A124340 Number of solutions to n = x^2 + 2*y^2 + 4*(T(z) + T(w)) + 1 where x and y are integers, z and w are nonnegative integers and T(x) = (x^2+x)/2.

1, 2, 2, 4, 4, 4, 8, 8, 7, 8, 10, 8, 12, 16, 8, 16, 18, 14, 18, 16, 16, 20, 24, 16, 21, 24, 20, 32, 28, 16, 32, 32, 20, 36, 32, 28, 36, 36, 24, 32, 42, 32, 42, 40, 28, 48, 48, 32, 57, 42, 36, 48, 52, 40, 40, 64, 36, 56, 58, 32, 60, 64, 56, 64, 48, 40, 66

Offset: 1

Author

Michael Somos, Oct 26 2006

Keywords

• nonn
• mult

Comments

Number 18 of the 74 eta-quotients listed in Table I of Martin (1996).

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

Examples

			G.f. = q + 2*q^2 + 2*q^3 + 4*q^4 + 4*q^5 + 4*q^6 + 8*q^7 + 8*q^8 + 7*q^9 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• David Broadhurst and Daniele Dorigoni, Resurgent Lambert series with characters, arXiv:2507.21352 [math.NT], 2025. See p. 37.
• Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
• Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers, 2016.
• Michael Somos, Introduction to Ramanujan theta functions, 2019.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Crossrefs

Cf. A091337, A328895.

Cf. A000122, A000700, A010054, A121373.

Programs

Maple

with(numtheory):
A091337 := n -> [0, 1, 0, -1, 0, -1, 0, 1][`mod`(n, 8)+1]:
seq(add(A091337(n/d)d, d in divisors(n)), n = 1..60); # Peter Bala, Jan 06 2021

Mathematica

a[n_] := Sum[JacobiSymbol[2, d]*n/d, {d, Divisors[n]}]; a /@ Range[80] (* Jean-François Alcover, Jan 10 2014 *)
a[ n_] := SeriesCoefficient[ q QPochhammer[ q^2]^3 QPochhammer[ q^4] QPochhammer[ q^8]^2 / QPochhammer[ q]^2, {q, 0, n}]; (* Michael Somos, Jul 09 2015 *)

PARI

{a(n) = if( n<1, 0, sumdiv( n, d, n / d * kronecker(2, d)))};

PARI

{a(n) = my(A, p, e, f); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; f = kronecker( 2, p); (p^(e+1) - f^(e+1)) / (p - f)))};

PARI

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^4 + A) * eta(x^8 + A)^2 / eta(x + A)^2, n))};

Formula

Expansion of q * phi(q) * phi(q^2) * psi(q^4)^2 in powers of q where phi(), psi() are Ramanujan theta functions.

Expansion of eta(q^2)^3 * eta(q^4) * eta(q^8)^2 / eta(q)^2 in powers of q.

Euler transform of period 8 sequence [ 2, -1, 2, -2, 2, -1, 2, -4, ...].

a(n) is multiplicative with a(2^e) = 2^e, a(p^e) = (p^(e+1) - 1)/(p - 1) if p == 1, 7 (mod 8), a(p^e) = (p^(e+1) + (-1)^e)/(p + 1) if p == 3, 5 (mod 8).

G.f.: Sum_{k>0} k * x^k * (1 - x^(2*k)) / (1 + x^(4*k)).

G.f.: x * Product_{k>0} (1 + x^k)^2 * (1 - x^(2*k)) * (1 - x^(4*k)) * (1 - x^(8*k))^2.

From Peter Bala, Jan 06 2021: (Start)

a(n) = Sum_{ d | n } X(n/d)*d, where X(k) = A091337(k) is a non-principal Dirichlet charcter modulo 8.

G.f.: A(x) = Sum_{n = -oo..oo} (-1)^n*x^(4*n+1)/(1 - x^(4*n+1))^2. (End)

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A328895. - Amiram Eldar, Feb 20 2024