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 A192383 #22 Jul 10 2023 08:21:04
%S A192383 1,0,6,8,60,160,744,2496,10064,36480,140512,522624,1983168,7439360,
%T A192383 28091520,105674752,398391552,1500057600,5652182528,21288560640,
%U A192383 80200784896,302101094400,1138045495296,4286942363648,16149041172480,60833034895360
%N A192383 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.
%C A192383 The polynomial p(n,x) is defined by ((x+d)^n - (x-d)^n)/(2*d), where d=sqrt(x+3). For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232.
%H A192383 G. C. Greubel, <a href="/A192383/b192383.txt">Table of n, a(n) for n = 1..1000</a>
%H A192383 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,8,-4,-4).
%F A192383 From _Colin Barker_, May 11 2014: (Start)
%F A192383 a(n) = 2*a(n-1) + 8*a(n-2) - 4*a(n-3) - 4*a(n-4).
%F A192383 G.f.: x*(1-2*x-2*x^2)/(1-2*x-8*x^2+4*x^3+4*x^4). (End)
%F A192383 From _G. C. Greubel_, Jul 10 2023: (Start)
%F A192383 T(n, k) = [x^k] ((x+sqrt(x+3))^n - (x-sqrt(x+3))^n)/(2*sqrt(x+3)).
%F A192383 a(n) = Sum_{k=0..n-1} T(n, k)*Fibonacci(k-1). (End)
%e A192383 The first five polynomials p(n,x) and their reductions are as follows:
%e A192383 p(0, x) = 1 -> 1
%e A192383 p(1, x) = 2*x -> 2*x
%e A192383 p(2, x) = 3 + x + 3*x^2 -> 6 + 4*x
%e A192383 p(3, x) = 12*x + 4*x^2 + 4*x^3 -> 8 + 24*x
%e A192383 p(4, x) = 9 + 6*x + 31*x^2 + 10*x^3 + 5*x^4 -> 60 + 72*x.
%e A192383 From these, read A192383 = (1, 0, 6, 8, 60, ...) and A192384 = (0, 2, 4, 24, 72, ...).
%t A192383 q[x_]:= x+1; d= Sqrt[x+3];
%t A192383 p[n_, x_]:= ((x+d)^n - (x-d)^n)/(2*d); (* suggested by A162517 *)
%t A192383 Table[Expand[p[n, x]], {n,6}]
%t A192383 reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t A192383 t = Table[FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 1, 30}];
%t A192383 Table[Coefficient[Part[t, n], x, 0], {n,30}] (* A192383 *)
%t A192383 Table[Coefficient[Part[t, n], x, 1], {n,30}] (* A192384 *)
%t A192383 Table[Coefficient[Part[t, n]/2, x, 1], {n,30}] (* A192385 *)
%t A192383 LinearRecurrence[{2,8,-4,-4}, {1,0,6,8}, 40] (* _G. C. Greubel_, Jul 10 2023 *)
%o A192383 (Magma)
%o A192383 R<x>:=PowerSeriesRing(Integers(), 41);
%o A192383 Coefficients(R!( x*(1-2*x-2*x^2)/(1-2*x-8*x^2+4*x^3+4*x^4) )) // _G. C. Greubel_, Jul 10 2023
%o A192383 (SageMath)
%o A192383 @CachedFunction
%o A192383 def a(n): # a = A192383
%o A192383 if (n<5): return (0,1,0,6,8)[n]
%o A192383 else: return 2*a(n-1) +8*a(n-2) -4*a(n-3) -4*a(n-4)
%o A192383 [a(n) for n in range(1,41)] # _G. C. Greubel_, Jul 10 2023
%Y A192383 Cf. A083084, A192232.
%Y A192383 Cf. A192384, A192385.
%K A192383 nonn
%O A192383 1,3
%A A192383 _Clark Kimberling_, Jun 30 2011