A110036 - OEIS

%I #9 Oct 21 2021 21:23:23

%S 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,

%T 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,

%U -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

%N 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.

%C Suggested by Ralf Stephan.

%C 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.

%H Paul Barry, <a href="https://arxiv.org/abs/2107.00442">Conjectures and results on some generalized Rueppel sequences</a>, arXiv:2107.00442 [math.CO], 2021.

%F 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)).

%e 1 + 1/x + 1/x^2 + 1/x^4 + 1/x^8 + 1/x^16 + ... =

%e [1; x - 1, x + 2, x, x, x - 2, x, x + 2, x, x - 2, ...].

%o (PARI) contfrac(1+sum(n=0,10,1/x^(2^n)))

%o (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)

%o (PARI) a(n)=subst(contfrac(1+sum(k=0,#binary(n+1),1/x^(2^k)))[n+1],x,0)

%Y Cf. A090678, A088567.

%K cofr,sign

%O 0,3

%A _Paul D. Hanna_, Jul 08 2005