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A008805 Triangular numbers repeated. 54
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

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 two-way infinite sequences

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Index entries for Molien series

FORMULA

G.f.: 1/((1-x)*(1-x^2)^2) = 1/((1+x)^2*(1-x)^3).

E.g.f.: exp(x)*(2x^2+12x+11)/16+exp(-x)*(-2x+5)/16.

a(-n) = a(-5+n).

a(n) = C(floor(n/2)+2, 2). - Vladimir Shevelev, May 03 2011

a(n) = (2*n+5)*(-1)^n/16+(2*n^2+10*n+11)/16; a(n) = Sum_{k=0..n} ((k+2)*(1+(-1)^k))/4. - Paul Barry, May 31 2003

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). - Paul Barry, Apr 16 2005

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

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

CROSSREFS

Cf. A000217, A002260, A006918 (partial sums), A054252.

Sequence in context: A049318 A079551 A182843 * A188270 A026925 A237665

Adjacent sequences:  A008802 A008803 A008804 * A008806 A008807 A008808

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified February 19 16:28 EST 2018. Contains 299356 sequences. (Running on oeis4.)