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A112603
Number of representations of n as the sum of a square and a triangular number.
12
1, 3, 2, 1, 4, 2, 1, 4, 0, 2, 5, 2, 2, 0, 2, 3, 4, 2, 0, 6, 0, 1, 4, 0, 2, 4, 4, 0, 3, 2, 2, 4, 2, 0, 0, 2, 3, 8, 0, 2, 4, 0, 2, 0, 2, 3, 6, 0, 0, 4, 2, 2, 4, 2, 2, 3, 2, 2, 0, 4, 0, 4, 0, 0, 8, 2, 1, 4, 0, 0, 8, 2, 2, 0, 2, 2, 0, 2, 1, 4, 2, 4, 6, 0, 2, 4, 0, 4, 0, 0, 0, 7, 4, 0, 4, 2, 2, 0, 0, 0, 6, 2, 4, 4, 2
OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
LINKS
M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
a(n) = A002325(8n+1). [Hirschhorn]
Expansion of q^(-1/8) * eta(q^2)^7 / (eta(q)^3 * eta(q^4)^2) in powers of q. - Michael Somos, Sep 29 2006
Expansion of phi(q) * psi(q) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Sep 29 2006
Euler transform of period 4 sequence [ 3, -4, 3, -2, ...]. - Michael Somos, Sep 29 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 2^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A139093. - Michael Somos, Mar 16 2011
G.f.: (Sum_{k} x^(k^2)) * (Sum_{k>0} x^((k^2 - k)/2)). - Michael Somos, Sep 29 2006
EXAMPLE
a(4) = 4 since we can write 4 = 2^2 + 0 = (-2)^2 + 0 = 1^2 + 3 = (-1)^2 + 3.
1 + 3*x + 2*x^2 + x^3 + 4*x^4 + 2*x^5 + x^6 + 4*x^7 + 2*x^9 + 5*x^10 + ...
q + 3*q^9 + 2*q^17 + q^25 + 4*q^33 + 2*q^41 + q^49 + 4*q^57 + 2*q^73 + ...
MATHEMATICA
a[n_] := DivisorSum[8n + 1, KroneckerSymbol[-2, #]&]; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Dec 06 2015, adapted from PARI *)
PROG
(PARI) {a(n) = if( n<0, 0, n = 8*n + 1; sumdiv( n, d, kronecker( -2, d)))} /* Michael Somos, Sep 29 2006 */
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 /(eta(x + A)^3 * eta(x^4 + A)^2), n))} /* Michael Somos, Sep 29 2006 */
CROSSREFS
Cf. A139093.
Sequence in context: A194520 A082727 A264597 * A097294 A060848 A265271
KEYWORD
nonn
AUTHOR
James A. Sellers, Dec 21 2005
STATUS
approved