A008437 -id:A008437 - cp's OEIS frontend

A290081 a(n) = number of ways of writing n as the sum of two odd positive squares.

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

Offset: 0

Antti Karttunen, Jul 24 2017

nonn

			a(2) = 1 as 2 = 1 + 1.
a(10) = 2 as 10 = 1 + 9 = 9 + 1.
a(50) = 3 as 50 = 1 + 49 = 49 + 1 = 25 + 25.

Antti Karttunen, Table of n, a(n) for n = 0..11050

Cf. A008437, A063725.

Bisections: A000004, A008442.

upto(n) = {my(m, v, res = vector(n)); m = (sqrtint(n)+1)\2; v = vector(m, i, (2*i-1)^2); forvec(x = vector(2, i, [1, #v]), s = v[x[1]] + v[x[2]]; if(s <= n, res[s]+=(1+(x[1]!=x[2]))), 1);concat(0, res)}  \\ David A. Corneth, Jul 24 2017

A008442(n) = if( n<1 || n%4!=1, 0, sumdiv(n, d, (d%4==1) - (d%4==3))); \\ This function from Michael Somos, Apr 24 2004
A290081(n) = if(n%2,0,A008442(n/2));

from sympy import divisors
def A290081(n): return 0 if n&1 else 0 if (m:=n>>1)&3!=1 else sum(((a:=d&3)==1)-(a==3) for d in divisors(m,generator=True)) # Chai Wah Wu, May 17 2023

(define (A290081 n) (cond ((< n 2) 0) ((odd? n) 0) (else (let loop ((k (- (A000196 n) (modulo (+ 1 (A000196 n)) 2))) (s 0)) (if (< k 1) s (loop (- k 2) (+ s (A010052 (- n (* k k))))))))))

a(2n) = A008442(n), a(2n+1) = 0.

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

Original entry on oeis.org

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

N. J. A. Sloane, Apr 25 2004

nonn

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

			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

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).

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

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

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

A349610 Number of solutions to x^2 + y^2 + z^2 <= n^2, where x, y, z are positive odd integers.

Original entry on oeis.org

0, 0, 1, 1, 4, 7, 17, 20, 35, 45, 69, 84, 114, 136, 184, 217, 272, 314, 389, 443, 528, 597, 702, 784, 901, 1018, 1166, 1268, 1442, 1589, 1791, 1926, 2157, 2332, 2584, 2800, 3058, 3293, 3596, 3872, 4194, 4485, 4878, 5184, 5590, 5950, 6388, 6761, 7232

Offset: 0

Ilya Gutkovskiy, Nov 23 2021

nonn

			a(4) = 4 since there are solutions (1,1,1), (3,1,1), (1,3,1), (1,1,3).

David A. Corneth, Table of n, a(n) for n = 0..10000
Index entries for sequences related to sums of squares

Cf. A000604, A000605, A008437, A053596, A085121, A253663, A349609, A349611.

Table[SeriesCoefficient[EllipticTheta[2, 0, x^4]^3/(8 (1 - x)), {x, 0, n^2}], {n, 0, 48}]

a(n) = [x^(n^2)] theta_2(x^4)^3 / (8 * (1 - x)).
a(n) = Sum_{k=0..n^2} A008437(k).
a(n) = A053596(n) / 8.

A372512 Number of solutions to x^2 + y^2 + z^2 <= n, where x, y, z are positive odd integers.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 4, 4, 4, 4, 4, 4, 7, 7, 7, 7, 7, 7, 7, 7, 11, 11, 11, 11, 11, 11, 11, 11, 17, 17, 17, 17, 17, 17, 17, 17, 20, 20, 20, 20, 20, 20, 20, 20, 26, 26, 26, 26, 26, 26, 26, 26, 35, 35, 35, 35, 35, 35, 35, 35, 38, 38, 38, 38, 38, 38, 38, 38, 45, 45, 45, 45, 45, 45

Offset: 0

Ilya Gutkovskiy, May 04 2024

nonn

Index entries for sequences related to sums of squares

Cf. A000606, A008437, A085121, A117609, A211639, A349610, A372511.

nmax = 80; CoefficientList[Series[EllipticTheta[2, 0, x^4]^3/(8 (1 - x)), {x, 0, nmax}], x]

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A290081 a(n) = number of ways of writing n as the sum of two odd positive squares.

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

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A349610 Number of solutions to x^2 + y^2 + z^2 <= n^2, where x, y, z are positive odd integers.

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A372512 Number of solutions to x^2 + y^2 + z^2 <= n, where x, y, z are positive odd integers.

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Mathematica