A109545 a(n) = 2*a(n-1) + a(n-2) + a(n-3).

1, 1, 2, 6, 15, 38, 97, 247, 629, 1602, 4080, 10391, 26464, 67399, 171653, 437169, 1113390, 2835602, 7221763, 18392518, 46842401, 119299083, 303833085, 773807654, 1970747476, 5019135691, 12782826512, 32555536191, 82913034585

Offset: 0

Author

Roger L. Bagula, Jun 20 2005

Keywords

• nonn
• easy

Links

• David Garth and Kevin G. Hare, Comments on the spectra of Pisot numbers, J. Number Theory 121 (2006), 187-203.
• Index entries for linear recurrences with constant coefficients, signature (2, 1, 1).

Programs

Mathematica

a = 2; b = -1; M = {{0, 1, 0, 0, 0}, { a - 2, a - 2, a - 2 - b, a - 2 - b, 0}, {1, 1, 1, 1, 0}, {0, 1, 1, 0, 0}, {0, 0, 0, 1, 1}} v[1] = {1, 1, 1, 1, 1} v[n_] := v[n] = M.v[n - 1] a0 = Table[Abs[v[n][[1]]], {n, 1, 50}]
LinearRecurrence[{2,1,1},{1,1,2},30] (* Harvey P. Dale, Aug 05 2015 *)
Lucas := 1 + x (1 + 2 x)/(1 - x - x^2); (* InvertTransform defined in A052987 *)
InvertTransform[Lucas, 28] (* Peter Luschny, Jan 10 2019 *)

Formula

lim_{n-> infinity} a(n)/a(n-1)= 2.54682...

G.f.: (1-x-x^2)/(1-2*x-x^2-x^3). [Sep 28 2009]

a(n) = A077939(n)-A116413(n-1).

G.f.: (-1+x+x^2)/(-1+2*x+x^2+x^3). a(n) = A077997(n)-A077939(n-2). [From R. J. Mathar, Sep 27 2009]

Extensions

Definition replaced by recurrence by the Associate Editors of the OEIS, Sep 28 2009