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.

A243174 Nonnegative integers of the form x^2 + 5*x*y - 5*y^2 (discriminant 45).

Original entry on oeis.org

0, 1, 4, 9, 16, 19, 25, 31, 36, 45, 49, 55, 61, 64, 76, 79, 81, 99, 100, 109, 121, 124, 139, 144, 145, 151, 169, 171, 180, 181, 196, 199, 205, 211, 220, 225, 229, 241, 244, 256, 261, 271, 279, 289, 295, 304, 316, 319, 324, 331, 349, 355, 361, 369, 379, 396, 400, 405, 409, 421, 436, 439, 441, 445, 451, 475, 484, 495, 496, 499, 505, 529, 531, 541, 549
Offset: 1

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Author

N. J. A. Sloane, Jun 01 2014

Keywords

Comments

Also numbers representable as x^2 + 7*x*y + y^2 with 0 <= x <= y. - Gheorghe Coserea, Jul 29 2018
Also numbers of the form x^2 - x*y - 11*y^2 with 0 <= x <= y (or x^2 + x*y - 11*y^2 with x, y nonnegative). - Jianing Song, Jul 31 2018
Also nonnegative integers of the form 9x^2 - 5y^2. - Jon E. Schoenfield, Jun 03 2022

Links

Crossrefs

Cf. A028957, A031363.
Primes: A033212.

A106406 Expansion of (eta(q) * eta(q^15))^2 / (eta(q^3) * eta(q^5)) in powers of q.

Original entry on oeis.org

1, -2, -1, 3, -1, 2, 0, -4, 1, 2, 0, -3, 0, 0, 1, 5, -2, -2, 2, -3, 0, 0, -2, 4, 1, 0, -1, 0, 0, -2, 2, -6, 0, 4, 0, 3, 0, -4, 0, 4, 0, 0, 0, 0, -1, 4, -2, -5, 1, -2, 2, 0, -2, 2, 0, 0, -2, 0, 0, 3, 2, -4, 0, 7, 0, 0, 0, -6, 2, 0, 0, -4, 0, 0, -1, 6, 0, 0, 2
Offset: 1

Views

Author

Michael Somos, May 02 2005

Keywords

Comments

Number 30 of the 74 eta-quotients listed in Table I of Martin (1996).

Examples

			G.f. = q - 2*q^2 - q^3 + 3*q^4 - q^5 + 2*q^6 - 4*q^8 + q^9 + 2*q^10 + ...

Links

Crossrefs

Cf. A028955, A028957, A035175, A123864.

Programs

  • Magma
    A := Basis( ModularForms( Gamma1(15), 1), 80); A[2] - 2*A[3] - A[4] + 3*A[5] - A[6] + 2*A[7]; /* Michael Somos, May 18 2015 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^15])^2 / (QPochhammer[ q^3] QPochhammer[ q^5]), {q, 0, n}]; (* Michael Somos, May 18 2015 *)
a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ #, 3] KroneckerSymbol[ n/#, 5] &]]; (* Michael Somos, May 18 2015 *)
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^15 + A)^2 / (eta(x^3 + A) * eta(x^5 + A)), n))};
  • PARI
    {a(n) = if( n<1, 0, sumdiv( n, d, kronecker( d, 3) * kronecker( n/d, 5)))};
  • PARI
    {a(n) = my(A, p, e, x); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==3 || p==5, (-1)^e, (p%15) != 2^(x = valuation( p%15, 2)), (e+1)%2, (e+1) * (-1)^(x*e))))};
  • PARI
    {a(n) = if( n<1, 0, (qfrep([2, 1;1, 8],n, 1) - qfrep([4, 1;1, 4], n, 1))[n])}; /* Michael Somos, Aug 25 2006 */

Formula

Euler transform of period 15 sequence [-2, -2, -1, -2, -1, -1, -2, -2, -1, -1, -2, -1, -2, -2, -2, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = -v^3 + 4 * u*v*w + 2 * u*w^2 + u^2*w.
a(n) is multiplicative with a(3^e) = a(5^e) = (-1)^e, a(p^e) = (1 + (-1)^e) / 2 if p == 7, 11, 13, 14 (mod 15), a(p^e) = e+1 if p == 1, 4 (mod 15), a(p^e) = (e+1) * (-1)^e if p == 2, 8 (mod 15). - Michael Somos, Oct 19 2005
G.f.: (1/2) * (Sum_{n,m in Z} x^(n^2 + n*m + 4*m^2) - x^(2*n^2 + n*m + 2 *m^2)). - Michael Somos, Aug 25 2006
G.f.: Sum_{k>0} Kronecker(k, 3) * x^k * (1 - x^k) * (1 - x^(2*k)) / (1 - x^(5*k)) = Sum_{k>0} Kronecker(k, 5) * x^k * (1 - x^k) / (1 - x^(3*k)).
G.f.: x * Product_{k>0} ((1 - x^k) * (1 - x^(15*k)))^2 / ((1 - x^(3*k)) * (1 - x^(5*k))).
a(15*n + 7) = a(15*n + 11) = a(15*n + 13) = a(15*n + 14) = 0. a(3*n) = a(5*n) = -a(n).
A035175(n) = |a(n)|. a(n)>0 iff n in A028957. a(n)<0 iff n in A028955.
G.f. is a period 1 Fourier series which satisfies f(-1 / (15 t)) = 15^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, May 18 2015

A128616 Expansion of q * psi(q^3) * psi(q^5) in powers of q where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 1, 0, 0, 0, 2, 0, 0
Offset: 1

Views

Author

Michael Somos, Mar 13 2007

Keywords

Comments

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

Examples

			G.f. = x + x^4 + x^6 + x^9 + x^10 + x^15 + x^16 + 2*x^19 + x^24 + x^25 + 2*x^31 + ...

References

  • B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 377, Entry 9(iv).

Links

Crossrefs

Cf. A082451, A121362.

Programs

  • Mathematica
    a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ -60, #] + KroneckerSymbol[ 20, #] KroneckerSymbol[ -3, n/#] &] / 2]; (* Michael Somos, Nov 12 2015 *)
a[ n_] := SeriesCoefficient[ q(QPochhammer[ q^6] QPochhammer[ q^10])^2 / (QPochhammer[ q^3] QPochhammer[ q^5]), {q, 0, n}]; (* Michael Somos, Nov 12 2015 *)
  • PARI
    {a(n) = if( n<1, 0, sumdiv(n, d, kronecker(-60, d) + kronecker(20, d) * kronecker(-3, n/d) )/2)};
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^6 + A) * eta(x^10 + A))^2 / (eta(x^3 + A) * eta(x^5 + A)), n))};

Formula

Expansion of (eta(q^6) * eta(q^10))^2 / (eta(q^3) * eta(q^5)) in powers of q.
Euler transform of period 30 sequence [ 0, 0, 1, 0, 1, -1, 0, 0, 1, -1, 0, -1, 0, 0, 2, 0, 0, -1, 0, -1, 1, 0, 0, -1, 1, 0, 1, 0, 0, -2, ...].
For n>0, n in A028957 equivalent to a(n) nonzero. If a(n) nonzero, a(n) = A082451(n) and a(n) = A121362(n).
a(n) = (A082451(n) + A121362(n))/2.
G.f.: x * Product_{k>0} (1 - x^(3*k)) * (1 - x^(5*k)) * (1 + x^(6*k))^2 * (1 + x^(10*k))^2.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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