A218984 Power floor sequence of 2+sqrt(6).

4, 17, 75, 333, 1481, 6589, 29317, 130445, 580413, 2582541, 11490989, 51129037, 227498125, 1012250573, 4503998541, 20040495309, 89169978317, 396760903885, 1765383572173, 7855056096461, 34950991530189, 155514078313677, 691958296315085, 3078861341887693

Offset: 0

Author

Clark Kimberling, Nov 11 2012

Keywords

• nonn
• easy

Comments

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...

Examples

			a(0) = [r] = 4, where r = 2+sqrt(6); a(1) = [4*r] = 17; a(2) = [17*r] = 75.
		

Links

• Clark Kimberling, Table of n, a(n) for n = 0..250
• Index entries for linear recurrences with constant coefficients, signature (5,-2,-2).

Crossrefs

Cf. A214992, A090017, A123347, A218985.

Programs

Mathematica

x = 2 + Sqrt[6]; z = 30; (* z = # terms in sequences *)
f[x_] := Floor[x]; c[x_] := Ceiling[x];
p1[0] = f[x]; p2[0] = f[x]; p3[0] = c[x]; p4[0] = c[x];
p1[n_] := f[x*p1[n - 1]]
p2[n_] := If[Mod[n, 2] == 1, c[x*p2[n - 1]], f[x*p2[n - 1]]]
p3[n_] := If[Mod[n, 2] == 1, f[x*p3[n - 1]], c[x*p3[n - 1]]]
p4[n_] := c[x*p4[n - 1]]
t1 = Table[p1[n], {n, 0, z}]  (* A218984 *)
t2 = Table[p2[n], {n, 0, z}]  (* A090017 *)
t3 = Table[p3[n], {n, 0, z}]  (* A123347 *)
t4 = Table[p4[n], {n, 0, z}]  (* A218985 *)

PARI

Vec((4 - 3*x - 2*x^2) / ((1 - x)*(1 - 4*x - 2*x^2)) + O(x^40)) \\ Colin Barker, Nov 13 2017

Formula

a(n) = [x*a(n-1)], where x=2+sqrt(6), a(0) = [x].

a(n) = 5*a(n-1) - 2*a(n-2) - 2*a(n-3).

G.f.: (4 - 3*x - 2*x^2)/(1 - 5*x + 2*x^2 + 2*x^3).

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