A195198 -id:A195198 - cp's OEIS frontend

A145377 a(n) = A002324(n) mod 2.

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

Offset: 1

N. J. A. Sloane, Mar 12 2009

Antti Karttunen, Table of n, a(n) for n = 1..65537
J. S. Rutherford, Generating functions for the cage isomers of the C_{20n} icosahedral fullerenes, J. Mathematical Chem., 14 (1993), 385-390. See Eq. (3).
J. S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices, Act. Cryst. A48 (1992), 500-508. [See Sect. (7), p. 505.]
Index entries for characteristic functions

Essentially same as A195198.

Cf. A002324, A154272, A003050.

Array[Boole@ OddQ@ If[# < 1, 0, DivisorSum[#, KroneckerSymbol[-3, #] &]] &, 105] (* Michael De Vlieger, Nov 05 2017, after Michael Somos at A002324 *)

A002324(n) = if( n<1, 0, sumdiv(n, d, (d%3==1) - (d%3==2)));
A145377(n) = (A002324(n)%2); \\ Antti Karttunen, Nov 06 2017

a(n) = A195198(n) for n >= 1.

a(n) = Sum_{ m: m^2|n } A154272(n/m^2). - Andrey Zabolotskiy, May 07 2018

More terms from Antti Karttunen, Nov 05 2017

A214284 Characteristic function of squares or five times squares.

Original entry on oeis.org

1, 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, 0

Offset: 0

Michael Somos, Jul 09 2012

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

A195198 is a similar sequence except with three instead of five. - Michael Somos, Oct 22 2017

			G.f. = 1 + x + x^4 + x^5 + x^9 + x^16 + x^20 + x^25 + x^36 + x^45 + x^49 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
S. Cooper and M. 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. A127693, A195198, A344212.

Cf. A000700, A000122, A010054, A121373.

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

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

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

Expansion of f(q, q^9) * f(-q^8, -q^12) / f(-q^4, -q^16) in powers of q where f(, ) is Ramanujan's general theta function.
Expansion of f(q^3, q^7) * f(-q^2, -q^3) / f(-q, -q^4) in powers of q where f(, ) is Ramanujan's general theta function.
Euler transform of period 20 sequence [1, -1, 0, 1, 0, 0, 0, -1, 1, -1, 1, -1, 0, 0, 0, 1, 0, -1, 1, -1, ...].
a(n) is multiplicative with a(0) = a(5^e) = 1, a(p^e) = 1 if e is even, 0 otherwise.
G.f.: (theta_3(q) + theta_3(q^5)) / 2 = 1 + (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 + 1) = A127693(n). a(5*n) = a(n).
Sum_{k=0..n} a(k) ~ c * sqrt(n), where c = 1+1/sqrt(5) = 1.447213... (A344212). - Amiram Eldar, Sep 14 2023

A127692 Expansion of psi(x^4) + x * psi(x^12) in powers of x where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, 1, 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, 1, 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, 1, 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

Offset: 0

Michael Somos, Jan 19 2007

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

a(n) = 1 if n is four times a triangular number or one more than twelve times a triangular number else 0. - Michael Somos, Jul 19 2012

			G.f. = 1 + x + x^4 + x^12 + x^13 + x^24 + x^37 + x^40 + x^60 + x^73 + x^84 + ...
G.f. = q + q^3 + q^9 + q^25 + q^27 + q^49 + q^75 + q^81 + q^121 + q^147 + q^169 + ...

Antti Karttunen, Table of n, a(n) for n = 0..10000
Richard Blecksmith, John Brillhart, and Irving Gerst, Some infinite product identities, Math. Comp. 51 (1988), no. 183, 301-314. MR0942157 (89f:05017)
Shaun Cooper and Michael Hirschhorn, On some infinite product identities, Rocky Mountain J. Math., 31 (2001), 131-139. see p. 134 Theorem 5.
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A005369, A010054, A080995, A089806, A145439.

Cf. A000122, A000700, A010054, A121373.

{a(n) = issquare(2*n + 1) + issquare(6*n + 3)};

{a(n) = n = 2*n + 1; issquare(n) || issquare(3*n)};

Euler transform of period 24 sequence [ 1, -1, 0, 1, -1, 1, -1, 0, 0, 0, 1, -1, 1, 0, 0, 0, -1, 1, -1, 1, 0, -1, 1, -1, ...].
a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = 1, else b(p^e) = (1 + (-1)^e)/2.
a(3*n + 1) = a(n), a(3*n + 2) = a(4*n + 2) = a(4*n + 3) = a(6*n + 3) = 0.
a(2*n) = A005369(n). a(4*n) = A010054(n). a(6*n) = A089806(n). a(12*n) = A080995(n).
G.f.: Sum_{k>0} x^(2k(k-1)) +x^(6k(k-1)+1) = Product_{k>0} (1-x^(24k)) (1-x^(24k-5)) (1-x^(24k-7)) (1-x^(24k-17)) (1-x^(24k-19)) (1+x^(12k-1)) (1+x^(12k-4)) (1+x^(12k-6)) (1+x^(12k-8)) (1+x^(12k-11)).
From Michael Somos, Jul 19 2012: (Start)
Expansion of f(x, -x^5) * f(-x^4, -x^8) / f(x, -x) in powers of x where f(,) is the Ramanujan two-variable theta function.
G.f.: (Sum_{k in Z} x^(2*k*(k + 1)) + x^(6*k*(k + 1) + 1)) / 2.
a(n) = A195198(2*n + 1). (End)
Sum_{k=1..n} a(k) ~ c * sqrt(n), where c = 1/sqrt(2) + 1/sqrt(6) = 1.115355... (A145439). - Amiram Eldar, Dec 29 2023

cp's OEIS Frontend

A145377 a(n) = A002324(n) mod 2.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A214284 Characteristic function of squares or five times squares.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A127692 Expansion of psi(x^4) + x * psi(x^12) in powers of x where psi() is a Ramanujan theta function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula