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

Author

Barry E. Williams, Jun 03 2000

Keywords

• easy
• nonn

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).

Crossrefs

Cf. A000032, A000045.

Programs

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

Magma

[Lucas(2*n-1) + 2*Lucas(2*n+1): n in [0..30]]; // G. C. Greubel, Jan 16 2020

Magma

R:=PowerSeriesRing(Integers(), 26); Coefficients(R!( (1+6*x)/(1-3*x+x^2) )); // Marius A. Burtea, 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 *)

PARI

vector(31, n, fibonacci(2*n) +6*fibonacci(2*n-2) ) \\ G. C. Greubel, Jan 16 2020

Sage

[fibonacci(2*n+2) + 6*fibonacci(2*n) for n in (0..30)] # G. C. Greubel, Jan 16 2020

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