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.

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A084154 Binomial transform of sinh(x)*cosh(sqrt(2)*x).

Original entry on oeis.org

0, 1, 2, 10, 32, 116, 392, 1352, 4608, 15760, 53792, 183712, 627200, 2141504, 7311488, 24963200, 85229568, 290992384, 993509888, 3392055808, 11581202432, 39540700160, 135000393728, 460920178688, 1573679923200, 5372879343616
Offset: 0

Views

Author

Paul Barry, May 16 2003

Keywords

Links

Crossrefs

Cf. A084155.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1-2*x+2*x^2)/((1-2*x^2)*(1-4*x+2*x^2)))); // G. C. Greubel, Aug 16 2018
  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[x]Sinh[x]Cosh[Sqrt[2]x],{x,0, nn}], x] Range[0,nn]!] (* or *) LinearRecurrence[{4,0,-8,4},{0,1,2, 10}, 30] (* Harvey P. Dale, Jun 19 2016 *)
  • PARI
    x='x+O('x^30); concat([0], Vec(x*(1-2*x+2*x^2)/((1-2*x^2)*(1-4*x+2*x^2)))) \\ G. C. Greubel, Aug 16 2018

Formula

a(n) = 4*a(n-1) - 8*a(n-3) + 4*a(n-4), a(0)=0, a(1)=1, a(2)=2, a(3)=10.
a(n) = ((2+sqrt(2))^n + (2-sqrt(2))^n - sqrt(2)^n - (-sqrt(2))^n)/4.
G.f.: x*(1-2*x+2*x^2)/((1-2*x^2)*(1-4*x+2*x^2)).
E.g.f.: exp(x)*sinh(x)*cosh(sqrt(2)*x).

A084157 a(n) = 8*a(n-1) - 16*a(n-2) + 12*a(n-4) with a(0)=0, a(1)=1, a(2)=4, a(3)=22.

Original entry on oeis.org

0, 1, 4, 22, 112, 556, 2704, 13000, 62080, 295312, 1401664, 6644320, 31472896, 149017792, 705395968, 3338614912, 15800258560, 74772443392, 353840161792, 1674425579008, 7923565146112, 37494981225472, 177428889407488
Offset: 0

Views

Author

Paul Barry, May 16 2003

Keywords

Comments

Binomial transform of A084156.

Links

Crossrefs

Cf. A026150, A083881, A084155, A084156.
Cf. A002605, A003462, A080040.

Programs

  • Magma
    I:=[0,1,4,22]; [n le 4 select I[n] else 8*Self(n-1) -16*Self(n-2) +12*Self(n-4): n in [1..41]]; // G. C. Greubel, Oct 11 2022
  • Mathematica
    LinearRecurrence[{8,-16,0,12},{0,1,4,22},30] (* Harvey P. Dale, Feb 19 2017 *)
  • SageMath
    A083881 = BinaryRecurrenceSequence(6,-6,1,3)
A026150 = BinaryRecurrenceSequence(2,2,1,1)
def A084157(n): return (A083881(n) - A026150(n))/2
[A084157(n) for n in range(41)] # G. C. Greubel, Oct 11 2022

Formula

a(n) = (A083881(n) - A026150(n))/2.
a(n) = 8*a(n-1) - 16*a(n-2) + 12*a(n-4).
a(n) = ((3+sqrt(3))^n + (3-sqrt(3))^n - (1+sqrt(3))^n - (1-sqrt(3))^n)/4.
G.f.: x*(1-4*x+6*x^2)/((1-2*x-2*x^2)*(1-6*x+6*x^2)).
E.g.f.: exp(2*x)*sinh(x)*cosh(sqrt(3)*x).
From G. C. Greubel, Oct 11 2022: (Start)
a(2*n) = A003462(n)*A026150(2*n) = A003462(n)*A080040(2*n)/2.
a(2*n+1) = (1/2)*(3^(n+1)*A002605(2*n+1) - A026150(2*n+1)). (End)
Showing 1-2 of 2 results.

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