A109545 - OEIS

%I #13 Jan 10 2019 22:54:42

%S 1,1,2,6,15,38,97,247,629,1602,4080,10391,26464,67399,171653,437169,

%T 1113390,2835602,7221763,18392518,46842401,119299083,303833085,

%U 773807654,1970747476,5019135691,12782826512,32555536191,82913034585

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

%H David Garth and Kevin G. Hare, <a href="http://dx.doi.org/10.1016/j.jnt.2006.02.003">Comments on the spectra of Pisot numbers</a>, J. Number Theory 121 (2006), 187-203.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2, 1, 1).

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

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

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

%F 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]

%t 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}]

%t LinearRecurrence[{2,1,1},{1,1,2},30] (* _Harvey P. Dale_, Aug 05 2015 *)

%t Lucas := 1 + x (1 + 2 x)/(1 - x - x^2); (* InvertTransform defined in A052987 *)

%t InvertTransform[Lucas, 28] (* _Peter Luschny_, Jan 10 2019 *)

%K nonn,easy

%O 0,3

%A _Roger L. Bagula_, Jun 20 2005

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