cp's OEIS Frontend

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.

%I A110036 #9 Oct 21 2021 21:23:23
%S A110036 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 A110036 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 A110036 -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 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.
%C A110036 Suggested by Ralf Stephan.
%C A110036 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 A110036 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 A110036 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 A110036 1 + 1/x + 1/x^2 + 1/x^4 + 1/x^8 + 1/x^16 + ... =
%e A110036 [1; x - 1, x + 2, x, x, x - 2, x, x + 2, x, x - 2, ...].
%o A110036 (PARI) contfrac(1+sum(n=0,10,1/x^(2^n)))
%o A110036 (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 A110036 (PARI) a(n)=subst(contfrac(1+sum(k=0,#binary(n+1),1/x^(2^k)))[n+1],x,0)
%Y A110036 Cf. A090678, A088567.
%K A110036 cofr,sign
%O A110036 0,3
%A A110036 _Paul D. Hanna_, Jul 08 2005