A218991 Power floor sequence of 3+sqrt(10).

6, 36, 221, 1361, 8386, 51676, 318441, 1962321, 12092366, 74516516, 459191461, 2829665281, 17437183146, 107452764156, 662153768081, 4080375372641, 25144406003926, 154946811396196, 954825274381101, 5883898457682801

Offset: 0

Author

Clark Kimberling, Nov 12 2012

Keywords

• nonn
• easy

Comments

See A214992 for a discussion of power floor sequence and the power floor function, p1(x) = lim_{n->oo} a(n,x)/x^n. The present sequence is a(n,r), where r = 3+sqrt(10), and the limit p1(r) = 5.815421188487681054332319082...

See A218992 for the power floor function, p4. For comparison with p1, we have lim_{r->oo} p4(r)/p1(r) = (3+sqrt(10))/5 = 1.23245553....

Examples

			a(0) = floor(r) = 6, where r = 3+sqrt(10);
a(1) = floor(6*r) = 36;
a(2) = floor(36*r) = 221.
		

Links

• Clark Kimberling, Table of n, a(n) for n = 0..250
• Index entries for linear recurrences with constant coefficients, signature (7,-5,-1).

Crossrefs

Cf. A214992, A005668, A015451, A218992.

Cf. A176398 (3+sqrt(10)).

Programs

Magma

[IsZero(n) select Floor(r) else Floor(r*Self(n)) where r is 3+Sqrt(10): n in [0..20]]; // Bruno Berselli, Nov 22 2012

Mathematica

x = 3 + Sqrt[10]; 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}]  (* A218991 *)
t2 = Table[p2[n], {n, 0, z}]  (* A005668 *)
t3 = Table[p3[n], {n, 0, z}]  (* A015451 *)
t4 = Table[p4[n], {n, 0, z}]  (* A218992 *)

Formula

a(n) = floor(r*a(n-1)), where r=3+sqrt(10), a(0) = floor(r).

a(n) = 7*a(n-1) - 5*a(n-2) - a(n-3).

G.f.: (6 - 6*x - x^2)/(1 - 7*x + 5*x^2 + x^3).

a(n) = ((5+sqrt(10))*(3-sqrt(10))^(n+2) + (5-sqrt(10))*(3+sqrt(10))^(n+2)+2)/12. - Bruno Berselli, Nov 22 2012