A159915 a(n) = floor((n+1)/4)*floor(n/2).

0, 0, 0, 1, 2, 2, 3, 6, 8, 8, 10, 15, 18, 18, 21, 28, 32, 32, 36, 45, 50, 50, 55, 66, 72, 72, 78, 91, 98, 98, 105, 120, 128, 128, 136, 153, 162, 162, 171, 190, 200, 200, 210, 231, 242, 242, 253, 276, 288, 288, 300, 325, 338, 338, 351, 378, 392, 392, 406, 435, 450, 450

Offset: 0

Author

M. F. Hasler, May 01 2009, May 03 2009

Keywords

• easy
• nonn

Comments

Half the number of (n-2)-element subsets of {1,...,n} with odd sum of the elements.

This is half the antepenultimate column of A159916, cf. formula.

The number of subsets of {1,...,n} with n-2 elements, adding up to an odd integer, is always even (cf. examples), so we divide it by 2.

We prefer to include a(0)=a(1)=a(2)=0, even if it might seem more natural to start only at n=2 or n=3.

From the rational g.f. it can be seen that the sequence is a linear recurrence with constant coefficients (3,-5,7,-7,5,-3,1) of order 7.

A quasipolynomial of order 4 and degree 2. - Charles R Greathouse IV, Sep 18 2024

Examples

			a(0)=a(1)=0 since there are no subsets with -2 or -1 elements.
a(2)=0 since the sum of the elements of a 0-element subset is zero.
a(3)=1 since for n=3 we have two singleton subsets of {1,2,3}, {1} and {3}, with odd sum of elements.
a(4)=2 since for n=4 we have four 2-element subsets of {1,2,3,4} with odd sum: {1,2}, {2,3}, {1,4}, {3,4}.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (3,-5,7,-7,5,-3,1).

Crossrefs

Cf. A093005, A093353, A159916.

Programs

Magma

A159915:= func< n | Floor((n+1)/4)*Floor(n/2) >;
[A159915(n): n in [0..70]]; // G. C. Greubel, Sep 18 2024

Mathematica

Table[Floor[(n+1)/4]*Floor[n/2], {n,0,70}] (* G. C. Greubel, Sep 18 2024 *)

PARI

A159915(n)= polcoeff( (1-x+x^2)/((1-x)^3*(1+x^2)^2) + O(x^(n-2)), n-3);
a(n,t=[0,0,0,1,2,2,3],c=[1,-3,5,-7,7,-5,3]~)=while(n-->5,t=concat(vecextract(t,"^1"),t*c));t[n+2] /* Note: a(n+1,[0,0,0,0,1,2,2]) gives the same result as a(n) */

PARI

A159915(n)=(n+1)\4*(n\2) \\ M. F. Hasler, May 03 2009

SageMath

def A159915(n): return ((n+1)//4)*(n//2)
[A159915(n) for n in range(71)] # G. C. Greubel, Sep 18 2024

Formula

G.f.: x^3*(1 - x + x^2)/(1 - 3*x + 5*x^2 - 7*x^3 + 7*x^4 - 5*x^5 + 3*x^6 - x^7) = x^3*(1-x+x^2)/((1-x)^3*(1+x^2)^2).

a(n) = 3*a(n-1) - 5*a(n-2) + 7*a(n-3) - 7*a(n-4) + 5*a(n-5) - 3*a(n-6) + a(n-7) for n > 7.

For n > 2, a(n) = A159916(n*(n-1)/2 + n - 2)/2 = T(n,n-2)/2 as defined there.

From M. F. Hasler, May 03 2009: (Start)

a(n) = floor((n+1)/4)*floor(n/2).

a(2n+1) = A093005(n).

a(2n) = A093353(n-1) = floor(n/2)*n. (End)

a(n) ~ n^2/8. - Charles R Greathouse IV, Sep 18 2024