OFFSET
0,2
COMMENTS
Similar sequences with the formula h*Fibonacci(n+k) + Fibonacci(n+k-h):
h= 1, k=-1: A000045;
h= 2, k= 1: A013655;
h= 5, k= 1: A022113;
h= 6, k= 2: A022125;
h= 7, k= 3: A097657;
h= 9, k= 3: 10, 32, 42, 74, 116, 190, 306, 496, 802, ... = 2*A022140;
h=10, k= 3: 33, 22, 55, 77, 132, 209, 341, 550, 891, ... = 11*A013655;
h=11, k= 3: this sequence.
LINKS
Indranil Ghosh, Table of n, a(n) for n = 0..4769
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (1,1).
FORMULA
G.f.: (1 + 45*x)/(1 - x - x^2).
a(n) = a(n-1) + a(n-2).
a(n) = a(i)*Fibonacci(n-i+1) + a(i-1)*Fibonacci(n-i). Examples:
for i= 3, a(3)=93, a(2)= 47: a(n) = 93*Fibonacci(n-2) + 47*Fibonacci(n-3);
for i=-1, a(-1)=45, a(-2)=-44: a(n) = 45*Fibonacci(n+2) - 44*Fibonacci(n+1).
Other formulae:
a(n) = 44*Fibonacci(n) + Fibonacci(n+2),
a(n) = 45*Fibonacci(n) + Fibonacci(n+1),
a(n) = 46*Fibonacci(n) + Fibonacci(n-1),
a(n) = 47*Fibonacci(n) - Fibonacci(n-2).
a(n) = ((91 + sqrt(5))*((1 + sqrt(5))/2)^n - (91 - sqrt(5))*((1 - sqrt(5))/2)^n)/sqrt(20).
MATHEMATICA
Table[11 Fibonacci[n + 3] + Fibonacci[n - 8], {n, 0, 40}]
LinearRecurrence[{1, 1}, {1, 46}, 36] (* or *) CoefficientList[Series[(1 + 45*x)/(1 - x - x^2) , {x, 0, 35}], x] (* Indranil Ghosh, Feb 22 2017 *)
PROG
(Magma) [11*Fibonacci(n+3)+Fibonacci(n-8): n in [0..40]];
(PARI) a(n) = 11*fibonacci(n+3) + fibonacci(n-8) \\ Indranil Ghosh, Feb 23 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Bruno Berselli, Feb 20 2017
STATUS
approved