A175372 Number of integer pairs (x,y) satisfying x^4 + y^4 = n.

1, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Author

R. J. Mathar, Apr 24 2010

Keywords

• nonn

Comments

A 4th-power variant of A004018 and A175362.

a(n) is nonzero when n appears in A004831. a(n) > 8 when n appears in A003824. - Mason Korb, Oct 06 2018

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• M. Korb, What's the connection between theta series and the number of integer solutions on a curve? , Math StackExchange, July 2018.

Crossrefs

Cf. A004018, A175362.

Cf. A003824, A004831 (where a(n) is nonzero).

Programs

Magma

m:=120; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+2*(&+[x^(j^4): j in [1..50]]))^2)); // G. C. Greubel, Oct 06 2018

Maple

seq(coeff(series((1+2*add(x^(j^4),j=1..n))^2,x,n+1), x, n), n = 0 .. 120); # Muniru A Asiru, Oct 07 2018

Mathematica

CoefficientList[Series[(1 + 2*Sum[x^(j^4), {j, 1, 100}])^2, {x, 0, 120}], x] (* G. C. Greubel, Oct 06 2018 *)

PARI

x='x+O('x^120); Vec((1+2*sum(j=1,50, x^(j^4)))^2) \\ G. C. Greubel, Oct 06 2018

Formula

G.f.: (1 + 2*Sum_{j>=1} x^(j^4))^2.