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A110399 Expansion of (theta_3(q)*theta_3(q^7) - 1)/2 in powers of q. 3
1, 0, 0, 1, 0, 0, 1, 2, 1, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 1, 2, 0, 0, 4, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 5, 0, 0, 2, 0, 0, 0, 2, 2, 0, 0, 0, 0, 2, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,8

COMMENTS

Half the number of integer solutions to x^2 + 7*y^2 = n. - Jianing Song, Nov 20 2019

REFERENCES

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 302 Entry 17(ii).

LINKS

Table of n, a(n) for n=1..105.

FORMULA

a(n) is multiplicative with a(2^e) = |e-1|, 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).

G.f.: Sum_{k>0} Kronecker(-7, k) x^k/(1-(-x)^k).

G.f.: (theta_3(q)*theta_3(q^7) - 1)/2 where theta_3(q) = 1 + 2*(q + q^4 + q^9 + ...).

a(2*n + 1) = A035162(2*n + 1) = A035182(2*n + 1). A033719(n) = 2*a(n) if n > 0.

EXAMPLE

G.f. = x + x^4 + x^7 + 2*x^8 + x^9 + 2*x^11 + 3*x^16 + 2*x^23 + ...

PROG

(PARI) {a(n) = my(x); if( n<1, 0, x = valuation(n, 2); abs(x -1) * sumdiv(n/2^x, d, kronecker(-28, d)))};

(PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, e-1,  p==7, 1, kronecker(-7, p)==-1, (1+(-1)^e)/2, e+1)))};

(PARI) {a(n) = my(A); if( n<1, 0, A = x *O(x^n); polcoeff( (eta(x + A)^-2 * eta(x^2 + A)^5 * eta(x^4 + A)^-2 * eta(x^7 + A)^-2 * eta(x^14 + A)^5 * eta(x^28 + A)^-2 - 1)/2, n))};

CROSSREFS

Cf. A033719 (number of integer solutions to x^2 + 7*y^2 = n).

Cf. A035162, A035182.

Similar sequences: A096936, A113406, A138806.

Sequence in context: A266886 A267546 A228817 * A193275 A182033 A112214

Adjacent sequences:  A110396 A110397 A110398 * A110400 A110401 A110402

KEYWORD

nonn,mult

AUTHOR

Michael Somos, Oct 22 2005

STATUS

approved

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Last modified January 28 13:34 EST 2022. Contains 350656 sequences. (Running on oeis4.)