A054492 a(n) = 3*a(n-1) - a(n-2), a(0)=1, a(1)=6.

1, 6, 17, 45, 118, 309, 809, 2118, 5545, 14517, 38006, 99501, 260497, 681990, 1785473, 4674429, 12237814, 32039013, 83879225, 219598662, 574916761, 1505151621, 3940538102, 10316462685, 27008849953, 70710087174, 185121411569, 484654147533, 1268841031030

Offset: 0

Author

Barry E. Williams, May 06 2000

Keywords

• easy
• nonn

References

• A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• I. Adler, Three Diophantine equations - Part II, Fib. Quart., 7 (1969), pp. 181-193.
• E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242.
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (3,-1).

Crossrefs

Cf. A002878, A054486.

Programs

Magma

I:=[1,6]; [n le 2 select I[n] else 3*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Mar 20 2015

Mathematica

CoefficientList[Series[(1 + 3 x) / (1 - 3 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 20 2015 *)
LinearRecurrence[{3, -1}, {1, 6}, 100] (* G. C. Greubel, Mar 26 2016 *)

PARI

Vec((1+3*x)/(1-3*x+x^2) + O(x^30)) \\ Michel Marcus, Mar 20 2015

Formula

a(n) = (6*(((3+sqrt(5))/2)^n - ((3-sqrt(5))/2)^n) - (((3+sqrt(5))/2)^(n-1) - ((3-sqrt(5))/2)^(n-1)))/sqrt(5).

a(n) = 2*Lucas(2*n+1) - Fibonacci(2*n+1).

G.f.: (1+3*x)/(1-3*x+x^2). - Philippe Deléham, Nov 03 2008

a(n) = 5*Fibonacci(2*n) + Fibonacci(2*n-1). - Ehren Metcalfe, Mar 26 2016

E.g.f.: (1/10) * exp((3-sqrt(5))*x/2) * ((5-9*sqrt(5)) + (5+9*sqrt(5)) * exp(sqrt(5)*x) ). - G. C. Greubel, Mar 26 2016

Extensions

More terms from Vincenzo Librandi, Mar 20 2015

Typo in name fixed by Karl V. Keller, Jr., Jun 23 2015