OFFSET
0,2
COMMENTS
This sequence is part of a family of Fibonacci-like sequences, where:
sum({k=0..n) binomial(n,k)*m^(n-k)*Fibonacci(n+k)) produces a sequence whose terms are divisible by (m+1); m>=1.
A recurrence relation for a(n) (m not equal to zero) is:
a(n) = (m+3)*a(n-1) + (m^2+m-1)*a(n-2); a(0)=0, a(1)=m+1.
Notable values of m include:
m = 1: Fibonacci(3n),
m = 0: Fibonacci(2n) (using recurrence relation only - the sum above is undefined for m=0),
m = -1: the zero sequence,
m = -2: (-1)*Fibonacci(n), or A152163(n+2).
For any value of m, the sequence gives a(nk) divisible by a(n); n>=1, k>=1, m not equal to -1 (zero is not divisible by zero).
Equivalent sequences are given by: sum_{k=0..n} binomial(n,k) * (m+1)^k * Fibonacci(k).
When these sequences are divided by m+1, we obtain the family of sequences A057088, A015553, A087567, A087579, A087584, A087603, and so on.
Another interesting value of m, m = -3, gives a(2n-1)= -2 * 5^(n-1); a(2n)=0.
LINKS
FORMULA
a(n) = ((13 + 11*sqrt(5))^n - (13 - 11*sqrt(5))^n)/(2^n*sqrt(5)).
a(n) = 13*a(n-1) + 109*a(n-2); a(0)=0, a(1)=11.
G.f.: 11*x*/(1 - 13*x - 109*x^2). - Corrected by Georg Fischer, May 10 2019
MATHEMATICA
Table[Sum[Binomial[n, k]*10^(n - k)*Fibonacci[n + k], {k, 0, n}], {n, 0, 25}]
FullSimplify[Table[((13 + 11 Sqrt[5])^n - (13 - 11 Sqrt[5])^n)/(2^n Sqrt[5]), {n, 0, 25}]]
LinearRecurrence[{13, 109}, {0, 11}, 30] (* Harvey P. Dale, Jul 31 2018 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
John Molokach, Jul 27 2013
STATUS
approved