OFFSET
1,2
COMMENTS
Series of the kind m^((k^2 + 3*k)/2) from k=1 to n were studied by Bernoulli and Euler.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..80
FORMULA
a(n) = 2^(n+1)*a(n-1) + 1, a(1) = 1. - Alexandre Herrera, Mar 23 2024
a(n) ~ c * A036442(n+1) = c * 2^(n*(n+3)/2), where c = 2^(1/8) * EllipticTheta[2, 0, 1/Sqrt[2]] - 3 [in Mathematica] = 2^(1/8) * JacobiTheta2(0, 1/sqrt(2)) - 3 [in Maple] = 0.2832651213103077325876855404508588684521230759134794956... - Vaclav Kotesovec, Apr 10 2024
MATHEMATICA
aa = {}; m = 1/2; sum = 0; Do[sum = sum + m^((n + 3) n/2); AppendTo[aa, Numerator[sum]], {n, 1, 20}]; aa
Numerator[Table[Sum[1/2^((k^2 + 3*k)/2), {k, 1, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Apr 10 2024 *)
PROG
(PARI) a(n) = numerator(sum(k=1, n, (1/2)^((k^2+3*k)/2))); \\ Michel Marcus, Sep 09 2013
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Artur Jasinski, May 21 2010
STATUS
approved