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 A008805 Triangular numbers repeated. 58
 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, 91, 105, 105, 120, 120, 136, 136, 153, 153, 171, 171, 190, 190, 210, 210, 231, 231, 253, 253, 276, 276, 300, 300, 325, 325, 351, 351, 378, 378, 406, 406, 435, 435 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Number of choices for nonnegative integers x,y,z such that x and y are even and x + y + z = n. Diagonal sums of A002260, when arranged as a number triangle. - Paul Barry, Feb 28 2003 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 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 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 Partial sums of positive terms of A142150. - Reinhard Zumkeller, Jul 07 2012 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 Number of the distinct symmetric pentagons in a regular n-gon, see illustration for some small n in links. - Kival Ngaokrajang, Jun 25 2013 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 a(n) = number of distinct opening moves in n X n tic-tac-toe. - I. J. Kennedy, Sep 04 2013 a(n) is the number of symmetry-allowed, linearly-independent terms at n-th order in the series expansion of the T2xt2 vibronic perturbation matrix, H(Q) (cf. Opalka & Domcke). - Bradley Klee, Jul 20 2015 a(n-1) gives also 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 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 REFERENCES H. D. Brunk, An Introduction to Mathematical Statistics, Ginn, Boston, 1960; p. 360. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..3000 G. E. Andrews, M. Beck, N. Robbins, Partitions with fixed differences between largest and smallest parts, arXiv preprint arXiv:1406.3374 [math.NT], 2014. P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 46. Kival Ngaokrajang, The distinct symmetric 5-gons in a regular n-gon for n = 6..13 D. Opalka and W. Domcke, High-order expansion of T2xt2 Jahn-Teller potential energy surfaces in tetrahedral molecules, J. Chem. Phys., 132, 154108 (2010). V. Shevelev, A problem of enumeration of two-color bracelets with several variations, arXiv:0710.1370 [math.CO], 2007-2011. Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1). FORMULA G.f.: 1/((1-x)*(1-x^2)^2) = 1/((1+x)^2*(1-x)^3). E.g.f.: (exp(x)*(2*x^2 +12*x+ 11) - exp(-x)*(2*x -5))/16. a(-n) = a(-5+n). a(n) = binomial(floor(n/2)+2, 2). - Vladimir Shevelev, May 03 2011 From Paul Barry, May 31 2003: (Start) a(n) = ((2*n +5)*(-1)^n + (2*n^2 +10*n +11))/16. a(n) = Sum_{k=0..n} ((k+2)*(1+(-1)^k))/4. (End) From Paul Barry, Apr 16 2005: (Start) a(n) = Sum_{k=0..n} floor((k+2)/2)*(1-(-1)^(n+k-1))/2. a(n) = Sum_{k=0..floor(n/2)} floor((n-2k+2)/2). (End) A signed version is given by Sum_{k=0..n} (-1)^k*floor(k^2/4). - Paul Barry, Aug 19 2003 a(n) = A108299(n-2,n)*(-1)^floor((n+1)/2) for n>1. - Reinhard Zumkeller, Jun 01 2005 a(n+1) = [Sum_{k=1..n} k mod (n+1)] + a(n), with n>=1 and a(1)=1. - Paolo P. Lava, Mar 19 2007 a(n) = A004125(n+3) - A049798(n+2). - Carl Najafi, Jan 31 2013 a(n) = Sum_{i=1..floor((n+2)/2)} i. - Wesley Ivan Hurt, Jun 08 2013 a(n) = (1/2)*floor((n+2)/2)*(floor((n+2)/2)+1). - Wesley Ivan Hurt, Jun 08 2013 From Wesley Ivan Hurt, Apr 22 2015: (Start) a(n) = a(n-1) +2*a(n-2) -2*a(n-3) -a(n-4) +a(n-5). a(n) = (2*n +3 +(-1)^n)*(2*n +7 +(-1)^n)/32. (End) a(n-1) = A054252(n,1) = A054252(n^2-1), n >= 1. See a Oct 03 2016 comment above. - Wolfdieter Lang, Oct 03 2016 a(n) = A000217(A008619(n)). - Guenther Schrack, Sep 12 2018 EXAMPLE 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 MAPLE 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 MATHEMATICA CoefficientList[Series[1/(1-x^2)^2/(1-x), {x, 0, 50}], x] Table[Binomial[Floor[n/2] + 2, 2], {n, 0, 57}] (* Michael De Vlieger, Oct 03 2016 *) PROG (PARI) a(n)=(n\2+2)*(n\2+1)/2 (Haskell) import Data.List (transpose) a008805 = a000217 . (`div` 2) . (+ 1) a008805_list = drop 2 \$ concat \$ transpose [a000217_list, a000217_list] -- Reinhard Zumkeller, Feb 01 2013 (MAGMA) [(2*n+3+(-1)^n)*(2*n+7+(-1)^n)/32 : n in [0..50]]; // Wesley Ivan Hurt, Apr 22 2015 (Sage) [(2*n +3 +(-1)^n)*(2*n +7 +(-1)^n)/32 for n in (0..60)] # G. C. Greubel, Sep 12 2019 (GAP) List([0..60], n-> (2*n +3 +(-1)^n)*(2*n +7 +(-1)^n)/32); # G. C. Greubel, Sep 12 2019 CROSSREFS Cf. A000217, A002260, A006918 (partial sums), A054252. Sequence in context: A325861 A079551 A182843 * A188270 A026925 A237665 Adjacent sequences:  A008802 A008803 A008804 * A008806 A008807 A008808 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified October 21 15:00 EDT 2019. Contains 328301 sequences. (Running on oeis4.)