A055849 a(n) = 3*a(n-1) - a(n-2) with a(0)=1, a(1)=9.

1, 9, 26, 69, 181, 474, 1241, 3249, 8506, 22269, 58301, 152634, 399601, 1046169, 2738906, 7170549, 18772741, 49147674, 128670281, 336863169, 881919226, 2308894509, 6044764301, 15825398394, 41431430881, 108468894249

Offset

0,2

References

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

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) = (9*(((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+6*x)/(1-3*x+x^2).

a(n) = L(2*n-1) + 2*L(2*n+1), where L(n) is the n-th Lucas number. - Rigoberto Florez, Dec 24 2018

a(n) = Fibonacci(2*n+2) + 6*Fibonacci(2*n). - G. C. Greubel, Jan 16 2020

Maple

with(combinat); seq( fibonacci(2*n+2) + 6*fibonacci(2*n), n=0..30); # G. C. Greubel, Jan 16 2020

Mathematica

LinearRecurrence[{3, -1}, {1, 9}, 30] (* Harvey P. Dale, Jan 20 2013 *)
Table[LucasL[2n-1]+2LucasL[2n+1], {n, 0, 30}] (* Rigoberto Florez, Dec 24 2018 *)

Prog

(PARI) vector(31, n, fibonacci(2*n) +6*fibonacci(2*n-2) ) \\ G. C. Greubel, Jan 16 2020
(Magma) [Lucas(2*n-1) + 2*Lucas(2*n+1): n in [0..30]]; // G. C. Greubel, Jan 16 2020
(Magma) R<x>:=PowerSeriesRing(Integers(), 26); Coefficients(R!( (1+6*x)/(1-3*x+x^2) )); // Marius A. Burtea, Jan 16 2020
(Sage) [fibonacci(2*n+2) + 6*fibonacci(2*n) for n in (0..30)] # G. C. Greubel, Jan 16 2020
(GAP) List([0..30], n->  Lucas(1, -1, 2*n-1)[2] + 2*Lucas(1, -1, 2*n+1)[2] ); # G. C. Greubel, Jan 16 2020

Crossrefs

Cf. A000032, A000045.

Sequence in context: A048468 A255108 A048771 * A235163 A372668 A084813

Adjacent sequences: A055846 A055847 A055848 * A055850 A055851 A055852

Keyword

easy,nonn

Author

Barry E. Williams, Jun 03 2000

Status

approved