A214999 Power floor sequence of sqrt(5).

2, 4, 8, 17, 38, 84, 187, 418, 934, 2088, 4668, 10437, 23337, 52183, 116684, 260913, 583419, 1304564, 2917093, 6522818, 14585464, 32614088, 72927317, 163070438, 364636584, 815352188, 1823182917, 4076760937, 9115914583

Offset: 0

Author

Clark Kimberling, Nov 10 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 = sqrt(5), and the limit p1(r) = 1.4935514451954997630823098687087959696356...

Examples

			a(0) = [r] = 2, where r = sqrt(5); a(1) = [2*r] = 4; a(2) = [4*r] = 8.
		

Links

• Clark Kimberling, Table of n, a(n) for n = 0..250

Crossrefs

Cf. A214992, A215091, A218982, A218983.

Programs

Mathematica

x = Sqrt[5]; 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]]
Table[p1[n], {n, 0, z}]  (* A214999 *)
Table[p2[n], {n, 0, z}]  (* A215091 *)
Table[p3[n], {n, 0, z}]  (* A218982 *)
Table[p4[n], {n, 0, z}]  (* A218983 *)

Formula

a(n) = [x*a(n-1)], where x=sqrt(5), a(0) = [x].