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