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%I A225799 #42 Nov 02 2024 06:49:35
%S A225799 0,11,143,3058,55341,1052755,19717984,371084087,6973353387,
%T A225799 131101759514,2464418392865,46327530894271,870879506447808,
%U A225799 16371134451297043,307750614069672631,5785211638097121890,108752568228856901349,2044371455527726003547,38430858858805840293152
%N A225799 a(n) = Sum_{k=0..n} binomial(n,k) * 10^(n-k) * Fibonacci(n+k).
%C A225799 This sequence is part of a family of Fibonacci-like sequences, where:
%C A225799 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.
%C A225799 A recurrence relation for a(n) (m not equal to zero) is:
%C A225799 a(n) = (m+3)*a(n-1) + (m^2+m-1)*a(n-2); a(0)=0, a(1)=m+1.
%C A225799 Notable values of m include:
%C A225799   m = 1: Fibonacci(3n),
%C A225799   m = 0: Fibonacci(2n) (using recurrence relation only - the sum above is undefined for m=0),
%C A225799   m = -1: the zero sequence,
%C A225799   m = -2: (-1)*Fibonacci(n), or A152163(n+2).
%C A225799 For any value of m, the sequence gives a(n*k) divisible by a(n); n>=1, k>=1, m not equal to -1 (zero is not divisible by zero).
%C A225799 Equivalent sequences are given by: Sum_{k=0..n} binomial(n,k) * (m+1)^k * Fibonacci(k).
%C A225799 When these sequences are divided by m+1, we obtain the family of sequences A057088, A015553, A087567, A087579, A087584, A087603, and so on.
%C A225799 Another interesting value of m, m = -3, gives a(2n-1)= -2 * 5^(n-1); a(2n)=0.
%H A225799 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (13,109).
%F A225799 a(n) = ((13 + 11*sqrt(5))^n - (13 - 11*sqrt(5))^n)/(2^n*sqrt(5)).
%F A225799 a(n) = 13*a(n-1) + 109*a(n-2); a(0)=0, a(1)=11.
%F A225799 G.f.: 11*x*/(1 - 13*x - 109*x^2). - Corrected by _Georg Fischer_, May 10 2019
%t A225799 Table[Sum[Binomial[n, k]*10^(n - k)*Fibonacci[n + k], {k, 0, n}], {n, 0, 25}]
%t A225799 FullSimplify[Table[((13 + 11 Sqrt[5])^n - (13 - 11 Sqrt[5])^n)/(2^n Sqrt[5]), {n, 0, 25}]]
%t A225799 LinearRecurrence[{13,109},{0,11},30] (* _Harvey P. Dale_, Jul 31 2018 *)
%Y A225799 Cf. A000045, A027941, A152163, A014445, A057088, A015553, A087567, A087579, A087584, A087603.
%K A225799 nonn,easy
%O A225799 0,2
%A A225799 _John Molokach_, Jul 27 2013

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