A214994 Power ceiling sequence of (golden ratio)^5.

12, 134, 1487, 16492, 182900, 2028393, 22495224, 249475858, 2766729663, 30683502152, 340285253336, 3773821288849, 41852319430676, 464149335026286, 5147495004719823, 57086594386944340, 633100033261107564, 7021186960259127545, 77866156596111510560

Offset: 0

Author

Clark Kimberling, Nov 09 2012

Keywords

• nonn
• easy

Comments

See A214992 for a discussion of power ceiling sequence and the power ceiling function, p4(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = (golden ratio)^5, and the limit p4(r) = (1/30)*(105+47*sqrt(5)).

See A214993 for the power floor sequence and power floor function, p1. For comparison with p4, we have p4(r)/p1(r) = (5 + 3*sqrt(5))/10.

Examples

			a(0) = ceiling(r) = [11.0902]=12, where r=(1+sqrt(5))^5.
a(1) = ceiling(12) = 134; a(2) = ceiling(134 ) = 1487.
		

Links

• Clark Kimberling, Table of n, a(n) for n = 0..250
• Index entries for linear recurrences with constant coefficients, signature (12,-10,-1).

Crossrefs

Cf. A214992, A214993, A049666, A015457.

Programs

Magma

I:=[12,134,1487]; [n le 3 select I[n] else 12*Self(n-1) - 10*Self(n-2) - Self(n-3): n in [1..30]]; // G. C. Greubel, Feb 01 2018

Mathematica

(See A214993.)
LinearRecurrence[{12,-10,-1}, {12,134,1487}, 30] (* G. C. Greubel, Feb 01 2018 *)

PARI

Vec((12 - 10*x - x^2) / ((1 - x)*(1 - 11*x - x^2)) + O(x^40)) \\ Colin Barker, Nov 13 2017

Formula

a(n) = ceiling(x*a(n-1)), x=((1+sqrt(5))/2)^5, a(0) = ceiling(x).

a(n) = 12*a(n-1) - 10*a(n-2) - a(n-3).

G.f.: (12 - 10*x - x^2)/(1 - 12*x + 10*x^2 + x^3).

a(n) = (1/550)*(-50 + (3325-1487*sqrt(5))*((11-5*sqrt(5))/2)^n + ((11+5*sqrt(5))/2)^n*(3325+1487*sqrt(5))). - Colin Barker, Nov 13 2017