A085121 -id:A085121 - cp's OEIS frontend

A005878 Theta series of cubic lattice with respect to deep hole.

8, 24, 24, 32, 48, 24, 48, 72, 24, 56, 72, 48, 72, 72, 48, 48, 120, 72, 56, 96, 24, 120, 120, 48, 96, 96, 72, 96, 120, 48, 104, 168, 96, 48, 120, 72, 96, 192, 72, 144, 96, 72, 144, 120, 96, 104, 192, 72, 120, 192, 48, 144, 216, 48, 96, 120, 144, 192, 168, 120, 96, 216, 72

Offset: 0

N. J. A. Sloane

Number of ways of writing 8*n+3 as the sum of three odd squares. - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 23 2008

Expansion of Jacobi theta constant theta_2^3. - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 23 2008

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 107.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..99 from Herman Jamke (hermanjamke(AT)fastmail.fm))
G. Nebe and N. J. A. Sloane, Home page for this lattice

Equals 8 times A008443. Cf. A085121.

QP = QPochhammer; CoefficientList[(2 QP[q^2]^2/QP[q])^3 + O[q]^63, q] (* Jean-François Alcover, Jul 04 2017 *)

{a(n)=if(n<0, 0, 8*polcoeff( sum(k=0, (sqrtint(8*n+1)-1)\2, x^((k^2+k)/2), x*O(x^n))^3, n))} {a(n)=local(A); if(n<0, 0, A=x*O(x^n); 8*polcoeff( (eta(x^2+A)^2/eta(x+A))^3, n))} \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 23 2008

G.f.: Form (Sum_{n=-inf..inf} q^((2n+1)^2))^3, then divide by q^3 and set q = x^(1/8).

a(n) = 8 * A008443(n).

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 23 2008

A008437 Expansion of Jacobi theta constant theta_2^3 /8.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

nonn

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

			From _Antti Karttunen_, Jul 24 2017: (Start)
a(19) = 3 as 19 = 1+9+9 = 9+1+9 = 9+9+1.
a(27) = 4 as 27 = 1+1+25 = 1+25+1 = 25+1+1 = 9+9+9.
(End)

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.

Antti Karttunen, Table of n, a(n) for n = 0..10000
J. E. Jones [Lennard-Jones] and A. E. Ingham, On the calculation of certain crystal potential constants and on the cubic crystal of least potential energy, Proc. Royal Soc., A 107 (1925), 636-653 (see p. 650).

Equals A085121/8.

Cf. A000004 (the even bisection), A000196, A290081.

(define (A008437 n) (cond ((< n 3) 0) ((even? n) 0) (else (let loop ((k (- (A000196 n) (modulo (+ 1 (A000196 n)) 2))) (s 0)) (if (< k 1) s (loop (- k 2) (+ s (A290081 (- n (* k k)))))))))) ;; Antti Karttunen, Jul 24 2017

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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A005878 Theta series of cubic lattice with respect to deep hole.

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A008437 Expansion of Jacobi theta constant theta_2^3 /8.

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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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Author

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Formula

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