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.

A099255 Expansion of g.f. (7+6*x-6*x^2-3*x^3)/((x^2+x-1)*(x^2-x-1)).

Original entry on oeis.org

7, 6, 15, 15, 38, 39, 99, 102, 259, 267, 678, 699, 1775, 1830, 4647, 4791, 12166, 12543, 31851, 32838, 83387, 85971, 218310, 225075, 571543, 589254, 1496319, 1542687, 3917414, 4038807, 10255923, 10573734, 26850355, 27682395, 70295142, 72473451
Offset: 0

Views

Author

Creighton Dement, Oct 09 2004

Keywords

Comments

One of two sequences involving the Lucas/Fibonacci numbers.
This sequence consists of pairs of numbers more or less close to each other with "jumps" in between pairs. "pos((Ex)^n)" sums up over all floretion basis vectors with positive coefficients for each n. The following relations appear to hold: a(2n) - (a(2n-1) + a(2n-2)) = 2*Luc(2n) a(2n+1) - a(2n) = Fib(2n), apart from initial term a(2n+1)/a(2n-1) -> 2 + golden ratio phi a(2n)/a(2n-2) -> 2 + golden ratio phi An identity: (1/2)a(n) - (1/2)A099256(n) = ((-1)^n)A000032(n)

Links

Crossrefs

Cf. A099256, A000032.

Programs

  • Mathematica
    LinearRecurrence[{0,3,0,-1},{7,6,15,15},40] (* Harvey P. Dale, Dec 29 2012 *)

Formula

a(n) = 2*pos((Ex)^n)
a(0) = 7, a(1) = 6, a(2) = a(3) = 15, a(n+4) = 3a(n+2) - a(n).
a(2n) = A022097(2n+1), a(2n+1) = A022086(2n+3).
a(n) = A061084(n+1)+A013655(n+2). [R. J. Mathar, Nov 30 2008]

Extensions

More terms from Creighton Dement, Apr 19 2005

A122012 G.f.: x^2*(3+3*x-2*x^2)/ ( (x^2-x-1) * (x^2+x-1)).

Original entry on oeis.org

0, 3, 3, 7, 9, 18, 24, 47, 63, 123, 165, 322, 432, 843, 1131, 2207, 2961, 5778, 7752, 15127, 20295, 39603, 53133, 103682, 139104, 271443, 364179, 710647, 953433, 1860498, 2496120, 4870847, 6534927, 12752043, 17108661, 33385282, 44791056
Offset: 1

Views

Author

Roger L. Bagula, Sep 11 2006

Keywords

Links

Programs

  • Mathematica
    M = {{0, 1, 1, 0}, {1, 0, 0, 1}, {1, 0, 0, 0}, {0, 1, 0, 0}} v[1] = {0, 1, 2, 3} v[n_] := v[n] = M.v[n - 1] a = Table[Floor[v[n][[1]]], {n, 1, 50}]

Formula

a(n)= 3*a(n-2) -a(n-4).
a(n)= A022096(n-1)/2 + (-1)^n*A000045(n-2)/2, n >1.
a(2n+1)= A099256(2n-2), n>=1. [Mar 27 2010]

Extensions

Definition replaced with generating function by the Assoc. Eds. of the OEIS, Mar 27 2010
Showing 1-2 of 2 results.

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