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 A051959 #29 Jan 05 2025 19:51:36
%S A051959 1,10,36,104,273,686,1688,4112,9969,24114,58268,140728,339809,820438,
%T A051959 1980784,4782112,11545121,27872474,67290196,162453000,392196337,
%U A051959 946845822,2285888136,5518622256,13323132817,32164888066,77652909132,187470706520,452594322369,1092659351462,2637913025504,6368485402688
%N A051959 Expansion of (1+6*x)/((1-2*x-x^2)*(1-x)^2).
%H A051959 Vincenzo Librandi, <a href="/A051959/b051959.txt">Table of n, a(n) for n = 0..1000</a>
%H A051959 A. F. Horadam, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/5-5/horadam.pdf">Special Properties of the Sequence W(n){a,b; p,q}</a>, Fib. Quart., Vol. 5, No. 5 (1967), pp. 424-434.
%H A051959 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,0,1).
%F A051959 a(n) = 2*a(n-1) + a(n-2) + (7*n+1), with a(0)=1, a(1)=10.
%F A051959 a(n) = 4*a(n-1) - 4*a(n-2) + a(n-4).
%F A051959 a(n) = ( (25 + 17*sqrt(2))*(1+sqrt(2))^n - (25 - 17*sqrt(2))*(1-sqrt(2))^n )/(4*sqrt(2)) - (7*n + 15)/2.
%F A051959 a(n) = (1/2)*(4*Pell(n+2) - 3*Pell(n) - 7*n - 15), with Pell(n) = A000129(n). - _Ralf Stephan_, May 15 2007
%F A051959 E.g.f.: (1/4)*exp(x)*(-30 - 14*x + 25*sqrt(2)*sinh(sqrt(2)*x) + 34*cosh(sqrt(2)*x)). - _G. C. Greubel_, Nov 11 2024
%t A051959 LinearRecurrence[{4,-4,0,1},{1,10,36,104},40] (* _Vincenzo Librandi_, Jun 22 2012 *)
%o A051959 (Magma) I:=[1,10,36,104]; [n le 4 select I[n] else 4*Self(n-1)-4*Self(n-2) +Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, Jun 22 2012
%o A051959 (SageMath)
%o A051959 def A051959(n):
%o A051959 @CachedFunction
%o A051959 def a(n):
%o A051959 if n<4: return (1,10,36,104)[n]
%o A051959 else: return 4*a(n-1) -4*a(n-2) +a(n-4)
%o A051959 return a(n)
%o A051959 [A051959(n) for n in range(41)] # _G. C. Greubel_, Nov 11 2024
%Y A051959 Cf. A000129, A048771, A048695.
%K A051959 easy,nonn
%O A051959 0,2
%A A051959 _Barry E. Williams_, Jan 04 2000