

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(n4) = A097364(n,2) for n>3.  Reinhard Zumkeller, Aug 09 2004
For n>=i,i=4,5, a(ni) is the number of incongruent twocolor 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 ngon, 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 tictactoe.  I. J. Kennedy, Sep 04 2013
a(n) is the number of symmetryallowed, linearlyindependent terms at nth order in the series expansion of the T2xt2 vibronic perturbation matrix, H(Q) (cf. Opalka & Domcke).  Bradley Klee, Jul 20 2015
a(n1) 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 5gons in a regular ngon for n = 6..13
D. Opalka and W. Domcke, Highorder expansion of T2xt2 JahnTeller potential energy surfaces in tetrahedral molecules, J. Chem. Phys., 132, 154108 (2010).
V. Shevelev, A problem of enumeration of twocolor bracelets with several variations, arXiv:0710.1370 [math.CO], 20072011.
Index entries for twoway 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/((1x)*(1x^2)^2) = 1/((1+x)^2*(1x)^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+k1))/2.
a(n) = Sum_{k=0..floor(n/2)} floor((n2k+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(n2,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(n1) +2*a(n2) 2*a(n3) a(n4) +a(n5).
a(n) = (2*n +3 +(1)^n)*(2*n +7 +(1)^n)/32. (End)
a(n1) = A054252(n,1) = A054252(n^21), 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/(1x^2)^2/(1x), {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

N. J. A. Sloane


STATUS

approved



