cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A002175 Excess of number of divisors of 12n+1 of form 4k+1 over those of form 4k+3.

Original entry on oeis.org

1, 2, 3, 2, 1, 2, 2, 4, 2, 2, 1, 0, 4, 2, 3, 2, 2, 4, 0, 2, 2, 0, 4, 2, 3, 0, 2, 6, 2, 2, 1, 2, 0, 2, 2, 2, 2, 4, 2, 0, 4, 4, 4, 0, 1, 2, 0, 4, 2, 0, 2, 2, 5, 2, 0, 2, 2, 4, 4, 2, 0, 2, 4, 2, 2, 0, 4, 0, 0, 2, 3, 2, 4, 2, 0, 4, 0, 6, 2, 4, 1, 0, 4, 2, 2, 2, 2, 0, 0, 2, 0, 2, 8, 2, 2, 0, 2, 4, 0, 4, 2, 2, 3, 2, 2
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number of ways to write n as an ordered sum of 2 generalized pentagonal numbers. - Ilya Gutkovskiy, Aug 14 2017

Examples

			G.f. = 1 + 2*x + 3*x^2 + 2*x^3 + x^4 + 2*x^5 + 2*x^6 + 4*x^7 + 2*x^8 + 2*x^9 + ...
G.f. = q + 2*q^13 + 3*q^25 + 2*q^37 + q^49 + 2*q^61 + 2*q^73 + 4*q^85 + 2*q^97 + ...

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002654, A008441, A008442, A035154, A035181, A035184, A112301, A113406.
Cf. A113652, A116604, A121363, A121450, A122856, A122864, A122865, A125061.
Cf. A125079, A129447, A129448, A132004, A134013, A134015, A138741, A138746.
Cf. A138950, A138952, A163746, A258210, A258228, A258256, A258279, A258292.

Programs

  • Maple
    series(mul( ( (1 + q^n)*(1 - q^(3*n))/(1 + q^(3*n)) )^2, n = 1..100), q, 101):
seq(coeftayl(%, q = 0, n), n = 0..100); # Peter Bala, Jan 05 2025
  • Mathematica
    ed[n_]:=Module[{divs=Divisors[12n+1]},Count[divs,?(Mod[#,4] == 1&)]- Count[divs,?(Mod[#,4]==3&)]]; Array[ed,110,0] (* Harvey P. Dale, Jul 01 2012 *)
a[ n_] := If[ n < 0, 0, With[ {m = 12 n + 1}, Sum[ KroneckerSymbol[ 4, d], {d, Divisors[m]}]]]; (* Michael Somos, Apr 23 2014 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2] QPochhammer[ x^3]^2 / (QPochhammer[ x] QPochhammer[ x^6]))^2, {x, 0, n}]; (* Michael Somos, Apr 23 2014 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, x^3] / QPochhammer[ x, x^2])^2, {x, 0, n}]; (* Michael Somos, May 25 2015 *)
  • PARI
    {a(n) = if( n<0, 0, n = 12*n + 1; sumdiv( n, d, (d%4==1) - (d%4==3)))}; /* Michael Somos, Sep 19 2005 */
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A)^2 / (eta(x + A) * eta(x^6 + A)))^2, n))}; /* Michael Somos, Jun 02 2012 */

Formula

Expansion of (phi(-x^3) / chi(-x))^2 in powers of x where phi(), chi() are Ramanujan theta functions.
Expansion of q^(-1/12) * (eta(q^2) * eta(q^3)^2 / (eta(q) * eta(q^6)))^2 in powers of q. - Michael Somos, Sep 19 2005
Euler transform of period 6 sequence [ 2, 0, -2, 0, 2, -2, ...]. - Michael Somos, Sep 19 2005
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 2 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A258279. - Michael Somos, May 25 2015
From Michael Somos, Jun 02 2012: (Start)
a(n) = A008441(3*n) = A121363(3*n) = A122865(4*n) = A122856(8*n).
a(n) = A116604(6*n) = A125079(6*n) = A129447(6*n) = A138741(6*n).
a(n) = A(12*n+1) where A = A002654, A008442, A035154, A035181, A035184, A112301, A113406, A113652, A121450, A122864, A125061, A129448, A132004, A134013, A134015, A138746, A138950, A138952, A163746. (End)
From Michael Somos, May 25 2015: (Start)
a(n) = A258277(4*n) = A258278(8*n) = A258291(3*n).
a(n) = - A258210(12*n + 1) = A258228(12*n + 1) = A258256(12*n + 1).
2*a(n) = A258279(12*n + 1) = - A258292(12*n + 1). (End)
G.f.: (Sum_{k = -oo..oo} x^(k*(3*k-1)/2))^2. - Ilya Gutkovskiy, Aug 14 2017
G.f.: ( Product_{n >= 1} (1 + q^n)*(1 - q^(3*n))/(1 + q^(3*n)) )^2. - Peter Bala, Jan 05 2025

A112301 Expansion of (eta(q) * eta(q^16))^2 / (eta(q^2) * eta(q^8)) in powers of q.

Original entry on oeis.org

1, -2, 0, 0, 2, 0, 0, 0, 1, -4, 0, 0, 2, 0, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0, 3, -4, 0, 0, 2, 0, 0, 0, 0, -4, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, -6, 0, 0, 2, 0, 0, 0, 0, -4, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, -4, 0, 0, 0, 0, 0, 0, 1, -4, 0, 0, 4, 0, 0, 0, 2, -4, 0, 0, 0, 0, 0, 0, 2, -2, 0, 0, 2, 0, 0, 0, 0
Offset: 1

Views

Author

Michael Somos, Sep 02 2005, Oct 02 2007

Keywords

Comments

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

Examples

			G.f. = q - 2*q^2 + 2*q^5 + q^9 - 4*q^10 + 2*q^13 + 2*q^17 - 2*q^18 + 3*q^25 - ...

Links

Crossrefs

Cf. A008441, A053692, A113407, A134013.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 2, 0, q^4] / 2, {q, 0, n}]; (* Michael Somos, Oct 19 2013 *)
QP = QPochhammer; s = (QP[q]*QP[q^16])^2/(QP[q^2]*QP[q^8]) + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015 *)
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^16 + A))^2 / (eta(x^2 + A) * eta(x^8 + A)), n))};
  • PARI
    {a(n) = if( n>0 & (n+1)%4\2, (n%2*3 - 2) * sumdiv( n / gcd(n, 2), d, (-1)^(d\2)))};

Formula

Expansion of q * phi(-q) * psi(q^8) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of (phi(-q^2)^2 - phi(-q)^2) / 4 in powers of q where phi() is a Ramanujan theta function.
Euler transform of period 16 sequence [ -2, -1, -2, -1, -2, -1, -2, 0, -2, -1, -2, -1, -2, -1, -2, -2, ...].
a(n) is ultiplicative with a(2) = -2, a(2^e) = 0 if e>1, a(p^e) = e+1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4).
Moebius transform is period 16 sequence [ 1, -3, -1, 2, 1, 3, -1, 0, 1, -3, -1, -2, 1, 3, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 4 (t/i) f(t) where q = exp(2 Pi i t).
G.f.: x * Product_{k>0} (1 - x^k)^2 * (1 + x^(8*k))^2 * (1 + x^(2*k)) * (1 + x^(4*k)).
G.f.: Sum_{k>0} Kronecker(-4, k) * x^k * (1 - x^k)^2 / (1 - x^(4*k)).
a(4*n) = a(4*n + 3) = a(8*n + 6) = 0. a(8*n + 2) = -2 * A008441(n).
a(n) = -(-1)^n * A134013(n). a(4*n + 1) = A008441(n). a(8*n + 1) = A113407(n). a(8*n + 5) = 2 * A053692(n).

A134014 Expansion of phi(-q) * phi(q^4) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 0, 0, 4, -4, 0, 0, 4, -2, 0, 0, 0, -4, 0, 0, 4, -4, 0, 0, 8, 0, 0, 0, 0, -6, 0, 0, 0, -4, 0, 0, 4, 0, 0, 0, 4, -4, 0, 0, 8, -4, 0, 0, 0, -4, 0, 0, 0, -2, 0, 0, 8, -4, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 4, -8, 0, 0, 8, 0, 0, 0, 4, -4, 0, 0, 0, 0, 0, 0, 8, -2
Offset: 0

Views

Author

Michael Somos, Oct 02 2007

Keywords

Comments

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

Examples

			G.f. = 1 - 2*q + 4*q^4 - 4*q^5 + 4*q^8 - 2*q^9 - 4*q^13 + 4*q^16 - 4*q^17 + ...

Links

Crossrefs

Cf. A004018, A004020, A004531, A134015.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q^4], {q, 0, n}]; (* Michael Somos, Oct 30 2015 *)
  • PARI
    {a(n) = if( n<1, n==0, if( n%4 < 2, (n%2*-6 + 4) * sumdiv(n, d, kronecker(-4, d))))};
  • PARI
    {a(n) = (-1)^n * if( n<1, n==0, 2 * qfrep([1, 0; 0, 4], n)[n])};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^8 + A)^5 / eta(x^2 + A) / eta(x^4 + A)^2 / eta(x^16 + A)^2, n))};

Formula

Expansion of eta(q)^2 * eta(q^8)^5 / (eta(q^2) * eta(q^4)^2 * eta(q^16)^2) in powers of q.
Euler transform of period 16 sequence [ -2, -1, -2, 1, -2, -1, -2, -4, -2, -1, -2, 1, -2, -1, -2, -2, ...].
Moebius transform is period 16 sequence [ -2, 2, 2, 4, -2, -2, 2, 0, -2, 2, 2, -4, -2, -2, 2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 8 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A134013.
a(4*n + 2) = a(4*n + 3) = 0.
G.f.: 1 - 2 * ( x / (1 + x^2) + x^3 / (1 + x^6) - 2 * x^4 / (1 + x^8) + ... ).
a(n) = (-1)^n * A004531(n). a(n) = -2 * A134015(n) unless n=0. a(4*n) = A004018(n). a(4*n+1) = - A004020(n).
Showing 1-3 of 3 results.

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