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.

A098648 Expansion of (1-3*x)/(1 - 6*x + 4*x^2).

Original entry on oeis.org

1, 3, 14, 72, 376, 1968, 10304, 53952, 282496, 1479168, 7745024, 40553472, 212340736, 1111830528, 5821620224, 30482399232, 159607914496, 835717890048, 4375875682304, 22912382533632, 119970792472576, 628175224700928, 3289168178315264, 17222308171087872
Offset: 0

Views

Author

Paul Barry, Sep 18 2004

Keywords

Comments

Binomial transform of A001077. Second binomial transform of A084057. Third binomial transform of 1/(1-5*x^2). Let A=[1,1,1,1;3,1,-1,-3;3,-1,-1,3;1,-1,1,-1], the 4 X 4 Krawtchouk matrix. Then a(n)=trace((16(A*A`)^(-1))^n)/4.

Links

Crossrefs

Cf. A098647.

Programs

  • Mathematica
    a[n_]:=(MatrixPower[{{5,1},{1,1}},n].{{2},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
CoefficientList[Series[(1-3x)/(1-6x+4x^2),{x,0,30}],x] (* or *) LinearRecurrence[{6,-4},{1,3},31] (* Harvey P. Dale, Jun 06 2011 *)
Table[2^(n - 1) LucasL[2 n], {n, 0, 20}] (* Eric W. Weisstein, Mar 31 2017 *)
  • PARI
    Vec((1-3*x)/(1 - 6*x + 4*x^2) + O(x^25)) \\ Jinyuan Wang, Jul 24 2021

Formula

E.g.f.: exp(3*x)*cosh(sqrt(5)*x).
a(n) = ((3-sqrt(5))^n + (3+sqrt(5))^n)/2.
a(n) = 2*(3*a(n-1) - 2*a(n-2)). - Lekraj Beedassy, Oct 22 2004
a(n) = A084326(n+1) - 3*A084326(n). - R. J. Mathar, Nov 10 2013
a(n) = 2^(n-1)*Lucas(2*n) = 2^(n-1)*A005248(n), n>0. - Eric W. Weisstein, Mar 31 2017

A098646 Trace sequence of 3 X 3 Krawtchouk matrix.

Original entry on oeis.org

3, 2, 12, 8, 48, 32, 192, 128, 768, 512, 3072, 2048, 12288, 8192, 49152, 32768, 196608, 131072, 786432, 524288, 3145728, 2097152, 12582912, 8388608, 50331648, 33554432, 201326592, 134217728, 805306368, 536870912, 3221225472, 2147483648
Offset: 0

Views

Author

Paul Barry, Sep 18 2004

Keywords

Comments

Let A=[1,1,1;2,0,-2;1,-1,1], the 3 X 3 Krawtchouk matrix. Then a(n)=trace(A^n).

Links

Crossrefs

Cf. A074535, A098647.

Programs

Formula

G.f.: (3+2*x)/((1+2*x)*(1-2*x)).
a(n) = (-2)^n+2*2^n.
Recurrence: a(n) = 4a(n-2), a(0)=3, a(1)=2. - Ralf Stephan, Jul 17 2013
a(2n+1)=A081294(n+1). a(2n)=A002001(n+1). - R. J. Mathar, Nov 11 2013

A228843 a(n) = 4^n*A228842(n).

Original entry on oeis.org

2, 24, 448, 9216, 192512, 4030464, 84410368, 1767899136, 37027315712, 775510032384, 16242492571648, 340187179646976, 7124972786941952, 149227367389200384, 3125458558976524288, 65460453902527758336, 1371021545886168645632, 28715048051506270961664
Offset: 0

Views

Author

R. J. Mathar, Nov 10 2013

Keywords

Comments

Bhadouria et al. call this the 4-binomial transform of the 4-Lucas sequence.
Binomial transform of the binomial transform of the binomial transform of A087215.

Links

Programs

  • Mathematica
    LinearRecurrence[{24,-64},{2,24},20] (* Harvey P. Dale, Jul 04 2022 *)
  • PARI
    Vec(2*(1 - 12*x) / (1 - 24*x + 64*x^2 ) + O(x^30)) \\ Colin Barker, Sep 23 2017

Formula

G.f.: 2*( 1-12*x ) / ( 1-24*x+64*x^2 ).
a(n) = 2*A098647(n).
a(n) = A000302(n)*A228842(n). - Omar E. Pol, Nov 10 2013
From Colin Barker, Sep 23 2017: (Start)
a(n) = 24*a(n-1) - 64*a(n-2) for n>1.
a(n) = (12-4*sqrt(5))^n + (4*(3+sqrt(5)))^n.
(End)
Showing 1-3 of 3 results.

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