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.

A016244 Expansion of 1/((1-x)*(1-6*x)*(1-9*x)).

Original entry on oeis.org

1, 16, 187, 1942, 19033, 180628, 1681639, 15470674, 141251605, 1283357680, 11622778531, 105040363246, 947975408017, 8547451504972, 77021100541663, 693754126856458, 6247172473597069, 56244864253707304
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A003464, A024346, A024347.

Programs

  • Magma
    [(1 -96*6^n +135*9^n)/40: n in [0..40]]; // G. C. Greubel, Jan 30 2022
  • Mathematica
    LinearRecurrence[{16,-69,54}, {1,16,187}, 41] (* G. C. Greubel, Jan 30 2022 *)
  • PARI
    Vec(1/((1-x)*(1-6*x)*(1-9*x)) + O(x^40)) \\ Michel Marcus, Sep 04 2017
  • Sage
    [(1 -16*6^(n+1) +15*9^(n+1))/40 for n in (0..40)] #  G. C. Greubel, Jan 30 2022

Formula

a(n) = (1 - 96*6^n + 135*9^n)/40. - Neven Juric, Oct 22 2009
a(0)=1, a(1)=16, a(n) = 15*a(n-1) - 54*a(n-2) + 1. - Vincenzo Librandi, Feb 10 2011
E.g.f.: (1/40)*(exp(x) - 96*exp(6*x) + 135*exp(9*x)). - G. C. Greubel, Jan 30 2022

A024346 Expansion of 1/((1-x)*(1-6*x)*(1-9*x)*(1-11*x)).

Original entry on oeis.org

1, 27, 484, 7266, 98959, 1269177, 15642586, 187539120, 2204181925, 25529358855, 292445725936, 3321943348542, 37489352241979, 420930326166741, 4707254688375814, 52473555698990412, 583456285162491601
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A003464, A016244, A024347.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 25); Coefficients(R!(1/((1-x)*(1-6*x)*(1-9*x)*(1-11*x)))); // Vincenzo Librandi, Jul 16 2013
  • Magma
    I:=[1,27,484,7266]; [n le 4 select I[n] else 27*Self(n-1)-245*Self(n-2)+813*Self(n-3)-594*Self(n-4): n in [1..25]]; // Vincenzo Librandi, Jul 16 2013
  • Mathematica
    CoefficientList[Series[1/((1-x)(1-6x)(1-9x)(1-11x)), {x, 0, 20}], x] (* Vincenzo Librandi, Jul 16 2013 *)
LinearRecurrence[{27,-245,813,-594},{1,27,484,7266},20] (* Harvey P. Dale, Oct 13 2016 *)
  • PARI
    a(n) = (12*11^(n+3) - 25*9^(n+3) + 16*6^(n+3) - 3)/1200; \\ Joerg Arndt, Aug 13 2013
  • Sage
    [(4*11^(n+3) -75*9^(n+2) +32*6^(n+2) -1)/400 for n in (0..20)] # G. C. Greubel, Jan 30 2022

Formula

a(n) = 27*a(n-1) - 245*a(n-2) + 813*a(n-3) - 594*a(n-4) for n > 3; a(0)=1, a(1)=27, a(2)=484, a(3)=7266. - Vincenzo Librandi, Jul 16 2013
a(n) = (12*11^(n+3) - 25*9^(n+3) + 16*6^(n+3) - 3)/1200. - Yahia Kahloune, Aug 13 2013
E.g.f.: (1/400)*(-exp(x) + 1152*exp(6*x) - 6075*exp(9*x) + 5324*exp(11*x)). - G. C. Greubel, Jan 30 2022

A009687 Expansion of e.g.f. tan(sinh(x)*sin(x))/2 in odd powers of x^2.

Original entry on oeis.org

1, 116, 221776, 2035313216, 57114906593536, 3831406959743783936, 525282700343846052745216, 132012923141044737907744587776, 56143990156951992731353775556591616, 38005061403111079268112619476719676686336, 39010496884020339788347064320442556327459291136
Offset: 0

Views

Author

R. H. Hardin

Keywords

Examples

			x^2/2! + 116*x^6/6! + 221776*x^10/10! + 2035313216*x^14/14! ...

Links

Crossrefs

Cf. A024347.

Programs

  • Mathematica
    Tan[ Sinh[ x ]*Sin[ x ] ]/2 (* [ x^(4n+2) ] *)

Extensions

Extended and signs tested 03/97.
a(9) onwards by Andrew Howroyd, Jan 12 2025
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.