OFFSET
0,2
COMMENTS
f satisfies the linear recursion f(n+1) = (t+2)*f(n)-t*f(n-1). For t=3 this gives a(n+1) = 5*a(n)-3*a(n-1).
Given the 3 X 3 matrix [1,1,1; 1,1,2; 1,1,3] = M, a(n) = term (1,1) in M^(n+1). - Gary W. Adamson, Aug 06 2010
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-3).
FORMULA
a(n+1) = (a(n)^2 + 3^n) / a(n-1).
From Philippe Deléham, Nov 14 2008: (Start)
G.f.: (1-2*x)/(1-5*x+3*x^2).
a(n) = Sum_{k, 0<=k<=n} A147703(n,k)*2^k. (End)
a(n) = (2^(-1-n)*((5-sqrt(13))^n*(-1+sqrt(13)) + (1+sqrt(13))*(5+sqrt(13))^n))/sqrt(13). - Colin Barker, Nov 26 2016
E.g.f.: exp(5*x/2)*(sqrt(13)*cosh(sqrt(13)*x/2) + sinh(sqrt(13)*x/2))/sqrt(13). - Stefano Spezia, Jul 09 2022
MAPLE
f := proc(n) if n=0 then 1 elif n=1 then t else sort(simplify((f(n-1)^2+t^(n-1))/f(n-2)), t) fi end; a := i->subs(t=3, f(i));
MATHEMATICA
a[0]=1; a[1]=3; a[n_] := a[n]=5a[n-1]-3a[n-2]; Array[a, 25, 0]
LinearRecurrence[{5, -3}, {1, 3}, 30] (* Harvey P. Dale, Jul 28 2013 *)
PROG
(PARI) Vec((1-2*x)/(1-5*x+3*x^2) + O(x^30)) \\ Colin Barker, Nov 26 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Victor Ufnarovski (ufn(AT)maths.lth.se), Apr 02 2003
STATUS
approved