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A110036
Constant terms of the partial quotients of the continued fraction expansion of 1 + Sum_{n>=0} 1/x^(2^n), where each partial quotient has the form {x + a(n)} after the initial constant term of 1.
2
1, -1, 2, 0, 0, -2, 0, 2, 0, -2, 2, 0, -2, 0, 0, 2, 0, -2, 2, 0, 0, -2, 0, 2, -2, 0, 2, 0, -2, 0, 0, 2, 0, -2, 2, 0, 0, -2, 0, 2, 0, -2, 2, 0, -2, 0, 0, 2, -2, 0, 2, 0, 0, -2, 0, 2, -2, 0, 2, 0, -2, 0, 0, 2, 0, -2, 2, 0, 0, -2, 0, 2, 0, -2, 2, 0, -2, 0, 0, 2, 0, -2, 2, 0, 0, -2, 0, 2, -2, 0, 2, 0, -2, 0, 0, 2, -2, 0, 2, 0, 0, -2, 0, 2, 0, -2, 2, 0, -2, 0, 0, 2, -2, 0
OFFSET
0,3
COMMENTS
Suggested by Ralf Stephan.
For n>1, |a(n)| = 2*A090678(n) where A090678(n) = A088567(n) mod 2 and A088567(n) = number of "non-squashing" partitions of n into distinct parts.
FORMULA
G.f. (1-x+3*x^2+x^3)/(1+x^2) - 2*Sum_{k>=1} x^(3*2^(k-1))/Product_{j=0..k} (1+x^(2^j)).
EXAMPLE
1 + 1/x + 1/x^2 + 1/x^4 + 1/x^8 + 1/x^16 + ... =
[1; x - 1, x + 2, x, x, x - 2, x, x + 2, x, x - 2, ...].
PROG
(PARI) contfrac(1+sum(n=0, 10, 1/x^(2^n)))
(PARI) a(n)=polcoeff((1-x+3*x^2+x^3)/(1+x^2)- 2*sum(k=1, #binary(n), x^(3*2^(k-1))/prod(j=0, k, 1+x^(2^j)+x*O(x^n))), n)
(PARI) a(n)=subst(contfrac(1+sum(k=0, #binary(n+1), 1/x^(2^k)))[n+1], x, 0)
CROSSREFS
Sequence in context: A029305 A339441 A226914 * A308046 A289323 A086937
KEYWORD
cofr,sign
AUTHOR
Paul D. Hanna, Jul 08 2005
STATUS
approved