(-1)^n)*(2*n +7 +(-1)^n)/32); # _G. C. Greubel_, Sep 12 2019
%o A008805 (Python)
%o A008805 def A008805(n): return (m:=(n>>1)+1)*(m+1)>>1 # _Chai Wah Wu_, Oct 20 2023
%Y A008805 Cf. A000217, A002260, A002620, A006918 (partial sums), A054252, A135276, A142150, A158920 (binomial trans.).
%K A008805 nonn,easy
%O A008805 0,3
%A A008805 _N. J. A. Sloane_