|
Search: id:A158610
|
|
|
| A158610 |
|
A sequence from a vector Markov with the matrix:t=4;M={{0,t},{t,1/t}} and characteristic polynomial :x^2-x/t-t^2. |
|
+0
1
|
|
| 1, 16, 272, 4368, 74000, 1192208, 20136208, 325341456, 5480210704, 88767623440, 1491701563664, 24216213164304, 406091813462288, 6605442383524112, 110564946629869840, 1801558196812042512, 30106184534058721552
(list; graph; listen)
|
|
|
OFFSET
|
0,2
|
|
|
COMMENT
|
Quadratic equation associated with group [3,3,5]
which instead of t=phi uses Integer t.
Phi(t)=(1+Sqrt[1+4*t^4])/(2*t);
Phi_star(t)=(1-Sqrt[1+4*t^4])/(2*t);
t=1:Phi(1)=(1+Sqrt[5])/2;
t=2:Phi(2)=(1 + Sqrt[65])/4;
t=3:Phi(3)=(1+5*Sqrt[13))/6;
t=4;Phi(4)=(1+5*Sqrt[41])/8.
General rule is:
Phi(t)+Phi_star(t)=1/t.
Table[(x /. NSolve[CharacteristicPolynomial[{{0, t}, {t, 1/t}}, x] == 0, x][[2]])
+ (x /. NSolve[CharacteristicPolynomial[{{0, t}, {t, 1/t}}, x] == 0, x][[1]])
- 1/ t, {t, 1, 20}]->zero
|
|
REFERENCES
|
H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover, NY, 1973,page 221.
|
|
FORMULA
|
t=4;M={{0,t},{t,1/t}};
and characteristic polynomial :x^2-x/t-t^2;
v(0)={1,1);v(n)=M.v(n-1);
out_(n)=t^n*v(n)[[1]]
a(n)=a(n-1)+256*a(n-2), a(0)=1, a(1)=16 . G.f.: (1+15x)/(1-x-256*x^2). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 26 2009]
a(n)=(1/2)*{[(1/2)-(5/2)*sqrt(41)]^n+[(1/2)+(5/2)*sqrt(41)]^n}+(31/410)*sqrt(41)*{[(1/2)+(5/2)*sqrt(41)]^n-[(1/2)-(5/2)*sqrt(41)]^n}, with n>=0 [From Paolo P. Lava (ppl(AT)spl.at), Mar 30 2009]
|
|
MATHEMATICA
|
Clear[M, v, t, n];
M = {{0, t}, {t, 1/t}};
v[0] = {1, 1};
v[n_] := v[n] = M.v[n - 1];
t = 4;
a = Table[t^n*v[n][[1]], {n, 0, 30}]
|
|
CROSSREFS
|
Sequence in context: A158574 A000487 A002303 this_sequence A004382 A038758 A027776
Adjacent sequences: A158607 A158608 A158609 this_sequence A158611 A158612 A158613
|
|
KEYWORD
|
nonn,uned
|
|
AUTHOR
|
Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 22 2009
|
|
|
Search completed in 0.002 seconds
|
