%I #22 Mar 25 2024 22:27:33

%S 4,19,91,439,2119,10231,49399,238519,1151671,5560759,26849719,

%T 129641911,625966519,3022433719,14593600951,70464138679,340230958519,

%U 1642780388791,7932045389239,38299303112119,184925394005431,892898788470199,4311296729902519

%N Power floor sequence of 2+sqrt(8).

%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(8), and the limit p1(r) = 3.8983688904482395322594950087206...

%C See A218989 for the power floor function, p4. For comparison with p1, limit(p4(r)/p1(r) = 4/3.

%H Clark Kimberling, <a href="/A218988/b218988.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 (5,0,-4).

%F a(n) = floor(x*a(n-1)), where x=2+sqrt(8), a(0) = floor(x).

%F a(n) = 5*a(n-1) - 4*a(n-3).

%F G.f.: (4 - x - 4*x^2) / ((1 - x)*(1 - 4*x - 4*x^2)). [Corrected by _Colin Barker_, Nov 13 2017]

%F a(n) = (1/28)*(4 + (54-39*sqrt(2))*(2-2*sqrt(2))^n + (2*(1+sqrt(2)))^n*(54+39*sqrt(2))). - _Colin Barker_, Nov 13 2017

%F From _Philippe Deléham_, Mar 18 2024: (Start)

%F a(n) = 4*a(n-1) + 4*a(n-2) - 1.

%F a(n-1) = Sum_{k = 0..n} A370174(n,k)*3^k. (End)

%e a(0) = [r] = 4, where r = 2+sqrt(8).

%e a(1) = [4*r] = 19; a(2) = [19*r] = 91.

%t x = 2 + Sqrt[8]; 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}] (* this sequence *)

%t t2 = Table[p2[n], {n, 0, z}] (* A057087 *)

%t t3 = Table[p3[n], {n, 0, z}] (* A086347 *)

%t t4 = Table[p4[n], {n, 0, z}] (* A218989 *)

%o (PARI) Vec((4 - x - 4*x^2) / ((1 - x)*(1 - 4*x - 4*x^2)) + O(x^40)) \\ _Colin Barker_, Nov 13 2017

%Y Cf. A214992, A057087, A086347, A218989.

%K nonn,easy

%O 0,1

%A _Clark Kimberling_, Nov 11 2012