A035162 Number of positive odd solutions to equation x^2 + 7y^2 = 8n.

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

Offset

1,11

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Also number of solutions to the equation x + 7y = n in triangular numbers give the same sequence offset by 1. E.g., for n = 10, 3 + 7*1 = 10 + 7*0 = 10 so there are two solutions.

References

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 303.

J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 346.

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Michael Somos, Introduction to Ramanujan theta functions.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Formula

Expansion of q * psi(q) * psi(q^7) in powers of q where psi() is a Ramanujan theta function. - Michael Somos, Apr 24 2015

Expansion of (eta(q^2) * eta(q^14))^2 / (eta(q) * eta(q^7)) in powers of q.

Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -28.

G.f.: Sum_{K>0} (x^k - x^(3*k) - x^(5*k) + x^(9*k) + x^(11*k) - x^(13*k)) / (1 - x^(14*k)).

Multiplicative with a(2^e) = a(7^e) = 1, a(p^e) = e+1 if p == 1, 2, 4 (mod 7), a(p^e) = (1 + (-1)^e)/2 if p == 3, 5, 6 (mod 7). - Michael Somos, Sep 10 2005

Euler transform of period 14 sequence [ 1, -1, 1, -1, 1, -1, 2, -1, 1, -1, 1, -1, 1, -2, ...].

a(2*n) = a(7*n) = a(n). a(7*n + 3) = a(7*n + 5) = a(7*n + 6) = 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*u6 - u2^3*u3 - 3*u1*u6^3 + 3*u2*u3^3 + 3*u2*u6*(u1*(u2-u1) + 3*u3*(u6-u3)). - Michael Somos, Sep 10 2005

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u*w*(u-2*v) - v*(v-2*w)^2. - Michael Somos, Sep 10 2005

G.f.: Sum_{k>0} x^k * (1 - x^(2*k)) * (1 - x^(4*k)) * (1 - x^(6*k)) / (1 - x^(14*k)) = x * Product_{k>0} (1 - x^(2*k)) * (1 - x^(14*k)) / ((1 - x^(2*k-1)) * (1 - x^(14*k-7))).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(7)) = 0.593705... . - Amiram Eldar, Oct 25 2022

Example

For n=11, 5^2 + 7*3^2 = 9^2 + 7*1^2 = 8*11 so a(11)=2.
G.f. = q + q^2 + q^4 + q^7 + q^8 + q^9 + 2*q^11 + q^14 + q616 + q^18 + 2*q^22 + ...

Mathematica

a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ -28, d], { d, Divisors[ n]}]]; (* Michael Somos, Apr 24 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^(1/2)] EllipticTheta[ 2, 0, q^(7/2)] / 4, {q, 0, n}]; (* Michael Somos, Apr 24 2015 *)
QP = QPochhammer; s = (QP[q^2]*QP[q^14])^2/(QP[q]*QP[q^7]) + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015, adapted from PARI *)

Prog

(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, kronecker(-28, d)))};
(PARI) {a(n) = if( n<0, 0, sum(i=1, sqrtint(8*n \ 7), (i%2) * issquare(8*n - 7*i^2)))};
(PARI) {a(n) = if( n<1, 0, direuler(p=2, n, 1 / (1 - X) / (1 - kronecker(-28, p) * X))[n])};
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^14 + A)^2 / (eta(x + A) * eta(x^7 + A)), n))}; /* Michael Somos, Apr 24 2015 */

Crossrefs

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A119395 A087476 A307505 * A121454 A025462 A024879

Adjacent sequences: A035159 A035160 A035161 * A035163 A035164 A035165

Keyword

nonn,mult

Author

N. J. A. Sloane

Extensions

Entry revised by N. J. A. Sloane, Jul 31 2006

Status

approved