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.

A097134 a(n) = 3*Fibonacci(2*n) + 0^n.

Original entry on oeis.org

1, 3, 9, 24, 63, 165, 432, 1131, 2961, 7752, 20295, 53133, 139104, 364179, 953433, 2496120, 6534927, 17108661, 44791056, 117264507, 307002465, 803742888, 2104226199, 5508935709, 14422580928, 37758807075, 98853840297, 258802713816
Offset: 0

Views

Author

Paul Barry, Jul 26 2004

Keywords

Comments

Binomial transform of A097133.
Image of 1/(1-3x) under the mapping g(x)->g(x/(1+x^2)). - Paul Barry, Jan 16 2005

Links

Crossrefs

Cf. A000045.

Programs

Formula

G.f.: (1+x^2)/(1-3*x+x^2).
a(n) = 3*a(n-1) - a(n-2) for n > 2.
a(n) = Sum_{k=0..n} binomial(n, k)*(3*Fibonacci(k)+(-1)^k).
a(n) = A097135(2*n).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k-1,k)*(-1)^k*3^(n-2*k). - Paul Barry, Jan 16 2005
a(n) = Fibonacci(n+2)^2 - Fibonacci(n-2)^2. - Gary Detlefs, Dec 03 2010
a(n) = Fibonacci(6*n) - 5*Fibonacci(2*n)^3 for n > 0. - Gary Detlefs, Oct 18 2011
E.g.f.: 1 + 6*exp(3*x/2)*sinh(sqrt(5)*x/2)/sqrt(5). - Stefano Spezia, Aug 19 2019

A097136 a(n) = 3*Fibonacci(2*n) + 1.

Original entry on oeis.org

1, 4, 10, 25, 64, 166, 433, 1132, 2962, 7753, 20296, 53134, 139105, 364180, 953434, 2496121, 6534928, 17108662, 44791057, 117264508, 307002466, 803742889, 2104226200, 5508935710, 14422580929, 37758807076, 98853840298, 258802713817, 677554301152
Offset: 0

Views

Author

Paul Barry, Jul 26 2004

Keywords

Comments

Binomial transform of A097135.

Links

Crossrefs

Cf. A000045.

Programs

  • Mathematica
    Table[3*Fibonacci[2n]+1,{n,0,30}] (* or *) LinearRecurrence[{4,-4,1},{1,4,10},30] (* Harvey P. Dale, May 25 2018 *)
  • PARI
    Vec((1-2*x^2)/((1-x)*(1-3*x+x^2)) + O(x^30)) \\ Colin Barker, Nov 02 2016

Formula

G.f.: (1-2*x^2) / ((1-x)*(1-3*x+x^2)).
a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3).
a(n) = 1+3((3+sqrt(5))/2)^n/sqrt(5)-3((3-sqrt(5))/2)^n/sqrt(5).
a(n) = A097132(2*n) = A097133(2*n).

A371843 a(n) = 5*Fibonacci(n) + (-1)^n.

Original entry on oeis.org

1, 4, 6, 9, 16, 24, 41, 64, 106, 169, 276, 444, 721, 1164, 1886, 3049, 4936, 7984, 12921, 20904, 33826, 54729, 88556, 143284, 231841, 375124, 606966, 982089, 1589056, 2571144, 4160201, 6731344, 10891546, 17622889, 28514436, 46137324, 74651761, 120789084, 195440846
Offset: 0

Views

Author

Paul Curtz, Apr 08 2024

Keywords

Examples

			a(3) = 2*4 + 1 = 9. Also a(3) = -1 + 10*1 = 9.

Links

Crossrefs

Cf. A000045, A022088, A033999, A097133.

Programs

  • Mathematica
    LinearRecurrence[{0, 2, 1}, {1, 4, 6}, 50] (* Amiram Eldar, Apr 11 2024 *)

Formula

a(n) = a(n-2) + A022088(n-1).
a(n) = 2*a(n-2) + a(n-3).
a(n) = A022088(n) + A033999(n).
a(n) = - a(n-3) + 10*A000045(n-1) for n >= 3.
G.f.: (1+2*x)^2/((1+x)*(1-x-x^2)). - Joerg Arndt, Apr 13 2024
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.