|
| |
|
|
A033715
|
|
Number of integer solutions (x, y) to the equation x^2 + 2y^2 = n.
|
|
40
|
|
|
|
1, 2, 2, 4, 2, 0, 4, 0, 2, 6, 0, 4, 4, 0, 0, 0, 2, 4, 6, 4, 0, 0, 4, 0, 4, 2, 0, 8, 0, 0, 0, 0, 2, 8, 4, 0, 6, 0, 4, 0, 0, 4, 0, 4, 4, 0, 0, 0, 4, 2, 2, 8, 0, 0, 8, 0, 0, 8, 0, 4, 0, 0, 0, 0, 2, 0, 8, 4, 4, 0, 0, 0, 6, 4, 0, 4, 4, 0, 0, 0, 0, 10, 4, 4, 0, 0, 4, 0, 4, 4, 0, 0, 0, 0, 0, 0, 4, 4, 2, 12, 2, 0, 8, 0
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
Theta series of lattice C2 with Gram matrix [ 1, 0; 0, 2]. a(n) is nonzero if and only if n is in A002479. - Michael Somos, Dec 15 2011
Number 17 of the 74 eta-quotients listed in Table I of Martin (1996).
Denoted by |a_4(n)| in Kassel and Reutenauer 2015. - Michael Somos, Jun 16 2015
|
|
|
REFERENCES
|
Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 114 Entry 8(iii).
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, 1999, p. 102, eq. 9.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 3, p. 19.
Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.24).
J. W. L. Glaisher, Table of the excess of the number of (8k+1)- and (8k+3)-divisors of a number over the number of (8k+5)- and (8k+7)-divisors, Messenger Math., 31 (1901), 82-91.
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 346.
|
|
|
LINKS
|
|
|
|
FORMULA
|
Fine gives an explicit formula for a(n) in terms of the divisors of n.
Euler transform of period 8 sequence [ 2, -1, 2, -4, 2, -1, 2, -2, ...].
Expansion of (eta(q^2) * eta(q^4))^3 / (eta(q) * eta(q^8))^2 in powers of q.
Coefficients in expansion of Sum_{i,j=-inf..inf} q^(i^2 + 2*j^2).
G.f. = s(2)^3*s(4)^3/(s(1)^2*s(8)^2), where s(k) := subs(q=q^k, eta(q)), where eta(q) is Dedekind's function, cf. A010815. [Fine]
G.f.: 1 + 2 * Sum_{k>0} Kronecker(-2, n) * x^k / (1 - x^k) = 1 + 2 * Sum_{k>0} (x^k + x^(3*k)) / (1 + x^(4*k)).
G.f.: theta_3(q) * theta_3(q^2) = Product_{k>0} (1 + x^(2*k)) * ((1 + x^k) * (1 - x^(2*k)) / (1 + x^(4*k)))^2.
Moebius transform is period 8 sequence [ 2, 0, 2, 0, -2, 0, -2, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = (u1 - 3*u3) * (u1 - u2 - u3 + u6) - (u2 - 3*u6) * (u1 - 2*u2 - u3 + 2*u6). (End)
a(n) = 2 * A002325(n) unless n = 0.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 8^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 09 2012
Expansion of phi(q) * phi(q^2) in powers of q where phi() is a Ramanujan theta function.
a(2*n) = a(n). a(2*n + 1) = 2 * A113411(n). (End)
a(8*n + 5) = a(8*n + 7) = 0. a(8*n + 1) = 2 * A112603(n). a(8*n + 3) = 4 * A033761(n). (End)
a(0) = 1, a(n) = 2 * b(n) for n > 0, where b() is multiplicative with a(2^e) = 1, a(p^e) = e + 1 if p == 1, 3 (mod 8), a(p^e) = (1 + (-1)^e)/2 if p == 5, 7 (mod 8). - Jianing Song, Sep 04 2018
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = Pi/sqrt(2) = 2.221441... (A247719). - Amiram Eldar, Dec 16 2023
|
|
|
EXAMPLE
|
G.f. = 1 + 2*q + 2*q^2 + 4*q^3 + 2*q^4 + 4*q^6 + 2*q^8 + 6*q^9 + 4*q^11 + 4*q^12 + ...
|
|
|
MAPLE
|
d:=proc(r, m, n) local i, t1; t1:=0; for i from 1 to n do if n mod i = 0 and i-r mod m = 0 then t1:=t1+1; fi; od: t1; end; [seq(2*(d(1, 8, n)+d(3, 8, n)-d(5, 8, n)-d(7, 8, n)), n=1..120)];
|
|
|
MATHEMATICA
|
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2], {q, 0, n}]; (* Michael Somos, Sep 09 2012 *)
a[ n_] := If[ n < 1, Boole[ n == 0], 2 DivisorSum[ n, KroneckerSymbol[ -2, #] &]]; (* Michael Somos, Aug 29 2014 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2] QPochhammer[ q^4])^3 / (QPochhammer[ q] QPochhammer[ q^8])^2, {q, 0, n}]; (* Michael Somos, Aug 29 2014 *)
|
|
|
PROG
|
(PARI) {a(n) = if( n<1, n==0, 2 * (issquare(n) - issquare(2*n) + 2 * sum( i=1, sqrtint(n\2), issquare(n - 2*i^2))))};
(PARI) {a(n) = if( n<1, n==0, 2 * sumdiv( n, d, kronecker( -2, d)))}; /* Michael Somos, Aug 23 2005 */
(PARI) {a(n) = if( n<1, n==0, 2 * qfrep([ 1, 0; 0, 2], n)[n])}; /* Michael Somos, Aug 23 2005 */
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^-2 * eta(x^2 + A)^3 * eta(x^4 + A)^3 * eta(x^8 + A)^-2, n))};
(Sage) Q = DiagonalQuadraticForm(ZZ, [1, 2]); Q.representation_number_list(104); # Peter Luschny, Jun 20 2014
(Magma) A := Basis( ModularForms( Gamma1(8), 1), 105); A[1] + 2*A[2] + 2*A[3]; /* Michael Somos, Aug 29 2014 */
|
|
|
CROSSREFS
|
Cf. A002325, A002479, A028572, A033761, A082564, A112603, A113411, A133692, A139093, A226225, A226240, A242609, A245572, A247719.
Number of integer solutions to f(x,y) = n where f(x,y) is the principal binary quadratic form with discriminant d: A004016 (d=-3), A004018 (d=-4), A002652 (d=-7), this sequence (d=-8), A028609 (d=-11), A028641 (d=-19), A138811 (d=-43).
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
