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A214998 Power ceiling-floor sequence of 2 + sqrt(3).

%I A214998 #29 May 31 2025 05:19:00
%S A214998 4,14,53,197,736,2746,10249,38249,142748,532742,1988221,7420141,
%T A214998 27692344,103349234,385704593,1439469137,5372171956,20049218686,
%U A214998 74824702789,279249592469,1042173667088,3889445075882,14515606636441,54172981469881,202176319243084
%N A214998 Power ceiling-floor sequence of 2 + sqrt(3).
%C A214998 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 = 2 + sqrt(3), and the limit p3(r) = (23 + 13*sqrt(3))/12.
%H A214998 Clark Kimberling, <a href="/A214998/b214998.txt">Table of n, a(n) for n = 0..250</a>
%H A214998 R. C. Alperin, <a href="https://www.fq.math.ca/Papers/57-4/alperin07132019.pdf">A family of nonlinear recurrences and their linear solutions</a>, Fib. Q., 57:4 (2019), 318-321.
%H A214998 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,3,-1).
%F A214998 a(n) = floor(x*a(n-1)) if n is odd, a(n) = ceiling(x*a(n-1)) if n is even, where x = 2+sqrt(3) and a(0) = ceiling(x).
%F A214998 a(n) = 3*a(n-1) + 3*a(n-2) - a(n-3).
%F A214998 G.f.: (4 + 2*x - x^2)/(1 - 3*x - 3*x^2 + x^3).
%F A214998 a(n) = (-1)^n + 4*a(n-1) - a(n-2) with a(0) = 4 and a(1) = 14. - _Peter Bala_, Nov 12 2017
%F A214998 a(n) = (1/12)*(2*(-1)^n + (23-13*sqrt(3))*(2-sqrt(3))^n + (2+sqrt(3))^n*(23+13*sqrt(3))). - _Colin Barker_, Nov 13 2017
%e A214998 a(0) = ceiling(r) =  4, where r = 2+sqrt(3);
%e A214998 a(1) = floor(4*r) = 14; a(2) = ceiling(14*r) = 53.
%t A214998 x = 2 + Sqrt[3]; z = 30; (* z = # terms in sequences *)
%t A214998 z1 = 100; (* z1 = # digits in approximations *)
%t A214998 f[x_] := Floor[x]; c[x_] := Ceiling[x];
%t A214998 p1[0] = f[x]; p2[0] = f[x]; p3[0] = c[x]; p4[0] = c[x];
%t A214998 p1[n_] := f[x*p1[n - 1]]
%t A214998 p2[n_] := If[Mod[n, 2] == 1, c[x*p2[n - 1]], f[x*p2[n - 1]]]
%t A214998 p3[n_] := If[Mod[n, 2] == 1, f[x*p3[n - 1]], c[x*p3[n - 1]]]
%t A214998 p4[n_] := c[x*p4[n - 1]]
%t A214998 Table[p1[n], {n, 0, z}]  (* A001835 *)
%t A214998 Table[p2[n], {n, 0, z}]  (* A109437 *)
%t A214998 Table[p3[n], {n, 0, z}]  (* A214998 *)
%t A214998 Table[p4[n], {n, 0, z}]  (* A001353 *)
%o A214998 (PARI) Vec((4 + 2*x - x^2) / ((1 + x)*(1 - 4*x + x^2)) + O(x^30)) \\ _Colin Barker_, Nov 13 2017
%Y A214998 Cf. A214992, A001835, A109437, A001353, A001654.
%K A214998 nonn,easy
%O A214998 0,1
%A A214998 _Clark Kimberling_, Nov 10 2012