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

%I A027001 #19 Jul 25 2025 00:51:58
%S A027001 1,26,174,743,2552,7784,22193,60882,163430,433495,1142496,3001056,
%T A027001 7869649,20619098,54001422,141401879,370224248,969294632,2537687585,
%U A027001 6643800690,17393752166,45537499111,119218794624,312118940928,817138091617,2139295405274
%N A027001 a(n) = T(2*n, n+2), T given by A026998.
%H A027001 Colin Barker, <a href="/A027001/b027001.txt">Table of n, a(n) for n = 2..1000</a>
%H A027001 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (7,-19,26,-19,7,-1).
%F A027001 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
%F A027001 From _Colin Barker_, Feb 18 2016: (Start)
%F A027001 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.
%F A027001 G.f.: x^2*(1+x)*(1+18*x-7*x^2) / ((1-x)^4*(1-3*x+x^2)). (End)
%F A027001 From _G. C. Greubel_, Jul 20 2025: (Start)
%F A027001 a(n) = Lucas(2*n+5) - (4*(n+1)^3 - 10*n^2 + 7), n >= 2.
%F A027001 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)
%t A027001 LinearRecurrence[{7, -19, 26, -19, 7, -1}, {1, 26, 174, 743, 2552, 7784}, 30] (* _Vincenzo Librandi_, Feb 19 2016 *)
%o A027001 (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
%o A027001 (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
%o A027001 (SageMath)
%o A027001 def A027001(n): return lucas_number2(2*n+5,1,-1) -(4*(n+1)**3 -10*n**2 +7)
%o A027001 print([A027001(n) for n in range(2,41)]) # _G. C. Greubel_, Jul 20 2025
%Y A027001 Bisection of A027964.
%Y A027001 Cf. A000032, A000045, A026998.
%K A027001 nonn,easy
%O A027001 2,2
%A A027001 _Clark Kimberling_