|
%I #13 Nov 13 2017 07:06:35
%S 4,17,75,333,1481,6589,29317,130445,580413,2582541,11490989,51129037,
%T 227498125,1012250573,4503998541,20040495309,89169978317,396760903885,
%U 1765383572173,7855056096461,34950991530189,155514078313677,691958296315085,3078861341887693
%N Power floor sequence of 2+sqrt(6).
%C See A214992 for a discussion of power floor sequence and the power floor function, p1(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = 2+sqrt(6), and the limit p1(r) = 3.77794213613376987528458445727451673384055973517...
%H Clark Kimberling, <a href="/A218984/b218984.txt">Table of n, a(n) for n = 0..250</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-2,-2).
%F a(n) = [x*a(n-1)], where x=2+sqrt(6), a(0) = [x].
%F a(n) = 5*a(n-1) - 2*a(n-2) - 2*a(n-3).
%F G.f.: (4 - 3*x - 2*x^2)/(1 - 5*x + 2*x^2 + 2*x^3).
%F a(n) = (1/30)*(6 + (57-23*sqrt(6))*(2-sqrt(6))^n + (2+sqrt(6))^n*(57+23*sqrt(6))). - _Colin Barker_, Nov 13 2017
%e a(0) = [r] = 4, where r = 2+sqrt(6); a(1) = [4*r] = 17; a(2) = [17*r] = 75.
%t x = 2 + Sqrt[6]; z = 30; (* z = # terms in sequences *)
%t f[x_] := Floor[x]; c[x_] := Ceiling[x];
%t p1[0] = f[x]; p2[0] = f[x]; p3[0] = c[x]; p4[0] = c[x];
%t p1[n_] := f[x*p1[n - 1]]
%t p2[n_] := If[Mod[n, 2] == 1, c[x*p2[n - 1]], f[x*p2[n - 1]]]
%t p3[n_] := If[Mod[n, 2] == 1, f[x*p3[n - 1]], c[x*p3[n - 1]]]
%t p4[n_] := c[x*p4[n - 1]]
%t t1 = Table[p1[n], {n, 0, z}] (* A218984 *)
%t t2 = Table[p2[n], {n, 0, z}] (* A090017 *)
%t t3 = Table[p3[n], {n, 0, z}] (* A123347 *)
%t t4 = Table[p4[n], {n, 0, z}] (* A218985 *)
%o (PARI) Vec((4 - 3*x - 2*x^2) / ((1 - x)*(1 - 4*x - 2*x^2)) + O(x^40)) \\ _Colin Barker_, Nov 13 2017
%Y Cf. A214992, A090017, A123347, A218985.
%K nonn,easy
%O 0,1
%A _Clark Kimberling_, Nov 11 2012
| 