OFFSET
0,1
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
a(n) = 1 if n is four times a triangular number or one more than twelve times a triangular number else 0. - Michael Somos, Jul 19 2012
LINKS
Antti Karttunen, Table of n, a(n) for n = 0..10000
Richard Blecksmith, John Brillhart, and Irving Gerst, Some infinite product identities, Math. Comp. 51 (1988), no. 183, 301-314. MR0942157 (89f:05017)
Shaun Cooper and Michael Hirschhorn, On some infinite product identities, Rocky Mountain J. Math., 31 (2001), 131-139. see p. 134 Theorem 5.
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
FORMULA
Euler transform of period 24 sequence [ 1, -1, 0, 1, -1, 1, -1, 0, 0, 0, 1, -1, 1, 0, 0, 0, -1, 1, -1, 1, 0, -1, 1, -1, ...].
a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = 1, else b(p^e) = (1 + (-1)^e)/2.
a(3*n + 1) = a(n), a(3*n + 2) = a(4*n + 2) = a(4*n + 3) = a(6*n + 3) = 0.
G.f.: Sum_{k>0} x^(2k(k-1)) +x^(6k(k-1)+1) = Product_{k>0} (1-x^(24k)) (1-x^(24k-5)) (1-x^(24k-7)) (1-x^(24k-17)) (1-x^(24k-19)) (1+x^(12k-1)) (1+x^(12k-4)) (1+x^(12k-6)) (1+x^(12k-8)) (1+x^(12k-11)).
From Michael Somos, Jul 19 2012: (Start)
Expansion of f(x, -x^5) * f(-x^4, -x^8) / f(x, -x) in powers of x where f(,) is the Ramanujan two-variable theta function.
G.f.: (Sum_{k in Z} x^(2*k*(k + 1)) + x^(6*k*(k + 1) + 1)) / 2.
a(n) = A195198(2*n + 1). (End)
Sum_{k=1..n} a(k) ~ c * sqrt(n), where c = 1/sqrt(2) + 1/sqrt(6) = 1.115355... (A145439). - Amiram Eldar, Dec 29 2023
EXAMPLE
1 + x + x^4 + x^12 + x^13 + x^24 + x^37 + x^40 + x^60 + x^73 + x^84 + ...
q + q^3 + q^9 + q^25 + q^27 + q^49 + q^75 + q^81 + q^121 + q^147 + q^169 + ...
PROG
(PARI) {a(n) = issquare(2*n + 1) + issquare(6*n + 3)}
(PARI) {a(n) = n = 2*n + 1; issquare(n) || issquare(3*n)}
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Michael Somos, Jan 19 2007
STATUS
approved