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URL: https://oeis.org/A004530

⇱ A004530 - OEIS

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A004530
Expansion of (theta_2(0, x) + theta_3(0, x) + theta_4(0, x)) / 2 in powers of x^(1/4).
1
1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
OFFSET
0,17
COMMENTS
From Robert Israel, Nov 16 2025: (Start)
a(n) = 2 if n > 0 is a square and is divisible by 16.
a(n) = 1 if n = 0 or n is an odd square.
a(n) = 0 otherwise. (End)
REFERENCES
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 120.
LINKS
FORMULA
G.f.: (theta_2(0, x^4) + theta_3(0, x^4) + theta_4(0, x^4)) / 2.
EXAMPLE
G.f. = 1 + x + x^9 + 2*x^16 + x^25 + x^49 + 2*x^64 + x^81 + x^121 + 2*x^144 + x^169 + ...
MAPLE
f:= proc(n)
if not issqr(n) then 0
else
if n::odd then 1
elif n mod 16 = 0 then 2
else 0
fi
fi
end proc:
f(0):= 1:
map(f, [$0..100]); # Robert Israel, Nov 16 2025
MATHEMATICA
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, x^4] + EllipticTheta[ 3, 0, x^4] + EllipticTheta[4, 0, x^4]) / 2, {x, 0, n}]; (* Michael Somos, Aug 28 2017 *)
a[ n_] := With[{m = Sqrt@n}, Which[ n <= 0, Boole[n == 0], IntegerQ[m] && (Mod[m, 4] != 2), 2 - Mod[m, 2], True, 0]]; (* Michael Somos, Aug 28 2017 *)
PROG
(PARI) {a(n) = my(m); if( n<=0, n==0, issquare(n, &m) && (m%4!=2), 2-m%2, 0)}; /* Michael Somos, Aug 28 2017 */
(PARI) {a(n) = my(A); if(n<0, 0, A = x * O(x^n); polcoeff( x * eta(x^16 + A)^2 / eta(x^8 + A) + (eta(x^8 + A)^5 / (eta(x^4 + A) * eta(x^16 + A))^2 + eta(x^4 + A)^2 / eta(x^8 + A)) / 2, n))}; /* Michael Somos, Aug 28 2017 */
CROSSREFS
Sequence in context: A069849 A138045 A072507 * A368980 A355549 A347438
KEYWORD
nonn,mult
EXTENSIONS
Data section extended up to a(121) by Antti Karttunen, Aug 28 2017
STATUS
approved