A004020 - OEIS

A004020

Theta series of square lattice with respect to edge.
(Formerly M0931)

2, 4, 2, 4, 4, 0, 6, 4, 0, 4, 4, 4, 2, 4, 0, 4, 8, 0, 4, 0, 2, 8, 4, 0, 4, 4, 0, 4, 4, 4, 2, 8, 0, 0, 4, 0, 8, 4, 4, 4, 0, 0, 6, 4, 0, 4, 8, 0, 4, 4, 0, 8, 0, 0, 0, 8, 6, 4, 4, 0, 4, 4, 0, 0, 4, 4, 8, 4, 0, 4, 4, 0, 6, 4, 0, 0, 8, 0, 4, 4, 0, 12, 0, 4, 4, 0, 0, 4, 4, 0, 2, 8, 4, 4, 8, 0, 0, 4, 0, 4, 4, 4, 4, 0

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OFFSET

0,1

COMMENTS

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

Number of solutions in integers of n = x^2 + y^2 + y.

REFERENCES

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 106.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions.

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

FORMULA

G.f.: 2 * (Sum_{k>0} x^((k^2 - k)/2))^2 = (Sum_{k in Z} x^(k^2 + k)) * (Sum_{k in Z} x^(k^2)).

Expansion of q^(-1/2) * c(q) / 2 in powers of q^2 where c(q) is the third function in the quadratic Gauss AGM. - Michael Somos, Feb 10 2006

Expansion of 2 * phi(x) * psi(x^2) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Feb 10 2006

a(n) = 2*A008441(n) = A004531(4*n + 1).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi (A000796). - Amiram Eldar, Oct 15 2022

EXAMPLE

G.f. = 2 + 4*x + 2*x^2 + 4*x^3 + 4*x^4 + 6*x^6 + 4*x^7 + 4*x^9 + 4*x^10 + ...

G.f. = 2*q + 4*q^5 + 2*q^9 + 4*q^13 + 4*q^17 + 6*q^25 + 4*q^29 + 4*q^37 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 2, 0, x] / x^(1/4), {x, 0, n}]; (* Michael Somos, Feb 22 2015 *)

s = 2*QPochhammer[q^2]^4/QPochhammer[q]^2+O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 09 2015 *)

PROG

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); 2 * polcoeff( eta(x^2 A)^4 / eta(x + A)^2, n))};

(PARI) {a(n) = 2 * if( n<1, n==0, polcoeff( sum(k=0, (sqrtint(8*n + 1) - 1)\2, x^(k*(k + 1)/2), x*O(x^n))^2, n))};

CROSSREFS

Cf. A000122, A000700, A000796, A004531, A008441, A010054, A121373.

Sequence in context: A032059 A074075 A184186 * A363290 A246436 A143235

Adjacent sequences: A004017 A004018 A004019 * A004021 A004022 A004023

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved