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 A027004 #19 Jul 23 2025 01:01:43 %S A027004 1,8,26,73,196,518,1361,3568,9346,24473,64076,167758,439201,1149848, %T A027004 3010346,7881193,20633236,54018518,141422321,370248448,969323026, %U A027004 2537720633,6643838876,17393795998,45537549121,119218851368,312119004986,817138163593,2139295485796 %N A027004 a(n) = T(2*n+1,n+1), T given by A026998. %H A027004 Colin Barker, <a href="/A027004/b027004.txt">Table of n, a(n) for n = 0..1000</a> %H A027004 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,1). %F A027004 a(n) = Fibonacci(2*n+3) + 2*Fibonacci(2*n+2) - 3. %F A027004 a(n) = A002878(n+1) - 3. %F A027004 From _Colin Barker_, Feb 18 2016: (Start) %F A027004 a(n) = 2^(-n)*((2-sqrt(5))*(3-sqrt(5))^n + (2+sqrt(5))*(3+sqrt(5))^n) - 3. %F A027004 a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3) for n > 2. %F A027004 G.f.: (1+4*x-2*x^2) / ((1-x)*(1-3*x+x^2)). (End) %F A027004 From _G. C. Greubel_, Jul 21 2025: (Start) %F A027004 a(n) = Lucas(2*n+3) - 3. %F A027004 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) %t A027004 LucasL[2*Range[0,40] +3] -3 (* _G. C. Greubel_, Jul 21 2025 *) %o A027004 (Magma) [Fibonacci(2*n+3) + 2*Fibonacci(2*n+2) - 3: n in [0..30]]; // _Vincenzo Librandi_, Apr 18 2011 %o A027004 (PARI) Vec((1+4*x-2*x^2)/((1-x)*(1-3*x+x^2)) + O(x^40)) \\ _Colin Barker_, Feb 18 2016 %o A027004 (SageMath) %o A027004 def A027004(n): return lucas_number2(2*n+3,1,-1) -3 # _G. C. Greubel_, Jul 21 2025 %Y A027004 Cf. A000032, A002878, A026998. %K A027004 nonn,easy %O A027004 0,2 %A A027004 _Clark Kimberling_