OFFSET
1,3
COMMENTS
It seems that after a(1) = 1, -1's occur only at the positions 2^k (with k >= 1) and -2's only at positions 3*2^k (with k >= 0, A007283), with everything else being 1. It would be nice to know whether this is true. - Antti Karttunen, May 27 2017
From Wolfdieter Lang, May 29 2017: (Start)
The preceding conjecture can be checked by using for even n Theorem 1A, eq. (41), and for odd n Theorem 2A, eq. (50) of the W. Lang arXiv link given in A187360 putting x = -1.
One uses for the polynomials that (A127672) and q (A130777) appearing there the result that(n, -1) = A099837(n+3), i.e., = 2 if n == 0 (mod 3), = -1 if n == 1 or 2 (mod 3), and q(n, -1) = A061347(n+2), i.e., = 1 if n == 0 or 2 (mod 3) and = -2 if n == 1 (mod 3).
E.g., n = 2^k, k >= 1: C(2^k, -1) = that(2^(k-1), -1) = -1 because 2^(k-1) == 1 or 2 (mod 3).
n = 3*2^k, k >= 1: C(2^k*3) = that(2^(k-1)*3, -1) / that(2^(k-1), -1) = 2/(-1) = -2 because 2^(k-1)*3 == 0 (mod 3), and the previous congruence. C(3, -1) = -2 also, by theorem 2A, see the next example.
n = 3^k, k >= 1: C(3^k, -1) = q((3^k-1)/2, -1) / q((3^(k-1)-1)/2, -1) = (-2)/1 = -2 if k = 1, and = (-2)/(-2) = +1 if k >= 2. (End)
LINKS
R. J. Mathar, Table of n, a(n) for n = 1..220
FORMULA
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Wolfdieter Lang, Jul 14 2011
STATUS
approved