A027000 a(n) = Lucas(2*n+3) - (6*n+4).

1, 13, 54, 171, 487, 1324, 3525, 9297, 24418, 64015, 167691, 439128, 1149769, 3010261, 7881102, 20633139, 54018415, 141422212, 370248333, 969322905, 2537720506, 6643838743, 17393795859, 45537548976, 119218851217, 312119004829

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Links

• Nathaniel Johnston, Table of n, a(n) for n = 1..600
• Index entries for linear recurrences with constant coefficients, signature (5,-8,5,-1).

Crossrefs

Cf. A000032, A000045, A026998.

Programs

Magma

[ Lucas(2*n+3) - (6*n+4): n in [1..100]]; // Vincenzo Librandi, Apr 15 2011

Mathematica

Table[LucasL[2n + 3] - (6n + 4), {n, 30}] (* Alonso del Arte, Dec 16 2011 *)

SageMath

def A027000(n): return lucas_number2(2*n+3,1,-1) -2*(3*n+2) # G. C. Greubel, Jul 20 2025

Formula

a(n) = A026998(2*n, n+1).

a(n) = 4*F(2*n-1) + 7*F(2*n) - 6*n - 4, with F(n) = A000045(n) (Fibonacci). - Ralf Stephan, Feb 07 2004

G.f.: x*(1+8*x-3*x^2) / ((1-x)^2*(1-3*x+x^2)). - Franklin T. Adams-Watters and D. S. McNeil, Dec 16 2011

E.g.f.: 2*exp(3*x/2)*( 2*cosh(p*x) + 2*p*sinh(p*x) - (3*x+2)*exp(-x/2) ), where 2*p = sqrt(5). - G. C. Greubel, Jul 20 2025

Extensions

a(23) - a(26) from Vincenzo Librandi, Apr 15 2011