A238244 A recursive sequence: a(n) = Fibonacci(n)*a(n-1) + 3.

1, 4, 11, 36, 183, 1467, 19074, 400557, 13618941, 749041758, 66664716465, 9599719170963, 2236734566834382, 843248931696562017, 514381848334902830373, 507694884306549093578154, 810788730237558902444311941, 2095078078933852203916102055547

Offset: 1

Author

Vaclav Kotesovec, Feb 20 2014

Keywords

• nonn

Comments

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).

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..98

Crossrefs

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

Programs

Mathematica

RecurrenceTable[{a[n]==Fibonacci[n]*a[n-1]+3,a[1]==1},a,{n,1,20}]
nxt[{n_,a_}]:={n+1,a*Fibonacci[n+1]+3}; NestList[nxt,{1,1},20][[;;,2]] (* Harvey P. Dale, Sep 04 2024 *)

Formula

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).