

A014455


Theta series of quadratic form with Gram matrix [ 1, 0, 0; 0, 1, 0; 0, 0, 2 ]. Number of integer solutions to x^2 + y^2 + 2*z^2 = n.


10



1, 4, 6, 8, 12, 8, 8, 16, 6, 12, 24, 8, 24, 24, 0, 16, 12, 16, 30, 24, 24, 16, 24, 16, 8, 28, 24, 32, 48, 8, 0, 32, 6, 32, 48, 16, 36, 40, 24, 16, 24, 16, 48, 40, 24, 40, 0, 32, 24, 36, 30, 16, 72, 24, 32, 48, 0, 32, 72, 24, 48, 40, 0, 48, 12, 16, 48, 56, 48, 32, 48, 16, 30, 64
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,2


COMMENTS

This is the tetragonal P lattice (the classical holotype) of dimension 3.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).


LINKS

John Cannon, Table of n, a(n) for n = 0..10000
G. Nebe and N. J. A. Sloane, Home page for this lattice
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions


FORMULA

Expansion of phi(q)^2 * phi(q^2) = psi(q)^4 / psi(q^4) in powers of q where phi(), psi() are Ramanujan theta functions.  Michael Somos, Apr 07 2012
Expansion of eta(q^2)^8 * eta(q^4) / (eta(q)^4 * eta(q^8)^2) in powers of q.  Michael Somos, Jul 05 2005
Euler transform of period 8 sequence [4, 4, 4, 5, 4, 4, 4, 3, ...].  Michael Somos, Jul 07 2005
G.f.: theta_3(q)^2 * theta_3(q^2) = Product_{k>0} (1  x^(2*k))^8 * (1  x^(4*k)) / ((1  x^k)^4 * (1  x^(8*k))^2).
There is a classical formula (essentially due to Gauss): Write (uniquely) 2n=D(2^vf)^2, with D<0 fundamental discriminant, f odd, v>=1. Then a(n)=12L((D/.),0)(1(D/2))\sum_{d\mid f}\mu(d)(D/d)sigma(f/d) (the formula for A005875), except that the factor (1(D/2)) has to be replaced by 1/3 if v=1 and by 1 if v=0 (and kept if v>=1). Here mu() is the Moebius function, (D/2) and (D/d) are KroneckerLegendre symbols, sigma() is the sum of divisors function, L((D/.),0)=h(D)/(w(D)/2) is the value at 0 of the L() function of the quadratic character (D/.), equal to the class number h(D) divided by 2 or 3 in the special cases D=4 and 3.  Henri Cohen (Henri.Cohen(AT)math.ubordeaux1.fr), May 12 2010
a(2*n) = a(8*n) = A005875(n). a(2*n + 1) = A005877(n) = 4 * A045828(n). a(4*n) = A004015(n). a(4*n + 2) = 2 * A045826(n). a(8*n + 4) = 12 * A045828(n). a(8*n + 7) = 16 * A033763(n). a(16*n + 6) = 8 * A008443(n). a(16*n + 14) = 0.  Michael Somos, Apr 07 2012
G.f. is a period 1 Fourier series which satisfies f(1 / (8 t)) = 32^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246631.


EXAMPLE

G.f. = 1 + 4*q + 6*q^2 + 8*q^3 + 12*q^4 + 8*q^5 + 8*q^6 + 16*q^7 + 6*q^8 + 12*q^9 + ...


MATHEMATICA

r[n_, z_] := Reduce[x^2 + y^2 + 2*z^2 == n, {x, y}, Integers]; a[n_] := Module[{rn0, rnz, k0, k}, rn0 = r[n, 0]; k0 = If[rn0 === False, 0, If[Head[rn0] === And, 1, Length[rn0]]]; For[k = 0; z = 1, z <= Ceiling[Sqrt[n/2]], z++, rnz = r[n, z]; If[rnz =!= False, k = If[Head[rnz] === And, k+1, k + Length[rnz]]]]; k0 + 2*k]; Table[a[n], {n, 0, 100}] (* JeanFrançois Alcover, Oct 07 2013 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^2 EllipticTheta[ 3, 0, q^2], {q, 0, n}]; (* Michael Somos, Aug 31 2014 *)
QP = QPochhammer; s = QP[q^2]^8*(QP[q^4]/(QP[q]^4*QP[q^8]^2)) + O[q]^80; CoefficientList[s, q] (* JeanFrançois Alcover, Nov 25 2015, after Michael Somos *)


PROG

(PARI) {a(n) = if( n<1, n==0, 2 * qfrep([ 1, 0, 0; 0, 1, 0; 0, 0, 2], n)[n])}; /* Michael Somos, Jul 05 2005 */
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^8 * eta(x^4 + A) / (eta(x + A)^4 * eta(x^8 + A)^2), n))}; /* Michael Somos, Jul 05 2005 */
(MAGMA) A := Basis( ModularForms( Gamma1(8), 3/2), 40); A[1] + 4*A[2] + 6*A[3] + 8*A[4]; /* Michael Somos, Aug 31 2014 */


CROSSREFS

Cf. A004015, A005875, A005877, A008443, A033763, A045826, A045828, A156384, A213024, A246631.
Sequence in context: A293806 A310661 A139404 * A320136 A320135 A320134
Adjacent sequences: A014452 A014453 A014454 * A014456 A014457 A014458


KEYWORD

nonn,nice,easy


AUTHOR

N. J. A. Sloane.


STATUS

approved



