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.

A006646 Exponential self-convolution of Pell numbers.

Original entry on oeis.org

0, 0, 2, 12, 64, 320, 1568, 7616, 36864, 178176, 860672, 4156416, 20070400, 96911360, 467935232, 2259402752, 10909384704, 52675215360, 254338531328, 1228055248896, 5929575645184, 28630524624896
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A000129, A002203, A006668.

Programs

  • Magma
    [Floor(((2+Sqrt(8))^n+(2-Sqrt(8))^n-2^(n+1))/8): n in [0..30] ]; // Vincenzo Librandi, Aug 20 2011
  • Mathematica
    Table[2^(n-3)*(LucasL[n, 2] - 2), {n, 0, 20}] (* Vladimir Reshetnikov, Oct 07 2016 *)

Formula

a(n) = ((2+sqrt(8))^n+(2-sqrt(8))^n-2^(n+1))/8; E.g.f.: exp(2x)(sinh(sqrt(2)*x))^2/2 = (exp(x)*sinh(sqrt(2)*x)/sqrt(2))^2. - Paul Barry, May 16 2003
G.f.: 2*x^2 / ( (2*x-1)*(4*x^2+4*x-1) ). - R. J. Mathar, Nov 24 2012
a(n) = 2^(n-3)*(A002203(n) - 2). - Vladimir Reshetnikov, Oct 07 2016

A084150 A Pell related sequence.

Original entry on oeis.org

0, 0, 1, 3, 14, 50, 199, 749, 2892, 11028, 42301, 161799, 619706, 2372006, 9081955, 34767953, 133109592, 509594856, 1950956857, 7469077707, 28594853414, 109473250778, 419110475455, 1604533706357, 6142840740900, 23517417426300
Offset: 0

Views

Author

Paul Barry, May 16 2003

Keywords

Comments

Binomial transform of expansion of (sinh(sqrt(2)x))^2/4 = (0, 0, 1, 0, 8, 0, 64, ...). Inverse binomial transform of A006668.

Links

Crossrefs

Cf. A006646, A006668, A015519, A084058.

Programs

  • Magma
    [n le 3 select Floor((n-1)/2) else 3*Self(n-1) +5*Self(n-2) -7*Self(n-3): n in [1..41]]; // G. C. Greubel, Oct 11 2022
  • Mathematica
    LinearRecurrence[{3,5,-7}, {0,0,1}, 41] (* G. C. Greubel, Oct 11 2022 *)
  • SageMath
    A084058 = BinaryRecurrenceSequence(2,7,1,1)
def A084150(n): return (A084058(n) - 1)/8
[A084150(n) for n in range(41)] # G. C. Greubel, Oct 11 2022

Formula

a(n) = ( (1+sqrt(8))^n + (1-sqrt(8))^n - 2 )/16.
E.g.f.: (1/4)*exp(x)*( sinh(sqrt(2)*x) )^2.
G.f.: x^2 / ( (1-x)*(1-2*x-7*x^2) ). - R. J. Mathar, Feb 05 2011
a(n) = (A015519(n) - A015519(n-1) - 1)/8 = (A084058(n) - 1)/8. - G. C. Greubel, Oct 11 2022

A084151 Binomial transform of a Pell convolution.

Original entry on oeis.org

0, 0, 1, 9, 62, 390, 2359, 14007, 82412, 482652, 2820061, 16457397, 95983370, 559619970, 3262267891, 19015581699, 110836005272, 646014798840, 3765295834489, 21945889348257, 127910427675542, 745517838966462
Offset: 0

Views

Author

Paul Barry, May 16 2003

Keywords

Comments

Binomial transform of A006668. Second binomial transform of A084150.

Links

Crossrefs

Cf. A000244, A001541, A006668, A084150.

Programs

  • Magma
    [(Evaluate(ChebyshevFirst(n), 2) -3^n)/8: n in [0..40]]; // G. C. Greubel, Oct 11 2022
  • Mathematica
    LinearRecurrence[{9,-19,3},{0,0,1},30] (* Harvey P. Dale, Jun 06 2021 *)
  • SageMath
    [(chebyshev_T(n, 3) - 3^n)/8 for n in range(41)] # G. C. Greubel, Oct 11 2022

Formula

a(n) = ( (3+sqrt(8))^n + (3-sqrt(8))^n - 2*3^n )/16.
E.g.f.: (1/4)*exp(3*x)*( sinh(sqrt(2)*x) )^2.
G.f.: x^2 / ( (1-3*x)*(1-6*x+x^2) ). - R. J. Mathar, Sep 27 2012
a(n) = (A001541(n) - 3^n)/8. - R. J. Mathar, Sep 27 2012
a(n) = (1/8)*(ChebyshevT(n, 3) - 3^n) = (A001541(n) - A000244(n))/8. - G. C. Greubel, Oct 11 2022
Showing 1-3 of 3 results.

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

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