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A225799 Sum_{k=0..n} binomial(n,k) * 10^(n-k) * Fibonacci(n+k). 0
0, 11, 143, 3058, 55341, 1052755, 19717984, 371084087, 6973353387, 131101759514, 2464418392865, 46327530894271, 870879506447808, 16371134451297043, 307750614069672631, 5785211638097121890, 108752568228856901349, 2044371455527726003547, 38430858858805840293152 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..18.

Index entries for linear recurrences with constant coefficients, signature (13,109).

FORMULA

a(n) = ((13 + 11 sqrt(5))^n - (13 - 11 sqrt(5))^n)/(2^n sqrt(5)).

a(n) = 13a(n-1) + 109a(n-2); a(0)=0, a(1)=11.

G.f.: 11x/(-1+13x+109x^2).

MATHEMATICA

Table[Sum[Binomial[n, k]*10^(n - k)*Fibonacci[n + k], {k, 0, n}], {n, 25}]

FullSimplify[Table[((13 + 11 Sqrt[5])^n - (13 - 11 Sqrt[5])^n)/(2^n Sqrt[5]), {n, 25}]]

CROSSREFS

Cf. A000045, A027941, A152163, A014445, A057088, A015553, A087567, A087579, A087584, A087603.

Sequence in context: A214098 A015687 A051583 * A027771 A098310 A293610

Adjacent sequences:  A225796 A225797 A225798 * A225800 A225801 A225802

KEYWORD

nonn,easy

AUTHOR

John Molokach, Jul 27 2013

STATUS

approved

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Last modified February 19 11:04 EST 2018. Contains 299330 sequences. (Running on oeis4.)