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
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (4,-4,1).
Programs
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Magma
[Fibonacci(2*n+3) + 2*Fibonacci(2*n+2) - 3: n in [0..30]]; // Vincenzo Librandi, Apr 18 2011
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Mathematica
LucasL[2*Range[0,40] +3] -3 (* G. C. Greubel, Jul 21 2025 *)
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PARI
Vec((1+4*x-2*x^2)/((1-x)*(1-3*x+x^2)) + O(x^40)) \\ Colin Barker, Feb 18 2016
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SageMath
def A027004(n): return lucas_number2(2*n+3,1,-1) -3 # G. C. Greubel, Jul 21 2025
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)