A005931 -id:A005931 - cp's OEIS frontend

A049240 Smallest nonnegative value taken on by x^2 - n*y^2 for an infinite number of integer pairs (x, y).

0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1

Offset: 1

David W. Wilson

Encodes to 1,2,1,4,1,6,1,8,1,10,...: unsigned version of A009531. - Paul Barry, Oct 12 2005

Parity of inverse Moebius transform of Jacobsthal numbers J(k) less J(n). - Paul Barry, Oct 12 2005

Cf. A001045, A005931, A010052.

Characteristic function of A000037 (the nonsquares).

[Floor((1 + Ceiling(Sqrt(n)) - Floor(Sqrt(n)))/2) : n in [1..100]]; // Wesley Ivan Hurt, Sep 27 2014

A049240:=n->`if`(issqr(n),0,1): seq(A049240(n), n=1..100); # Wesley Ivan Hurt, Sep 27 2014

Differences[Table[n - Ceiling[Sqrt[n]], {n, 105}]] (* Arkadiusz Wesolowski, Oct 30 2012 *)
Table[Floor[(1 + Ceiling[Sqrt[n]] - Floor[Sqrt[n]])/2], {n, 70}] (* Wesley Ivan Hurt, Sep 27 2014 *)

from math import isqrt
def A049240(n): return int(isqrt(n)**2!=n) # Chai Wah Wu, Jun 14 2022

a(n) = 0 if n is square, 1 otherwise.
a(n) = (A001045(n) - Sum_{k|n} A001045(k)) mod 2. - Paul Barry, Oct 12 2005
a(n) = 1 - A010052(n). - R. J. Mathar, Jul 04 2009
a(n) = floor(1+ceiling(sqrt(n))-floor(sqrt((n)))/2). - Wesley Ivan Hurt, Sep 27 2014
G.f.: (1+x)/(2-2*x) - (1/2)*theta_3(0,x) where theta_3 is a Jacobi theta function. - Robert Israel, Oct 02 2014

A003781 Expansion of theta series of {E_7}* lattice in powers of q^(1/2).

Original entry on oeis.org

1, 0, 0, 56, 126, 0, 0, 576, 756, 0, 0, 1512, 2072, 0, 0, 4032, 4158, 0, 0, 5544, 7560, 0, 0, 12096, 11592, 0, 0, 13664, 16704, 0, 0, 24192, 24948, 0, 0, 27216, 31878, 0, 0, 44352, 39816, 0, 0, 41832, 55944, 0, 0, 72576, 66584, 0, 0, 67536, 76104, 0, 0, 100800

Offset: 0

N. J. A. Sloane

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

			G.f. = 1 + 56*x^3 + 126*x^4 + 576*x^7 + 756*x^8 + 1512*x^11 + 2072*x^12 + ...
G.f. = 1 + 56*q^(3/2) + 126*q^2 + 576*q^(7/2) + 756*q^4 + 1512*q^(11/2) + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 125.
M. Eichler and D. Zagier, The Theory of Jacobi Forms, Birkhauser, 1985, p. 141.

G. C. Greubel, Table of n, a(n) for n = 0..1000
N. Elkies and B. H. Gross, Embeddings into the integral octonions, Olga Taussky-Todd: in memoriam, Pacific J. Math. 1997, Special Issue, 147-158.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A004008, A005931, A030443, A038723.

Basis( ModularForms( Gamma0(4), 7/2), 19) [1] ; /* Michael Somos, Jun 10 2014 */

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^3 (EllipticTheta[ 3, 0, q]^4 + 7 EllipticTheta[ 4, 0, q]^4) / 8, {q, 0, n}]; (* Michael Somos, Aug 27 2013 *)

{a(n) = local(A, B, m); n++; m=n%4; n\=4; if( n<0 || m>1, 0, A = sum(k=1, sqrtint(n), 2*x^k^2, 1 + x * O(x^n)); B = subst(A, x, -x); polcoeff( if(m==0, (A^4 - B^4) * (8*A^4 - B^4) / 2 / sum(k=0, sqrtint( 4*n + 1)\2, x^(k^2 + k), x * O(x^n)), 8*A^7 - 7*A^3 * subst(A, x, -x)^4 ), n))}; /* Michael Somos, Jun 11 2007 */

{a(n) = local(A); if( n<0, 0, A = sum(k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n)); polcoeff( A^3 * (A^4 + 7 * subst(A, x, -x)^4) / 8, n))}; /* Michael Somos, Aug 27 2013 */

Theta series is given on page 125 of Conway and Sloane.
Can be determined from A023919 (A*_7): [1] A003781(4n)=A023919(16n) [2] A003781(4n+3)=A023919(16n+12). Let A_7+[1] be the generator of A*_7/A_7, then these correspond to [1]A004008=theta(E_7)=theta(A_7)+theta(A_7+[4]), [2]A005931=theta(E_7+[1])=theta(A_7+[2])+theta(A_7+[6]) - Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), May 03 2000
Expansion of phi(q)^3 * (phi(q)^4 + 7 * phi(-q)^4) / 8 in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Aug 27 2013
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 2^(13/2) (t / i)^(7/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A004008. - Michael Somos, Aug 27 2013
a(4*n + 1) = a(4*n + 2) = 0. - Michael Somos, Jun 11 2007
a(4*n) = A004008(n), a(4*n + 3) = A005931(n). - Michael Somos, Jun 11 2007.

cp's OEIS Frontend

A049240 Smallest nonnegative value taken on by x^2 - n*y^2 for an infinite number of integer pairs (x, y).

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