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A159915 a(n) = floor((n+1)/4)*floor(n/2). 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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.

LINKS

Table of n, a(n) for n=0..61.

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).

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.

a(n) = floor((n+1)/4)*floor(n/2); a(2n+1) = A093005(n); a(2n) = A093353(n-1) = floor(n/2)*n. - M. F. Hasler, May 03 2009

EXAMPLE

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}.

PROG

(PARI) A159915(n)=polcoeff((1-x+x^2)/(1-3*x+5*x^2-7*x^3+7*x^4-5*x^5+3*x^6-x^7)+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

CROSSREFS

Sequence in context: A032062 A303028 A011141 * A007801 A275493 A193595

Adjacent sequences:  A159912 A159913 A159914 * A159916 A159917 A159918

KEYWORD

easy,nonn

AUTHOR

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

STATUS

approved

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Last modified April 8 08:04 EDT 2020. Contains 333313 sequences. (Running on oeis4.)