This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A027002 #25 Jun 18 2025 01:05:02
%S A027002 1,43,431,2482,10636,38138,122069,362853,1027843,2822668,7601784,
%T A027002 20228876,53447609,140633575,369179479,967898846,2535852052,
%U A027002 6641420806,17390705661,45533644161,119213967867,312112955384,817130734512,2139286435768,5600737350897
%N A027002 a(n) = T(2*n, n+3), T given by A026998.
%H A027002 Robert Israel, <a href="/A027002/b027002.txt">Table of n, a(n) for n = 3..2370</a>
%H A027002 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (9,-34,71,-90,71,-34,9,-1).
%F A027002 G.f.: x^3*(1+34*x+78*x^2-6*x^3-11*x^4) / ((1-x)^6*(1-3*x+x^2)). - _Colin Barker_, Feb 18 2016
%F A027002 From _Robert Israel_, Feb 18 2016: (Start)
%F A027002 By definition, a(n) is the coefficient of x^(2*n-6) in the Maclaurin series of (1+2*x)/((1-x-x^2)*(1-x)^6). This can be written explicitly:
%F A027002 a(n) = ((29-13*sqrt(5))/2)*((3-sqrt(5))/2)^n + ((29+13*sqrt(5))/2)*((3+sqrt(5))/2)^n - (4/5)*n^5 + (4/3)*n^4 - (25/3)*n^3 - (31/3)*n^2 - (433/15)*n - 29.
%F A027002 This confirms Colin Barker's g.f. (End)
%F A027002 From _G. C. Greubel_, Jun 16 2025: (Start)
%F A027002 a(n) = A000032(2*n+7) - (1/15)*(435 + 433*n + 155*n^2 + 125*n^3 - 20*n^4 + 12*n^5).
%F A027002 E.g.f.: exp(3*x/2)*(29*cosh(sqrt(5)*x/2) + 13*sqrt(5)*sinh(sqrt(5)*x/2)) - (1/15)*(435 + 705*x + 570*x^2 + 305*x^3 + 100*x^4 + 12*x^5)*exp(x). (End)
%p A027002 gf:= x^3*(1+34*x+78*x^2-6*x^3-11*x^4) / ((1-x)^6*(1-3*x+x^2)):
%p A027002 S:= series(gf,x,100):
%p A027002 seq(coeff(S,x,n),n=3..100); # _Robert Israel_, Feb 18 2016
%t A027002 LinearRecurrence[{9, -34, 71, -90, 71, -34, 9, -1}, {1, 43, 431, 2482, 10636, 38138, 122069, 362853}, 30] (* _Vincenzo Librandi_, Feb 19 2016 *)
%o A027002 (PARI) Vec(x^3*(1+34*x+78*x^2-6*x^3-11*x^4)/((1-x)^6*(1-3*x+x^2)) + O(x^40)) \\ _Colin Barker_, Feb 19 2016
%o A027002 (Magma)
%o A027002 A027002 := func< n | Lucas(2*n+7) -(435+433*n+155*n^2+125*n^3-20*n^4+12*n^5)/15 >;
%o A027002 [A027002(n): n in [3..50]]; // _G. C. Greubel_, Jun 16 2025
%o A027002 (SageMath)
%o A027002 def A027002(n): return lucas_number2(2*n+7,1,-1) -(435+433*n+155*n^2+125*n^3 -20*n^4+12*n^5)/15
%o A027002 print([A027002(n) for n in range(3,51)]) # _G. C. Greubel_, Jun 16 2025
%Y A027002 Cf. A000032, A026998.
%K A027002 nonn
%O A027002 3,2
%A A027002 _Clark Kimberling_