

A008805


Triangular numbers repeated.


71



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
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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
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) is the 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 T2 X t2 vibronic perturbation matrix, H(Q) (cf. Opalka & Domcke).  Bradley Klee, Jul 20 2015
a(n1) 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
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
In an nperson symmetric matching pennies game (a zerosum normalform game) with n > 2 symmetric and indistinguishable players, each with two strategies (viz. heads or tails), a(n3) 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(n1).  Ambrosio ValenciaRomero, Apr 17 2022
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
a(n) is the number of 132avoiding odd Grassmannian permutations of size n+2.  Juan B. Gil, Mar 10 2023
Consider a regular ngon 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(n2). See illustration.  Christopher Scussel, Nov 07 2023


REFERENCES

H. D. Brunk, An Introduction to Mathematical Statistics, Ginn, Boston, 1960; p. 360.


LINKS



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) = ((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)
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) = 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(n) = a(n1) if n odd, a(n) = a(n1) + (n+2)/2 if n is even, for n > 0, a(0) = 1.
a(n) = (n+1)*(n+3)/8 if n odd, a(n) = (n+2)*(n+4)/8 if n is even, for n >= 0.
a(n) = A002620(n+2)  a(n1), for n > 0, a(0) = 1.
a(n) = A142150(n+2) + a(n1), for n > 0, a(0) = 1.


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



MATHEMATICA

CoefficientList[Series[1/(1x^2)^2/(1x), {x, 0, 50}], x]


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]
(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
(Python)


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



