A027004 a(n) = T(2*n+1,n+1), T given by A026998.

1, 8, 26, 73, 196, 518, 1361, 3568, 9346, 24473, 64076, 167758, 439201, 1149848, 3010346, 7881193, 20633236, 54018518, 141422321, 370248448, 969323026, 2537720633, 6643838876, 17393795998, 45537549121, 119218851368, 312119004986, 817138163593, 2139295485796

Offset

0,2

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Formula

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

a(n) = A002878(n+1) - 3.

From Colin Barker, Feb 18 2016: (Start)

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

a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3) for n > 2.

G.f.: (1+4*x-2*x^2) / ((1-x)*(1-3*x+x^2)). (End)

From G. C. Greubel, Jul 21 2025: (Start)

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

E.g.f.: 2*exp(3*x/2)*(2*cosh(p*x) + p*sinh(p*x)) - 3*exp(x), where 2*p = sqrt(5). (End)

Mathematica

LucasL[2*Range[0, 40] +3] -3 (* G. C. Greubel, Jul 21 2025 *)

Prog

(Magma) [Fibonacci(2*n+3) + 2*Fibonacci(2*n+2) - 3: n in [0..30]]; // Vincenzo Librandi, Apr 18 2011
(PARI) Vec((1+4*x-2*x^2)/((1-x)*(1-3*x+x^2)) + O(x^40)) \\ Colin Barker, Feb 18 2016
(SageMath)
def A027004(n): return lucas_number2(2*n+3, 1, -1) -3 # G. C. Greubel, Jul 21 2025

Crossrefs

Cf. A000032, A002878, A026998.

Sequence in context: A296112 A051669 A207101 * A194021 A245126 A278769

Adjacent sequences: A027001 A027002 A027003 * A027005 A027006 A027007

Keyword

nonn,easy

Author

Clark Kimberling

Status

approved