A096960
a(n) = Sum_{0
1, 32, 244, 1024, 3126, 7808, 16808, 32768, 59293, 100032, 161052, 249856, 371294, 537856, 762744, 1048576, 1419858, 1897376, 2476100, 3201024, 4101152, 5153664, 6436344, 7995392, 9768751, 11881408, 14408200, 17211392, 20511150
Offset: 1
Examples
G.f. = q + 32*q^2 + 244*q^3 + 1024*q^4 + 3126*q^5 + 7808*q^6 + 16808*q^7 + 32768*q^8 + ...
Links
- Seiichi Manyama, Table of n, a(n) for n = 1..10000
- H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares, arXiv:math/0407061 [math.NT], 2004.
- J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
- Index entries for sequences mentioned by Glaisher.
Programs
-
Magma
Basis( ModularForms( Gamma0(2), 6), 30) [2]; /* Michael Somos, Nov 30 2014 */
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Mathematica
a[ n_] := If[ n < 1, 0, Sum[ d^5 Boole[ OddQ[ n/d]], {d, Divisors[ n]}]]; (* Michael Somos, Jun 04 2013 *) a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^4 + EllipticTheta[ 2, 0, q]^4) EllipticTheta[ 2, 0, q^(1/2)]^8 / 256, {q, 0, n}]; (* Michael Somos, Jun 04 2013 *) -
PARI
{a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * d^5))}; -
Sage
ModularForms( Gamma0(2), 6, prec=33).gen(1).coefficients(30) # Michael Somos, Jun 04 2013
Formula
G.f.: Sum {k>0} k^5 * x^k / (1 - x^(2*k)).
From Amiram Eldar, Nov 01 2022: (Start)
Multiplicative with a(2^e) = 2^(5*e) and a(p^e) = (p^(5*e+5)-1)/(p^5-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^6, where c = 21*zeta(6)/128 = 0.166907... . (End)
Dirichlet g.f.: zeta(s)*zeta(s-5)*(1-1/2^s). - Amiram Eldar, Jan 08 2023