OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..200
FORMULA
For n>0, a(n) is odd iff n = A118113(k) for some k >= 1, where A118113(k) = 2*Fibbinary(k) + 1 (conjecture).
a(n) ~ 2^(3*n + 5) * n^(2*n + 3/2) / (Pi^(2*n + 3/2) * exp(2*n)). - Vaclav Kotesovec, Nov 12 2020
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 17*x^3 + 274*x^4 + 6749*x^5 + 231276*x^6 + 10465440*x^7 + 604220826*x^8 + 43388420549*x^9 + 3797054582794*x^10 + ...
where
1 = A(x) - x/(A(x) - 3*x/(A(x) - 6*x/(A(x) - 10*x/(A(x) - 15*x/(A(x) - 21*x/(A(x) - 28*x/(A(x) - 36*x/(A(x) - 45*x/(A(x) - 55*x/(A(x) - ...)))))))))), a continued fraction relation in which the triangular numbers appear as coefficients.
PROG
(PARI) {a(n) = my(A=[1], CF=1); for(i=1, n, A=concat(A, 0); for(i=1, #A, CF = Ser(A) - (#A-i+1)*(#A-i+2)/2*x/CF ); A[#A] = -polcoeff(CF, #A-1) ); A[n+1] }
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 04 2020
STATUS
approved