|
Search: id:A106570
|
|
|
| A106570 |
|
First entry of the vector (M^n)v, where M is the 2 X 2 matrix [[0,3],[1,4]] and v is the column vector [0,1]. |
|
+0
1
|
|
| 0, 3, 12, 57, 264, 1227, 5700, 26481, 123024, 571539, 2655228, 12335529, 57307800, 266237787, 1236874548, 5746211553, 26695469856, 124020514083, 576168465900, 2676735405849, 12435447021096, 57771994301931, 268394318271012
(list; graph; listen)
|
|
|
OFFSET
|
0,2
|
|
|
COMMENT
|
Real Pisot roots (the eigenvalues of M): 2-sqrt(7)=-0.645751, 2+sqrt(7)=4.64575. a(n)=3*A015530(n).
|
|
FORMULA
|
a(n)=first entry of v[n], where v[n]=Mv[n-1], M is the 2 X 2 matrix [[0, 3], [1, 4]] and v[0] is the column vector [0,1]. G.f.=3x/(1-4x-3x^2). a(n)=4a(n-1)+3a(n-2); a(0)=0, a(1)=3.
a(n)=-(3/14)*[2-sqrt(7)]^n*sqrt(7)+(3/14)*sqrt(7)*[2+sqrt(7)]^n, with n>=0 [From Paolo P. Lava (ppl(AT)spl.at), Oct 07 2008]
|
|
MAPLE
|
a[0]:=0: a[1]:=3: for n from 2 to 24 do a[n]:=4*a[n-1]+3*a[n-2] od: seq(a[n], n=0..24);
|
|
MATHEMATICA
|
M = {{0, 3}, {1, 4}} v[1] = {0, 1} v[n_] := v[n] = M.v[n - 1] a = Table[v[n][[1]], {n, 1, 50}]
|
|
CROSSREFS
|
Equals 3*A015530.
Sequence in context: A009656 A020016 A001277 this_sequence A027140 A110309 A101106
Adjacent sequences: A106567 A106568 A106569 this_sequence A106571 A106572 A106573
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
Roger L. Bagula (rlbagulatftn(AT)yahoo.com), May 30 2005
|
|
EXTENSIONS
|
Edited by N. J. A. Sloane (njas(AT)research.att.com), Apr 30 2006
|
|
|
Search completed in 0.002 seconds
|
