A214505 a(n) = 1 if n is four times a triangular number, -1 if one more than twelve times a triangular number else 0.

1, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Author

Michael Somos, Jul 19 2012

Keywords

• sign
• changed

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 - x + x^4 + x^12 - x^13 + x^24 - x^37 + x^40 + x^60 - x^73 + x^84 + ...
G.f. = q - q^3 + q^9 + q^25 - q^27 + q^49 - q^75 + q^81 + q^121 - q^147 + q^169 + ...
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..65537
• S. Cooper and M. 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
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A010052, A214295.

Programs

PARI

{a(n) = n = 2*n + 1; issquare(n) - issquare(3*n)};

Formula

Expansion of psi(x^4) - x * psi(x^12) in powers of x where psi() is a Ramanujan theta function.

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.

Euler transform of period 24 sequence [ -1, 0, 0, 1, 1, 1, 1, 0, 0, -1, -1, -1, -1, -1, 0, 0, 1, 1, 1, 1, 0, 0, -1, -1, ...].

G.f.: (Sum_{k} x^(2*k*(k + 1)) - x^(6*k*(k + 1) + 1)) / 2.

a(n) = A214295(2*n + 1).