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.

A049651 a(n) = (F(3*n+1) - 1)/2, where F=A000045 (the Fibonacci sequence).

Original entry on oeis.org

0, 1, 6, 27, 116, 493, 2090, 8855, 37512, 158905, 673134, 2851443, 12078908, 51167077, 216747218, 918155951, 3889371024, 16475640049, 69791931222, 295643364939, 1252365390980, 5305104928861, 22472785106426, 95196245354567, 403257766524696, 1708227311453353, 7236167012338110
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

This is the sequence A(0,1;4,1;2) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - Wolfdieter Lang, Oct 18 2010
For n>0, a(n) is the least number whose greedy Fibonacci-union-Lucas representation (as at A214973), has n terms. - Clark Kimberling, Oct 23 2012

References

  • A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 24.

Links

Crossrefs

Cf. A000045, A033887, A077259, A214973.
Pairwise sums of A049652.

Programs

Formula

From Ralf Stephan, Jan 23 2003: (Start)
a(n) = 4*a(n-1) + a(n-2) + 2, a(0)=0, a(1)=1.
G.f.: x*(1+x)/((1-x)*(1-4*x-x^2)).
a(n) is asymptotic to -1/2+(sqrt(5)+5)/20*(sqrt(5)+2)^n. (End)
a(n+1) = F(2) + F(5) + F(8) + ... + F(3n+2).
a(n) = 5*a(n-1) - 3*a(n-2) - a(n-3), a(0)=0, a(1)=1, a(2)= 6. Observation by G. Detlefs. See the W. Lang link. - Wolfdieter Lang, Oct 18 2010
a(2*n) = A077259(2*n); a(2*n+1) = A294262(2*n+1). See "6 interlaced bisections" link. - Hermann Stamm-Wilbrandt, Apr 18 2019
E.g.f.: exp(x)*(exp(x)*(5*cosh(sqrt(5)*x) + sqrt(5)*sinh(sqrt(5)*x)) - 5)/10. - Stefano Spezia, May 24 2024

A232970 Expansion of (1-3*x)/(1-5*x+3*x^2+x^3).

Original entry on oeis.org

1, 2, 7, 28, 117, 494, 2091, 8856, 37513, 158906, 673135, 2851444, 12078909, 51167078, 216747219, 918155952, 3889371025, 16475640050, 69791931223, 295643364940, 1252365390981, 5305104928862, 22472785106427, 95196245354568, 403257766524697, 1708227311453354, 7236167012338111, 30652895360805796
Offset: 0

Views

Author

N. J. A. Sloane, Dec 05 2013

Keywords

Comments

For n > 2, a(n) is the number of tilings of (a 2 X (n+1) rectangle missing the top right and top left 1 X 1 cells) using 1 X 1 squares, dominoes and right trominoes. Compare with similar tiling sequences A001076 and A110679. - Greg Dresden and Yilin Zhu, Jul 10 2025

Links

Crossrefs

Cf. A001076, A110679.

Programs

  • Magma
    I:=[1,2,7]; [n le 3 select I[n] else 5*Self(n-1)- 3*Self(n-2)-Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 24 2017
  • Mathematica
    LinearRecurrence[{5, -3, -1}, {1, 2, 7}, 30] (* Vincenzo Librandi, Jun 24 2017 *)
CoefficientList[Series[(1-3x)/(1-5x+3x^2+x^3),{x,0,30}],x] (* Harvey P. Dale, Oct 19 2024 *)
  • PARI
    Vec((1-3*x)/(1-5*x+3*x^2+x^3) + O(x^30)) \\ Felix Fröhlich, Apr 15 2019
  • Sage
    [(fibonacci(3*n+1) +1)/2 for n in (0..30)] # G. C. Greubel, Apr 19 2019

Formula

a(n) = 5*a(n-1) - 3*a(n-2) - a(n-3). - N. J. A. Sloane, Jun 23 2017
a(n) = (Fibonacci(3*n+1) + 1)/2 = Sum_{k=0..n} Fibonacci(3*k-1). - Ehren Metcalfe, Apr 15 2019
a(2*n) = A294262(2*n); a(2*n+1) = A254627(2*n+2). See "6 interlaced bisections" link. - Hermann Stamm-Wilbrandt, Apr 18 2019
Showing 1-2 of 2 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.