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 1-2 of 2 results.

A161731 Expansion of (1-3*x)/(1-8*x+14*x^2).

Original entry on oeis.org

1, 5, 26, 138, 740, 3988, 21544, 116520, 630544, 3413072, 18476960, 100032672, 541583936, 2932214080, 15875537536, 85953303168, 465368899840, 2519604954368, 13641675037184, 73858930936320, 399887996969984
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Jun 17 2009

Keywords

Comments

Fourth binomial transform of A016116.
Inverse binomial transform of A161734. Binomial transform of A086351. - R. J. Mathar, Jun 18 2009

Links

Crossrefs

Cf. A016116, A086351, A161734.

Programs

  • Magma
    [Floor(((2+Sqrt(2))*(4+Sqrt(2))^n+(2-Sqrt(2))*(4-Sqrt(2))^n)/4): n in [0..30]]; // Vincenzo Librandi, Aug 18 2011
  • Mathematica
    CoefficientList[Series[(1-3x)/(1-8x+14x^2),{x,0,30}],x] (* or *) LinearRecurrence[{8,-14},{1,5},30] (* Harvey P. Dale, Feb 29 2024 *)
  • PARI
    F=nfinit(x^2-2); for(n=0, 20, print1(nfeltdiv(F, ((2+x)*(4+x)^n+(2-x)*(4-x)^n), 4)[1], ",")) \\ Klaus Brockhaus, Jun 19 2009

Formula

a(n) = ((2+sqrt(2))*(4+sqrt(2))^n+(2-sqrt(2))*(4-sqrt(2))^n)/4.
a(n) = 8*a(n-1)-14*a(n-2). - R. J. Mathar, Jun 18 2009
a(n) = A081180(n+1) -3*A081180(n). - R. J. Mathar, Jul 19 2012

Extensions

Extended by R. J. Mathar and Klaus Brockhaus, Jun 18 2009
Edited by Klaus Brockhaus, Jul 05 2009

A163458 a(n) = 12*a(n-1) - 34*a(n-2) for n > 1; a(0) = 1, a(1) = 7.

Original entry on oeis.org

1, 7, 50, 362, 2644, 19420, 143144, 1057448, 7822480, 57916528, 429034016, 3179246240, 23563798336, 174671207872, 1294885351040, 9599803144832, 71171535802624, 527665122707200, 3912149255197184, 29005176890321408
Offset: 0

Views

Author

Klaus Brockhaus, Jul 28 2009

Keywords

Comments

Binomial transform of A161734. Inverse binomial transform of A163459.

Links

Crossrefs

Cf. A161734, A163459.

Programs

  • Magma
    [ n le 2 select 6*n-5 else 12*Self(n-1)-34*Self(n-2): n in [1..20] ];
  • Mathematica
    LinearRecurrence[{12,-34},{1,7},40] (* Harvey P. Dale, Aug 25 2015 *)
  • PARI
    Vec((1-5*x)/(1-12*x+34*x^2) + O(x^50)) \\ G. C. Greubel, Dec 24 2016

Formula

a(n) = ((2+sqrt(2))*(6+sqrt(2))^n + (2-sqrt(2))*(6-sqrt(2))^n)/4.
G.f.: (1-5*x)/(1-12*x+34*x^2).
E.g.f.: (1/2)*exp(6*x)*(sqrt(2)*sinh(sqrt(2)*x) + 2*cosh(sqrt(2)*x)). - G. C. Greubel, Dec 24 2016
Showing 1-2 of 2 results.

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