A128975 a(n) = the number of unordered triples of integers (a,b,c) with a+b+c=n, whose bitwise XOR is zero. Equivalently, the number of three-heap nim games with n stones which are in a losing position for the first player.

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 4, 0, 0, 0, 1, 0, 1, 0, 4, 0, 1, 0, 4, 0, 4, 0, 13, 0, 0, 0, 1, 0, 1, 0, 4, 0, 1, 0, 4, 0, 4, 0, 13, 0, 1, 0, 4, 0, 4, 0, 13, 0, 4, 0, 13, 0, 13, 0, 40, 0, 0, 0, 1, 0, 1, 0, 4, 0, 1, 0, 4, 0, 4, 0, 13, 0, 1, 0, 4, 0, 4, 0, 13, 0, 4, 0, 13, 0, 13, 0, 40, 0, 1, 0, 4, 0, 4, 0

Offset: 1

Author

Jacob A. Siehler, Apr 29 2007

Keywords

• easy
• nonn

Comments

The following sequences all appear to have the same parity: A003071, A029886, A061297, A092524, A093431, A102393, A104258, A122248, A128975. - Jeremy Gardiner, Dec 28 2008

Examples

			For example, a(14)=4; the four 3-tuples are (1,6,7), (2,5,7), (3,4,7) and (3,5,6).
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Tanya Khovanova, There are no coincidences, arXiv preprint 1410.2193 [math.CO], 2014.

Crossrefs

Cf. A000120, A003987.

Programs

PARI

A128975(n) = if(n%2,0,(1/2)*((3^(hammingweight(n)-1))-1)); \\ Antti Karttunen, Sep 25 2018

Formula

a(n)=0 if n is odd; otherwise, a(n) = ( 3^(r-1) - 1)/2, where r is the number of 1's in the binary expansion of n.