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A085121 Number of ways of writing n as the sum of three odd squares. 3
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 (list; graph; refs; listen; history; text; internal format)
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=-inf..inf} 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: A180225 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

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Last modified August 20 10:04 EDT 2019. Contains 326145 sequences. (Running on oeis4.)