A097057 Number of integer solutions to a^2 + b^2 + 2*c^2 + 2*d^2 = n.

1, 4, 8, 16, 24, 24, 32, 32, 24, 52, 48, 48, 96, 56, 64, 96, 24, 72, 104, 80, 144, 128, 96, 96, 96, 124, 112, 160, 192, 120, 192, 128, 24, 192, 144, 192, 312, 152, 160, 224, 144, 168, 256, 176, 288, 312, 192, 192, 96, 228, 248, 288, 336, 216, 320, 288, 192, 320, 240, 240

Offset

0,2

Comments

a^2 + b^2 + 2*c^2 + 2*d^2 is another (cf. A000118) of Ramanujan's 54 universal quaternary quadratic forms. - Michael Somos, Apr 01 2008

References

B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 373 Entry 31.

Jesse Ira Deutsch, Bumby's technique and a result of Liouville on a quadratic form, Integers 8 (2008), no. 2, A2, 20 pp. MR2438287 (2009g:11047).

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.29).

S. Ramanujan, Collected Papers, Chap. 20, Cambridge Univ. Press 1927 (Proceedings of the Camb. Phil. Soc., 19 (1917), 11-21).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Jesse Ira Deutsch, A quaternionic proof of the representation formula of a quaternary quadratic form, J. Number Theory 113 (2005), no. 1, 149-174. MR2141762 (2006b:11033).

Y. Mimura, Almost Universal Quadratic Forms.

Olivia X. M. Yao and Ernest X. W. Xia, Combinatorial proofs of five formulas of Liouville, Discrete Math. 318 (2014), 1-9. MR3141622.

Formula

Euler transform of period 8 sequence [4, -2, 4, -8, 4, -2, 4, -4, ...]. - Michael Somos, Sep 17 2004

Multiplicative with a(n) = 4*b(n), b(2) = 2, b(2^e) = 6 if e > 1, b(p^e) = (p^(e+1) - 1) / (p - 1) if p > 2. - Michael Somos, Sep 17 2004

Expansion of (eta(q^2) * eta(q^4))^6 / (eta(q) * eta(q^8))^4 in powers of q.

G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 8 (t/i)^2 f(t) where q = exp(2 Pi i t). - Michael Somos, Jul 05 2011

G.f.: (theta_3(q) * theta_3(q^2))^2.

G.f.: Product_{k>0} ((1-x^(2k))(1-x^(4k)))^6/((1-x^k)(1-x^(8k)))^4.

G.f.: 1 + Sum_{k>0} 8 * x^(4*k) / (1 + x^(4*k))^2 + 4 * x^(2*k-1) / (1 - x^(2*k-1))^2 = 1 + Sum_{k>0} (2 + (-1)^k) * 4*k * x^(2*k) / (1 + x^(2*k)) + 4*(2*k - 1) * x^(2*k-1) / (1 - x^(2*k - 1)). - Michael Somos, Oct 22 2005

a(2*n) = A000118(n). a(2*n + 1) = 4 * A008438(n). a(4*n) = A004011(n). a(4*n + 1) = 4 * A112610(n). a(4*n + 2) = 8 * A008438(n). a(4*n + 3) = 16 * A097723(n). - Michael Somos, Jul 05 2011

Example

1 + 4*q + 8*q^2 + 16*q^3 + 24*q^4 + 24*q^5 + 32*q^6 + 32*q^7 + 24*q^8 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2])^2, {q, 0, n}] (* Michael Somos, Jul 05 2011 *)
f[p_, e_] := (p^(e+1)-1)/(p-1); f[2, 1] = 2; f[2, e_] := 6; a[0] = 1; a[1] = 4; a[n_] := 4 * Times @@ f @@@ FactorInteger[n]; Array[a, 100, 0] (* Amiram Eldar, Aug 22 2023 *)

Prog

(PARI) {a(n) = local(t); if( n<1, n>=0, t = 2^valuation( n, 2); 4 * sigma(n/t) * if( t>2, 6, t))} \\ Michael Somos, Sep 17 2004
(PARI) {a(n) = local(A = x * O(x^n)); polcoeff( (eta(x^2 + A) * eta(x^4 + A))^6 / (eta(x + A) * eta(x^8 + A))^4, n))} \\ Michael Somos, Sep 17 2004
(PARI) {a(n) = if( n<1, n==0, 2 * qfrep([ 1, 0, 0, 0; 0, 1, 0, 0; 0, 0, 2, 0; 0, 0, 0, 2], n)[n])} \\ Michael Somos, Oct 29 2005
(PARI) A097057(n)=if(n, sigma(n>>n=valuation(n, 2))*if(n>1, 24, 4<<n), 1) \\ M. F. Hasler, May 07 2018

Crossrefs

a^2 + b^2 + 2*c^2 + m*d^2 = n: this sequence (m=2), A320124 (m=3), A320125 (m=4), A320126 (m=5), A320127 (m=6), A320128 (m=7), A320130 (m=8), A320131 (m=9), A320132 (m=10), A320133 (m=11), A320134 (m=12), A320135 (m=13), A320136 (m=14).

Cf. A000118, A004011, A008438, A097723, A112610.

Sequence in context: A358435 A312823 A133690 * A347931 A354810 A333168

Adjacent sequences: A097054 A097055 A097056 * A097058 A097059 A097060

Keyword

nonn,mult

Author

N. J. A. Sloane, Sep 15 2004

Extensions

Added keyword mult and minor edits by M. F. Hasler, May 07 2018

Status

approved