|
| |
|
|
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].
|
|
0
|
|
|
|
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
|
|
|
|
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 (mathar(AT)strw.leidenuniv.nl), 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 (paoloplava(AT)gmail.com), 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
|
|
|
|
KEYWORD
|
nonn,changed
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
