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A121364
Convolution of A066983 with the double Fibonacci sequence A103609.
1
0, 0, 1, 2, 3, 6, 10, 18, 29, 50, 81, 136, 220, 364, 589, 966, 1563, 2550, 4126, 6710, 10857, 17622, 28513, 46224, 74792, 121160, 196041, 317434, 513619, 831430, 1345282, 2177322, 3522981, 5701290, 9224881, 14927768, 24153636, 39083988, 63239221, 102327390
OFFSET
1,4
COMMENTS
The convolution of 1,0,1,1,1,3,3,7,9,17,25,... (A066983 with 1,0 added to the front) with "A double Fibonacci sequence" (A103609) is the Fibonacci sequence (A000045), with an extra initial 0.
FORMULA
a(n) = F(n) - D(n+1), where F is the Fibonacci sequence (A000045) and D is "A double Fibonacci sequence" (A103609).
G.f.: x^3*(1+x-x^2) / ((1-x-x^2)*(1-x^2-x^4)). - Colin Barker, Oct 13 2014
EXAMPLE
a(7)=10 because F(7)=13 and D(8)=3 and a(7)=F(7)-D(8).
MATHEMATICA
LinearRecurrence[{1, 2, -1, 0, -1, -1}, {0, 0, 1, 2, 3, 6}, 40] (* James C. McMahon, Oct 17 2024 *)
PROG
(PARI) concat([0, 0], Vec(-x^3*(x^2-x-1)/((x^2+x-1)*(x^4+x^2-1)) + O(x^100))) \\ Colin Barker, Oct 13 2014
(Magma)
A121364:= func< n | Fibonacci(n) - Fibonacci(Floor((n+1)/2)) >;
[A121364(n): n in [1..70]]; // G. C. Greubel, Oct 23 2024
(SageMath)
def A121364(n): return fibonacci(n) - fibonacci((n+1)//2)
[A121364(n) for n in range(1, 71)] # G. C. Greubel, Oct 23 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Graeme McRae, Jul 23 2006
EXTENSIONS
More terms from Colin Barker, Oct 13 2014
STATUS
approved