A000925 Number of ordered ways of writing n as a sum of 2 squares of nonnegative integers.

1, 2, 1, 0, 2, 2, 0, 0, 1, 2, 2, 0, 0, 2, 0, 0, 2, 2, 1, 0, 2, 0, 0, 0, 0, 4, 2, 0, 0, 2, 0, 0, 1, 0, 2, 0, 2, 2, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 0, 2, 3, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 2, 4, 0, 0, 2, 0, 0, 0, 1, 2, 2, 0, 0, 0, 0, 0, 2, 2, 2, 0, 0, 4, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, 1, 0, 4

Offset: 0

Author

Jacques Haubrich (jhaubrich(AT)freeler.nl)

Keywords

• nonn
• nice

References

• A. Das and A. C. Melissinos, Quantum Mechanics: A Modern Introduction, Gordon and Breach, 1986, p. 47.
• E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985.

Links

• T. D. Noe, Table of n, a(n) for n = 0..10000
• Index entries for sequences related to sums of squares

Crossrefs

Cf. A000290, A010052, A000161, A247367.

Programs

Haskell

a000925 n = sum $ map (a010052 . (n -)) $ takeWhile (<= n) a000290_list
-- Reinhard Zumkeller, Sep 14 2014

Mathematica

a[n_] := (pr = PowersRepresentations[n, 2, 2]; Count[Union[Join[pr, Reverse /@ pr]], {j_ /; j >= 0, k_ /; k >= 0}]); a /@ Range[0, 100] (* Jean-François Alcover, Apr 05 2011 *)
nn = 100; t = CoefficientList[Series[Sum[x^k^2, {k, 0, Sqrt[nn]}]^2, {x, 0, nn}], x] (* T. D. Noe, Apr 05 2011 *)
SquareQ[n_] := IntegerQ[Sqrt[n]]; Table[Count[FrobeniusSolve[{1, 1}, n], {?SquareQ}], {n, 0, 100}] (* Robert G. Wilson v, Apr 15 2017 *)

PARI

a(n)=sum(i=0,n,sum(j=0,n,if(i^2+j^2-n,0,1)))

Formula

Coefficient of q^k in (1/4)*(1 + theta_3(0, q))^2.

a(A001481(n))>0; a(A022544(n))=0. - Benoit Cloitre, Apr 20 2003