A238244 - OEIS

%I #9 Sep 04 2024 17:30:51

%S 1,4,11,36,183,1467,19074,400557,13618941,749041758,66664716465,

%T 9599719170963,2236734566834382,843248931696562017,

%U 514381848334902830373,507694884306549093578154,810788730237558902444311941,2095078078933852203916102055547

%N A recursive sequence: a(n) = Fibonacci(n)*a(n-1) + 3.

%C Generally, sequence a(n) = Fibonacci(n)*a(n-1) + p, with a(1)=1 and fixed p, is asymptotic to  c(p) * ((1+sqrt(5))/2)^(n^2/2+n/2) / 5^(n/2), where constant c(p) = A062073 * (p*A101689 - p + 1).

%H Harvey P. Dale, <a href="/A238244/b238244.txt">Table of n, a(n) for n = 1..98</a>

%F a(n) ~ c * ((1+sqrt(5))/2)^(n^2/2+n/2) / 5^(n/2), where c = A062073 * (3*A101689-2) = 7.4996979520811499717534... is product of Fibonacci factorial constant (see A062073) and -2+3*sum_{n>=1} 1/product(A000045(k), k=1..n).

%t RecurrenceTable[{a[n]==Fibonacci[n]*a[n-1]+3,a[1]==1},a,{n,1,20}]

%t nxt[{n_,a_}]:={n+1,a*Fibonacci[n+1]+3}; NestList[nxt,{1,1},20][[;;,2]] (* _Harvey P. Dale_, Sep 04 2024 *)

%Y Cf. A176343, A238243, A003266, A101689, A062073, A000045, A139339.

%K nonn

%O 1,2

%A _Vaclav Kotesovec_, Feb 20 2014