A056123 a(n) = 3*a(n-1) - a(n-2) with a(0)=1, a(1)=11.
1, 11, 32, 85, 223, 584, 1529, 4003, 10480, 27437, 71831, 188056, 492337, 1288955, 3374528, 8834629, 23129359, 60553448, 158530985, 415039507, 1086587536, 2844723101, 7447581767, 19498022200, 51046484833, 133641432299
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Tanya Khovanova, Recursive Sequences
- Index entries for linear recurrences with constant coefficients, signature (3,-1).
Programs
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GAP
List([0..30], n-> Fibonacci(2*n+2) +8*Fibonacci(2*n) ); # G. C. Greubel, Jan 17 2020
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Magma
[Fibonacci(2*n+2) +8*Fibonacci(2*n): n in [0..30]]; // G. C. Greubel, Jan 17 2020
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Maple
with(combinat); seq( fiboacci(2*n+2) +8*fibonacci(2*n), n=0..30); # G. C. Greubel, Jan 17 2020
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Mathematica
Table[Fibonacci[2*n+2] +8*Fibonacci[2*n], {n,0,30}] (* G. C. Greubel, Jan 17 2020 *) -
PARI
vector(31, n, fibonacci(2*n) +8*fibonacci(2*n-2) ) \\ G. C. Greubel, Jan 17 2020
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Sage
[fibonacci(2*n+2) +8*fibonacci(2*n) for n in (0..30)] # G. C. Greubel, Jan 17 2020
Formula
a(n) = {11*[((3+sqrt(5))/2)^n - ((3-sqrt(5))/2)^n] - [((3+sqrt(5))/2)^(n-1) - ((3-sqrt(5))/2)^(n-1)]}/sqrt(5).
G.f.: (1+8*x)/(1-3*x+x^2).
a(n) = 6*Lucas(2n+1) - Fibonacci(2n+5).
From G. C. Greubel, Jan 17 2020: (Start)
a(n) = Fibonacci(2*n+2) + 8*Fibonacci(2*n).
E.g.f.: exp(3*t/2)*( cosh(sqrt(5)*t/2) + (19/sqrt(5))*sinh(sqrt(5)*t/2) ). (End)