A055273 a(n) = 3*a(n-1) - a(n-2) with a(0) = 1, a(1) = 8.

1, 8, 23, 61, 160, 419, 1097, 2872, 7519, 19685, 51536, 134923, 353233, 924776, 2421095, 6338509, 16594432, 43444787, 113739929, 297775000, 779585071, 2040980213, 5343355568, 13989086491, 36623903905, 95882625224, 251023971767, 657189290077, 1720543898464

Offset: 0

Author

Barry E. Williams, May 28 2000

Keywords

• easy
• nonn

References

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

Links

• Vincenzo Librandi, 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, A055267.

Programs

GAP

a:=[1,8];; for n in [3..30] do a[n]:=3*a[n-1]-a[n-2]; od; Print(a); # Muniru A Asiru, Dec 29 2018

Magma

[Fibonacci(2*n+2) + 5*Fibonacci(2*n): n in [0..30]]; // Vincenzo Librandi, Dec 25 2018

Maple

seq(coeff(series((1+5*x)/(1-3*x+x^2),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Dec 29 2018

Mathematica

Table[3Fibonacci[2n+2]-Fibonacci[2n-3], {n,0,20}] (* Rigoberto Florez, Dec 24 2018 *)
LinearRecurrence[{3, -1}, {1, 8}, 30] (* Vincenzo Librandi, Dec 25 2018 *)

PARI

vector(30, n, fibonacci(2*n) + 5*fibonacci(2*n-2) ) \\ G. C. Greubel, Jan 17 2020

Sage

[fibonacci(2*n+2) +5*fibonacci(2*n) for n in (0..30)] # G. C. Greubel, Jan 17 2020

Formula

a(n) = (8*(((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 + 5*x)/(1 - 3*x + x^2).

From Rigoberto Florez, Dec 24 2018: (Start)

a(n) = Fibonacci(2n+2) + 5*Fibonacci(2n),

a(n) = 3*Fibonacci(2n+2) - Fibonacci(2n-3). (End)

E.g.f.: (1/5)*exp(3*x/2)*(5*cosh(sqrt(5)*x/2) + 13*sqrt(5)*sinh(sqrt(5)*x/2)). - Franck Maminirina Ramaharo, Dec 26 2018

a(n) = ChebyshevT(n, 3/2) + (13/2)*ChebyshevU(n-1, 3/2) = ChebyshevU(n, 3/2) + 5*ChebyshevU(n-1, 3/2). - G. C. Greubel, Jan 17 2020