A245485 -id:A245485 - cp's OEIS frontend

A214295 a(n) = 1 if n is a square, -1 if n is three times a square, 0 otherwise.

1, 0, -1, 1, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 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, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0

Offset: 1

Michael Somos, Jul 10 2012

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

a(A092206(n)) = 0; a(A000290(n)) = 1; a(A033428(n)) = -1.

			G.f. = q - q^3 + q^4 + q^9 - q^12 + q^16 + q^25 - q^27 + q^36 - q^48 + q^49 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Shaun Cooper and Michael Hirschhorn, On some infinite product identities, Rocky Mountain J. Math., 31 (2001) 131-139. see p. 133 Theorem 3.
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A000700, A010052, A214293, A244612, A247133, A245485.

a214295 n = a010052 n - a010052 (3*n)  -- Reinhard Zumkeller, Jul 12 2012

Basis( ModularForms( Gamma1(12), 1/2), 50) [2] ; /* Michael Somos, Jun 10 2014 */

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] - EllipticTheta[ 3, 0, q^3]) / 2, {q, 0, n}];
a[ n_] := Boole[ IntegerQ[ Sqrt[ n]]] - Boole[ IntegerQ[ Sqrt[ 3 n]]]; (* Michael Somos, Jun 10 2014 *)
Table[Which[IntegerQ[Sqrt[n]],1,IntegerQ[Sqrt[n/3]],-1,True,0],{n,120}] (* Harvey P. Dale, Apr 08 2013 *)

PARI
```
{a(n) = issquare(n) - issquare(3*n)};
```

{a(n) = if( n<1, 0, direuler( p=2, n, if( p==3, 1 - X, 1) / (1 - X^2 ))[n])};

Expansion of q * psi(q^3) * f(-q^2, -q^10) / f(-q^5, -q^7) in powers of q where psi(), f() are Ramanujan theta functions.
Multiplicative with a(3^e) = (-1)^e, a(p^e) = 1 if e even, 0 otherwise.
G.f.: (theta_3(q) - theta_3(q^3)) / 2 = Sum_{k>0} x^(k^2) - x^(3*k^2).
Dirichlet g.f.: zeta(2*s) * (1 - 3^(-s)). [corrected by Amiram Eldar, Oct 24 2023]
a(3*n) = - a(n). - Reinhard Zumkeller, Jul 12 2012
Expansion of (phi(q) - phi(q^3)) / 2 = q * chi(q) * f(-q, -q^11) in powers fof q where phi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Jan 10 2015
Euler transform of period 12 sequence [ 0, -1, 1, 0, 1, -1, 1, 0, 1, -1, 0, -1, ...]. - Michael Somos, Jan 10 2015
Convolution product of A000700 and A247133. - Michael Somos, Jan 10 2015
Sum_{k=1..n} a(k) ~ c*sqrt(n), where c = 1 - 1/sqrt(3) = 0.42264973... . - Amiram Eldar, Oct 24 2023

A214293 a(n) = 1 if n is a square, -1 if n is five times a square.

Original entry on oeis.org

1, 0, 0, 1, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 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, -1, 1, 0, 0, 0, 0

Offset: 1

Michael Somos, Jul 10 2012

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

			G.f. = q + q^4 - q^5 + q^9 + q^16 - q^20 + q^25 + q^36 - q^45 + q^49 + q^64 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Shaun Cooper and Michael Hirschhorn, On some infinite product identities, Rocky Mountain J. Math., 31 (2001) 131-139. see p. 134 Theorem 4.
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A214284, A214295, A214960, A244612, A245485, A322159.

Basis( ModularForms( Gamma1(20), 1/2), 65) [2]; /* Michael Somos, Jul 01 2014 */

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] - EllipticTheta[ 3, 0, q^5]) / 2,  {q, 0, n}];
a[ n_] := If[ n < 0, 0, Boole[ OddQ [ Length @ Divisors @ n]] - Boole[ OddQ [ Length @ Divisors [5 n]]]];

PARI
```
{a(n) = issquare(n) - issquare(5*n)};
```

{a(n) = if( n<1, 0, direuler( p=2, n, if( p==5, 1 - X, 1) / (1 - X^2 ))[n])};

Expansion of (phi(q) - phi(q^5)) / 2 in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Sep 24 2013
Expansion of q * f(q^3, q^7) * f(-q^4, -q^16) / f(-q^8, -q^12) in powers of q where f() is Ramanujan's two-variable theta function.
Expansion of q * f(x, x^9) * f(-q, -q^4) / f(-q^2, -q^3) in powers of q where f() is Ramanujan's two-variable theta function. - Michael Somos, Sep 24 2013
Euler transform of period 20 sequence [ 0, 0, 1, -1, 0, -1, 1, 1, 0, -1, 0, 1, 1, -1, 0, -1, 1, 0, 0, -1, ...].
Multiplicative with a(5^e) = (-1)^e, a(p^e) = 1 if e even, 0 otherwise.
G.f.: (theta_3(q) - theta_3(q^5)) / 2 = Sum_{k>0} x^(k^2) - x^(5*k^2).
Dirichlet g.f.: zeta(2*s) * (1 - 5^-s).
a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = a(n). a(5*n) = -a(n).
a(4*n) = A214293(n). a(4*n+1) = A214960(n). - Michael Somos, Sep 24 2013
Sum_{k=1..n} a(k) ~ c*sqrt(n), where c = 1 - 1/sqrt(5) = 0.5527864... (A322159). - Amiram Eldar, Oct 24 2023

A244612 a(n) = 1 if n is a square, -1 if n is six times a square, 0 if n < 1.

Original entry on oeis.org

1, 0, 0, 1, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0

Offset: 1

Michael Somos, Jul 01 2014

sign

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

			G.f. = q + q^4 - q^6 + q^9 + q^16 - q^24 + q^25 + q^36 + q^49 - q^54 + ...

Antti Karttunen, Table of n, a(n) for n = 1..65537
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A089801, A214293, A214295, A245485.

Basis( ModularForms( Gamma1(24), 1/2), 64) [2];

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] - EllipticTheta[ 3, 0, q^6]) / 2,  {q, 0, n}];
a[ n_] := If[ n < 0, 0, Boole[ OddQ [ Length @ Divisors @ n]] - Boole[ OddQ [ Length @ Divisors [6 n]]]];

PARI
```
{a(n) = issquare(n) - issquare(6*n)};
```

Expansion of (phi(q) - phi(q^6)) / 2 in powers of q where phi() is a Ramanujan theta function.
G.f.: (theta_3(q) - theta_3(q^6)) / 2 = Sum_{k>0} x^(k^2) - x^(6*k^2).
a(3*n) = A089801(n). a(3*n + 2) = 0.
Sum_{k=1..n} a(k) ~ c*sqrt(n), where c = 1 - 1/sqrt(6) = 0.5917517... . - Amiram Eldar, Oct 24 2023

More terms from Antti Karttunen, Dec 15 2017

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A214295 a(n) = 1 if n is a square, -1 if n is three times a square, 0 otherwise.

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A214293 a(n) = 1 if n is a square, -1 if n is five times a square.

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A244612 a(n) = 1 if n is a square, -1 if n is six times a square, 0 if n < 1.

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