A085121 Number of ways of writing n as the sum of three odd squares.

0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 0, 0, 0, 0, 0, 72, 0, 0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 0, 56, 0, 0, 0, 0, 0, 0, 0, 72, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 0, 0, 0, 0, 0, 72

Offset

0,4

Comments

Number of ways of writing n as the sum of the squares of three odd numbers (see example). Equals 8*A008437 because each summand can be the square of either a positive or negative odd number, and there are three summands, thus 2^3 = 8. - Antti Karttunen & Michel Marcus, Jul 23 2018

Links

Antti Karttunen, Table of n, a(n) for n = 0..65537

J. E. Jones [Lennard-Jones] and A. E. Ingham, On the calculation of certain crystal potential constants and on the cubic crystal of least energy, Proc. Royal Soc., A 107 (1925), 636-653 (see p. 650).

Formula

G.f.: (Sum_{n=-oo..oo} q^((2n+1)^2))^3.

Example

a(3) = 8 because 3 = (+1)^2 + (+1)^2 + (+1)^2 = (-1)^2 + (+1)^2 + (+1)^2 = (+1)^2 + (-1)^2 + (+1)^2 = (+1)^2 + (+1)^2 + (-1)^2 = (-1)^2 + (-1)^2 + (+1)^2 = (-1)^2 + (+1)^2 + (-1)^2 = (+1)^2 + (-1)^2 + (-1)^2 = (-1)^2 + (-1)^2 + (-1)^2. - Antti Karttunen, Jul 23 2018

Prog

(PARI)
A008442(n) = if( n<1 || n%4!=1, 0, sumdiv(n, d, (d%4==1) - (d%4==3))); \\ From A008442.
A290081(n) = if(n%2, 0, A008442(n/2));
A008437(n) = if((n<3)||!(n%2), 0, my(s=0, k = sqrtint(n)); k -= ((1+k)%2); while(k>=1, s += A290081(n-(k*k)); k -= 2); (s));
A085121(n) = 8*A008437(n); \\ Antti Karttunen, Jul 22 2018

Crossrefs

Cf. A005875, A008437. The nonzero coefficients give A005878.

Sequence in context: A353294 A127886 A270033 * A228634 A229659 A306756

Adjacent sequences: A085118 A085119 A085120 * A085122 A085123 A085124

Keyword

nonn

Author

N. J. A. Sloane, Apr 25 2004

Status

approved