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A117110 The (1,1)-entry of the vector v[n]=Mv[n-1], where M is the 3 x 3 matrix [[0,-1/r,r],[ -1/r,-2/r,1],[r,1,2+2/r]], r being the golden ratio and v[0] is the column matrix [0,1,1]. 1
0, 1, 7, 22, 100, 376, 1552, 6112, 24640, 98176, 393472, 1572352, 6292480, 25163776, 100667392, 402644992, 1610629120, 6442418176, 25769869312, 103079084032, 412317122560, 1649266917376, 6597070815232, 26388276969472 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Characteristic polynomial of the matrix M is x(x^2-2x-8).

LINKS

Table of n, a(n) for n=0..23.

FORMULA

Recurrence relation: a(n)=2a(n-1)+8a(n-2) for n>=3; a(0)=0,a(1)=1,a(2)=7.

O.g.f.: -x*(1+5*x)/((2*x+1)*(4*x-1)). - R. J. Mathar, Dec 05 2007

a(n)=-(1/2)*(-2)^(n-1)+(3/2)*4^(n-1)-5/8*[C(2*n,n) mod 2], with n>=0 [From Paolo P. Lava, Oct 07 2008]

MAPLE

a[0]:=0: a[1]:=1: a[2]:=7: for n from 3 to 26 do a[n]:=2*a[n-1]+8*a[n-2] od: seq(a[n], n=0..26);

with(linalg): r:=(1+sqrt(5))/2: M:=matrix(3, 3, [0, -1/r, r, -1/r, -2/r, 1, r, 1, 2+2/r]): v[0]:=matrix(3, 1, [0, 1, 1]): for n from 1 to 26 do v[n]:=simplify(multiply(M, v[n-1])) od: seq(simplify(rationalize(v[n][1, 1])), n=0..26);

CROSSREFS

Sequence in context: A041913 A231339 A222178 * A027838 A194996 A000835

Adjacent sequences:  A117107 A117108 A117109 * A117111 A117112 A117113

KEYWORD

nonn

AUTHOR

Roger L. Bagula, Apr 18 2006

EXTENSIONS

Edited by N. J. A. Sloane, May 13 2006

STATUS

approved

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Last modified June 16 06:46 EDT 2019. Contains 324145 sequences. (Running on oeis4.)