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.

A093722 Integers of the form (k^2 - 1) / 120.

Original entry on oeis.org

0, 1, 3, 7, 8, 14, 20, 29, 31, 42, 52, 66, 69, 85, 99, 118, 122, 143, 161, 185, 190, 216, 238, 267, 273, 304, 330, 364, 371, 407, 437, 476, 484, 525, 559, 603, 612, 658, 696, 745, 755, 806, 848, 902, 913, 969, 1015, 1074, 1086, 1147, 1197, 1261, 1274, 1340
Offset: 1

Views

Author

Michael Somos, Apr 13 2004

Keywords

Comments

This is "one-fifteenth of triangular numbers (integers only)". - Vladimir Joseph Stephan Orlovsky, Mar 04 2009
The sequence terms are the exponents in the expansion of Product_{n >= 1} (1 - q^n)/( (1 - q^(10*n-2))*(1 - q^(10*n-8)) ) = 1 - q - q^3 + q^7 + q^8 - q^14 - q^20 + + - - ... . - Peter Bala, Dec 26 2024

Links

Crossrefs

Cf. A002381, A057538, A113430, A214429, A379210.

Programs

  • Maple
    A093722 := proc(q) local n;
for n from 0 to q do
 if type(sqrt(120*n+1), integer) then print(n);
fi; od; end:
A093722(1500); # Peter Bala, Dec 26 2024
  • Mathematica
    Select[Table[(n^2-1)/120,{n,0,700}],IntegerQ] (* Harvey P. Dale, Nov 26 2010 *)
  • PARI
    {a(n) = (((n\4 * 3 + n%4) * 10 + (-1)^(n\2))^2 - 1) / 120 } /* Michael Somos, Oct 17 2006 */

Formula

|A113430(n-1)| is the characteristic function of the numbers in A093722.
a(-1 - n) = a(n). a(n) = (A057538(n) * 2 - 1) / 120.
G.f.: -x^2*(1+2*x+4*x^2+x^3+4*x^4+x^6+2*x^5) / ( (1+x)^2*(x^2+1)^2*(x-1)^3 ). - R. J. Mathar, Jun 09 2013
From Peter Bala, Dec 26 2024: (Start)
a(n) is quasi-polynomial in n
a(4*n) = n*(15*n + 1)/2; a(4*n+1) = (3*n + 1)*(5*n + 2)/2;
a(4*n+2) = (3*n + 2)*(5*n + 3)/2; a(4*n+3) = (n + 1)*(15*n + 14)/2.
For 0 <= k <= 3, a(4*n+k) = (N_k(n)^2 - 1)/120, where N_0(n) = 30*n + 1, N_1(n) = 30*n + 11, N_2(n) = 30*n + 19 and N_3(n) = 30*n + 29. (End)

Extensions

More terms from Harvey P. Dale, Nov 26 2010
Offset corrected to 1 by Ray Chandler, Jul 29 2019

A113681 Expansion of f(-x^2, -x^3)^2 / f(-x, -x^2) in powers of x where f() is Ramanujan's two-variable theta function.

Original entry on oeis.org

1, 1, 0, -1, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 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, -1, 0, 0, -1, 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, 1, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Michael Somos, Nov 04 2005

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
f(a,b) = Sum_{k} a^((k^2+k)/2)*b^((k^2-k)/2) is Ramanujan's two-variable theta function.
|a(n)| is the characteristic function of A093722.
The exponents in the q-series for this sequence are the squares of the numbers of A057538.
This is an example of the quintuple product identity in the form f(a*b^4, a^2/b) - (a/b) * f(a^4*b, b^2/a) = f(-a*b, -a^2*b^2) * f(-a/b, -b^2) / f(a, b) where a = -x^4, b = -x.

Examples

			1 + x - x^3 - x^7 - x^8 - x^14 + x^20 + x^29 + x^31 + x^42 - x^52 - x^66 - ...
q + q^121 - q^361 - q^841 - q^961 - q^1681 + q^2401 + q^3481 + q^3721 + ...

Links

Crossrefs

Cf. A010815, A057538, A093722, A113430.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (QPochhammer[ q^5] QPochhammer[ q^2, q^5] QPochhammer[ q^3, q^5])^2 / QPochhammer[ q], {q, 0, n}] (* Michael Somos, Jul 17 2012 *)
  • PARI
    {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^k)^((k%5==0) - kronecker( 5, k)), 1 + x * O(x^n)), n))}
  • PARI
    {a(n) = n*=5; if( issquare( 24*n + 1, &n), kronecker( 12, n))}

Formula

Expansion of f(-x^7, -x^8) + x * f(-x^2, -x^13) where f() is Ramanujan's two-variable theta function.
Euler transform of period 5 sequence [ 1, -1, -1, 1, -1, ...].
G.f.: Sum_{k} (-1)^k * x^(5*k * (3*k + 1)/2) * (x^(-3*k) + x^(3*k + 1)).
G.f.: Product_{k>0} (1 - x^(5*k)) * (1 - x^(5*k - 2)) * (1 - x^(5*k - 3)) / ((1 - x^(5*k - 1)) * (1 - x^(5*k - 4))).
A010815(5*n) = a(n).

A207710 Expansion of f(x) * f(-x^10) / f(-x^2, -x^8) in powers of x where f() is Ramanujan's two-variable theta function.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 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, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Michael Somos, Feb 19 2012

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
This is an example of the quintuple product identity in the form f(a*b^4, a^2/b) - (a/b) * f(a^4*b, b^2/a) = f(-a*b, -a^2*b^2) * f(-a/b, -b^2) / f(a, b) where a = x^4, b = -x.

Examples

			1 + x + x^3 - x^7 + x^8 - x^14 - x^20 - x^29 - x^31 - x^42 - x^52 + x^66 + ...
q + q^121 + q^361 - q^841 + q^961 - q^1681 - q^2401 - q^3481 - q^3721 + ...

Links

Crossrefs

Cf. A057538, A093722, A113430.

Programs

  • Mathematica
    f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A207710[n_] := SeriesCoefficient[f[x^5, -x^10]*f[-x^2, x^3]/f[-x, x^4], {x, 0, n}]; Table[A207710[n], {n,0,50}] (* G. C. Greubel, Jun 18 2017 *)
  • PARI
    {a(n) = local(m); if( issquare( 120*n + 1, &m), kronecker( -120, m) * (-1)^(m \ 15))}

Formula

Expansion of f(x^5, -x^10) * f(-x^2, x^3) / f(-x, x^4) = f(-x^7, x^8) + x * f(x^2, -x^13) in powers of x where f() is Ramanujan's two-variable theta function.
Euler transform of period 20 sequence [ 1, -1, 1, -1, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, -1, 1, -1, 1, -1, ...].
|a(n)| is the characteristic function of A093722.
The exponents in the q-series q * A(q^120) are the squares of the numbers in A057538.
G.f.: Prod_{k>0} (1 - (-x)^k) / ((1 - x^(10*k - 2)) * (1 - x^(10*k - 8))).
G.f.: Sum_{k} (-1)^[-k/2] * x^(5*k * (3*k + 1)/2) * (x^(-3*k) + x^(3*k + 1)).
a(7*n + 2) = a(7*n + 4) = a(7*n + 5) = 0. a(n) * (-1)^n = A113430(n).
Showing 1-3 of 3 results.

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