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

0,2

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).

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)

Maple

with(combinat); seq( fiboacci(2*n+2) +8*fibonacci(2*n), n=0..30); # G. C. Greubel, Jan 17 2020

Mathematica

Table[Fibonacci[2*n+2] +8*Fibonacci[2*n], {n, 0, 30}] (* G. C. Greubel, Jan 17 2020 *)

Prog

(PARI) vector(31, n, fibonacci(2*n) +8*fibonacci(2*n-2) ) \\ G. C. Greubel, Jan 17 2020
(Magma) [Fibonacci(2*n+2) +8*Fibonacci(2*n): n in [0..30]]; // G. C. Greubel, Jan 17 2020
(Sage) [fibonacci(2*n+2) +8*fibonacci(2*n) for n in (0..30)] # G. C. Greubel, Jan 17 2020
(GAP) List([0..30], n-> Fibonacci(2*n+2) +8*Fibonacci(2*n) ); # G. C. Greubel, Jan 17 2020

Crossrefs

Cf. A000032, A000045, A055850.

Sequence in context: A007790 A195857 A048773 * A006655 A290640 A018956

Adjacent sequences: A056120 A056121 A056122 * A056124 A056125 A056126

Keyword

easy,nonn

Author

Barry E. Williams, Jul 06 2000

Status

approved