|
| |
|
|
A106707
|
|
First entry of the vector (M^n)v, where M is the 2 X 2 matrix [[0,-1],[1,4]] and v is the column vector [0,1].
|
|
5
|
|
|
|
0, -1, -4, -15, -56, -209, -780, -2911, -10864, -40545, -151316, -564719, -2107560, -7865521, -29354524, -109552575, -408855776, -1525870529, -5694626340, -21252634831, -79315912984, -296011017105, -1104728155436, -4122901604639, -15386878263120, -57424611447841
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
a(n) is the first entry of v[n], where v[n]=Mv[n-1], M is the 2 X 2 matrix [[0, -1], [1, 4]] and v[0] is the column vector [0,1].
Real Pisot roots (the eigenvalues of M): 2-sqrt(3)=0.267949, 2+sqrt(3)=3.73205.
|
|
|
LINKS
|
G. C. Greubel, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (4,-1).
|
|
|
FORMULA
|
G.f.: -x/(1-4*x+x^2).
a(n) = 4*a(n-1) - a(n-2); a(0)=0, a(1)=-1.
a(n) = (1/6)*sqrt(3)*[2-sqrt(3)]^n - (1/6)*sqrt(3)*[2+sqrt(3)]^n, with n>=0. - Paolo P. Lava, Oct 06 2008
|
|
|
MAPLE
|
a[0]:=0: a[1]:=-1: for n from 2 to 27 do a[n]:=4*a[n-1]-a[n-2] od: seq(a[n], n=0..27);
|
|
|
MATHEMATICA
|
M = {{0, -1}, {1, 4}} v[1] = {0, 1} v[n_] := v[n] = M.v[n - 1] a = Table[Abs[v[n][[1]]], {n, 1, 50}]
Table[Round[(1/6)*Sqrt[3]*(2 - Sqrt[3])^n - (1/6)*Sqrt[3]*(2 + Sqrt[3])^n ], {n, 0, 50}] (* G. C. Greubel, Feb 05 2018 *)
LinearRecurrence[{4, -1}, {0, -1}, 30] (* Harvey P. Dale, Nov 01 2019 *)
|
|
|
PROG
|
(PARI) x='x+O('x^30); Vec(-x/(1-4*x+x^2)) \\ G. C. Greubel, Feb 05 2018
(MAGMA) I:=[0, -1]; [n le 2 select I[n] else 4*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Feb 05 2018
|
|
|
CROSSREFS
|
Cf. A001076, A001353.
Sequence in context: A242495 A221859 A001353 * A125905 A195503 A010905
Adjacent sequences: A106704 A106705 A106706 * A106708 A106709 A106710
|
|
|
KEYWORD
|
sign,easy
|
|
|
AUTHOR
|
Roger L. Bagula, May 30 2005
|
|
|
EXTENSIONS
|
Edited by N. J. A. Sloane, Apr 30 2006
|
|
|
STATUS
|
approved
|
| |
|
|
