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A214996 Power floor-ceiling sequence of 2+sqrt(2).

%I A214996 #39 Oct 28 2024 02:00:49
%S A214996 3,11,37,127,433,1479,5049,17239,58857,200951,686089,2342455,7997641,
%T A214996 27305655,93227337,318298039,1086737481,3710353847,12667940425,
%U A214996 43251054007,147668335177,504171232695,1721348260425,5877050576311,20065505784393,68507921984951
%N A214996 Power floor-ceiling sequence of 2+sqrt(2).
%C A214996 See A214992 for a discussion of power floor-ceiling sequence and power floor-ceiling function, p2(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = 2+sqrt(2), and the limit p2(r) = (11 + 8*sqrt(2))/7.
%C A214996 From _Greg Dresden_, Jun 02 2020: (Start)
%C A214996 a(n) is the number of ways to tile a 2 X (n+1) strip, with one extra square at the top left corner, using 1 X 1 squares, 2 X 2 squares, and 1 X 2 dominoes (either horizontal or vertical). This picture shows a(1) = 11.
%C A214996    _     _     _     _     _     _     _     _     _     _     _
%C A214996   |_|_  |_|_  | |_  |_|_  |_|_  |_|_  |_|_  | |_  | |_  |_|_  |_|_
%C A214996   |_|_| |   | |_|_| | |_| |_| | |___| |_|_| |_|_| |_| | |___| | | |
%C A214996   |_|_| |___| |_|_| |_|_| |_|_| |_|_| |___| |___| |_|_| |___| |_|_|
%C A214996 (End)
%H A214996 Clark Kimberling, <a href="/A214996/b214996.txt">Table of n, a(n) for n = 0..250</a>
%H A214996 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,2,-2).
%F A214996 a(n) = ceiling(x*a(n-1)) if n is odd, a(n) = floor(x*a(n-1)) if n is even, where x = 2+sqrt(2) and a(0) = floor(x).
%F A214996 a(n) = 3*a(n-1) + 2*a(n-2) - 2*a(n-3).
%F A214996 G.f.: (3 + 2*x - 2*x^2)/(1 - 3*x - 2*x^2 + 2*x^3).
%F A214996 a(n) = (1/7)*((-1)^(1+n) + (11-8*sqrt(2))*(2-sqrt(2))^n + (2+sqrt(2))^n*(11+8*sqrt(2))). - _Colin Barker_, Nov 13 2017
%e A214996 a(0) = floor(r) = 3, where r = 2+sqrt(2).
%e A214996 a(1) = ceiling(3*r) = 11; a(2) = floor(11*r) = 37.
%t A214996 x = 2 + Sqrt[2]; z = 30; (* z = # terms in sequences *)
%t A214996 z1 = 100; (* z1 = # digits in approximations *)
%t A214996 f[x_] := Floor[x]; c[x_] := Ceiling[x];
%t A214996 p1[0] = f[x]; p2[0] = f[x]; p3[0] = c[x]; p4[0] = c[x];
%t A214996 p1[n_] := f[x*p1[n - 1]]
%t A214996 p2[n_] := If[Mod[n, 2] == 1, c[x*p2[n - 1]], f[x*p2[n - 1]]]
%t A214996 p3[n_] := If[Mod[n, 2] == 1, f[x*p3[n - 1]], c[x*p3[n - 1]]]
%t A214996 p4[n_] := c[x*p4[n - 1]]
%t A214996 Table[p1[n], {n, 0, z}]  (* A007052 *)
%t A214996 Table[p2[n], {n, 0, z}]  (* A214996 *)
%t A214996 Table[p3[n], {n, 0, z}]  (* A214997 *)
%t A214996 Table[p4[n], {n, 0, z}]  (* A007070 *)
%o A214996 (PARI) Vec((3 + 2*x - 2*x^2) / ((1 + x)*(1 - 4*x + 2*x^2)) + O(x^40)) \\ _Colin Barker_, Nov 13 2017
%o A214996 (Magma) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!((3+2*x-2*x^2)/(1-3*x-2*x^2+2*x^3))) // _G. C. Greubel_, Feb 02 2018
%Y A214996 Cf. A214992, A007052, A214997, A007070, A052543.
%K A214996 nonn,easy
%O A214996 0,1
%A A214996 _Clark Kimberling_, Nov 10 2012