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 A218990 #22 Nov 01 2024 05:11:06
%S A218990 6,34,199,1159,6756,39376,229501,1337629,7796274,45440014,264843811,
%T A218990 1543622851,8996893296,52437736924,305629528249,1781339432569,
%U A218990 10382407067166,60513102970426,352696210755391,2055664161561919,11981288758616124,69832068390134824
%N A218990 Power ceiling-floor sequence of 3+sqrt(8).
%C A218990 See A214992 for a discussion of power ceiling-floor sequence and power ceiling-floor function, p3(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = 3+sqrt(8), and the limit p3(r) = 5.854315472394508538153482993682502287049948...
%D A218990 R. C. Alperin, A family of nonlinear recurrences and their linear solutions, Fib. Q., 57:4 (2019), 318-321.~
%H A218990 Clark Kimberling, <a href="/A218990/b218990.txt">Table of n, a(n) for n = 0..250</a>
%H A218990 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,5,-1).
%F A218990 a(n) = floor(x*a(n-1)) if n is odd, a(n) = ceiling(x*a(n-1)) if n is even, where x=3+sqrt(8) and a(0) = ceiling(x).
%F A218990 a(n) = 5*a(n-1) + 5*a(n-2) - a(n-3).
%F A218990 G.f.: (6 + 4*x - x^2)/(1 - 5*x - 5*x^2 + x^3).
%F A218990 a(n) = (1/16)*(2*(-1)^n + (47-33*sqrt(2))*(3-2*sqrt(2))^n + (3+2*sqrt(2))^n*(47+33*sqrt(2))). - _Colin Barker_, Nov 13 2017
%e A218990 a(0) = ceiling(r) = 6, where r = 3+sqrt(8);
%e A218990 a(1) = floor(6*r) = 34; a(2) = ceiling(35*r) = 199.
%t A218990 x = 3 + Sqrt[8]; z = 30; (* z = # terms in sequences *)
%t A218990 f[x_] := Floor[x]; c[x_] := Ceiling[x];
%t A218990 p1[0] = f[x]; p2[0] = f[x]; p3[0] = c[x]; p4[0] = c[x];
%t A218990 p1[n_] := f[x*p1[n - 1]]
%t A218990 p2[n_] := If[Mod[n, 2] == 1, c[x*p2[n - 1]], f[x*p2[n - 1]]]
%t A218990 p3[n_] := If[Mod[n, 2] == 1, f[x*p3[n - 1]], c[x*p3[n - 1]]]
%t A218990 p4[n_] := c[x*p4[n - 1]]
%t A218990 t1 = Table[p1[n], {n, 0, z}] (* A001653 *)
%t A218990 t2 = Table[p2[n], {n, 0, z}] (* A084158 *)
%t A218990 t3 = Table[p3[n], {n, 0, z}] (* A218990 *)
%t A218990 t4 = Table[p4[n], {n, 0, z}] (* A001109 *)
%t A218990 LinearRecurrence[{5,5,-1},{6,34,199},30] (* _Harvey P. Dale_, Mar 21 2024 *)
%o A218990 (PARI) Vec((6 + 4*x - x^2) / ((1 + x)*(1 - 6*x + x^2)) + O(x^50)) \\ _Colin Barker_, Nov 13 2017
%Y A218990 Cf. A214992, A001653, A084158, A001109.
%K A218990 nonn,easy
%O A218990 0,1
%A A218990 _Clark Kimberling_, Nov 11 2012