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.
(See A192377.)
(See A192376.)
The first five polynomials p(n,x) and their reductions are as follows: p(0,x)=1 -> 1 p(1,x)=2x -> 2x p(2,x)=4+x+3x^2 -> 7+4x p(3,x)=16x+4x^2+4x^3 -> 16+20x p(4,x)=16+8x+41x^2+10x^3+5x^4 -> 73+68x. From these, read A192376=(1,0,7,16,73,...) and A192377=(0,2,4,20,68,...).
q[x_] := x + 2; d = Sqrt[x + 1]; p[n_, x_] := ((x + d)^n - (x - d)^ n )/(2 d) (* Cf. A162517 *) Table[Expand[p[n, x]], {n, 1, 6}] reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)}; t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 1, 30}] Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}] (* A192376 *) Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}] (* A192377 *) Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}] (* A192378 *)
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