A027001 a(n) = T(2*n, n+2), T given by A026998.

1, 26, 174, 743, 2552, 7784, 22193, 60882, 163430, 433495, 1142496, 3001056, 7869649, 20619098, 54001422, 141401879, 370224248, 969294632, 2537687585, 6643800690, 17393752166, 45537499111, 119218794624, 312118940928, 817138091617, 2139295405274

Offset

2,2

Links

Colin Barker, Table of n, a(n) for n = 2..1000

Index entries for linear recurrences with constant coefficients, signature (7,-19,26,-19,7,-1).

Formula

a(n+2) = 3*F(2*n+10) - 2*F(2*n+9) - F(2*n+8) -(4*n^3 +26*n^2 +68*n +75), n >= 0, F(n) = A000045(n). - Ralf Stephan, Feb 07 2004

From Colin Barker, Feb 18 2016: (Start)

a(n) = 2^(-1-n)*( (11-5*sqrt(5))*(3-sqrt(5))^n + (11+5*sqrt(5))*(3+sqrt(5))^n ) - 11 - 12*n - 2*n^2 - 4*n^3.

G.f.: x^2*(1+x)*(1+18*x-7*x^2) / ((1-x)^4*(1-3*x+x^2)). (End)

From G. C. Greubel, Jul 20 2025: (Start)

a(n) = Lucas(2*n+5) - (4*(n+1)^3 - 10*n^2 + 7), n >= 2.

E.g.f.: exp(3*x/2)*(11*cosh(p*x) + 10*p*sinh(p*x)) - (4*x^3 + 14*x^2 + 18*x + 11)*exp(x), where 2*p = sqrt(5). (End)

Mathematica

LinearRecurrence[{7, -19, 26, -19, 7, -1}, {1, 26, 174, 743, 2552, 7784}, 30] (* Vincenzo Librandi, Feb 19 2016 *)

Prog

(PARI) Vec(x^2*(1+x)*(1+18*x-7*x^2)/((1-x)^4*(1-3*x+x^2)) + O(x^40)) \\ Colin Barker, Feb 18 2016
(Magma) [3*Fibonacci(2*n+10)-2*Fibonacci(2*n+9)-Fibonacci(2*n+8)-4*n^3-26*n^2-68*n-75: n in [0..30]]; // Vincenzo Librandi, Feb 19 2016
(SageMath)
def A027001(n): return lucas_number2(2*n+5, 1, -1) -(4*(n+1)**3 -10*n**2 +7)
print([A027001(n) for n in range(2, 41)]) # G. C. Greubel, Jul 20 2025

Crossrefs

Bisection of A027964.

Cf. A000032, A000045, A026998.

Sequence in context: A321113 A126494 A374489 * A173893 A252936 A218083

Adjacent sequences: A026998 A026999 A027000 * A027002 A027003 A027004

Keyword

nonn,easy

Author

Clark Kimberling

Status

approved