A218987 Power ceiling sequence of 2+sqrt(7).

5, 24, 112, 521, 2421, 11248, 52256, 242769, 1127845, 5239688, 24342288, 113088217, 525379733, 2440783584, 11339273536, 52679444897, 244735600197, 1136980735480, 5282129742512, 24539461176489, 114004233933493, 529635319263440, 2460553978854240

Offset: 0

Author

Clark Kimberling, Nov 11 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 = 2+sqrt(7), and the limit p4(r) = 5.19758760498048832156707270895307875397561324042...

See A218986 for the power floor function, p1(x); for comparison of p1 and p4, limit(p4(r)/p1(r) = 4 - sqrt(7).

Examples

			a(0) = ceiling(r) = 5, where r = 2+sqrt(7);
a(1) = ceiling(5*r) = 24; a(2) = ceiling(24*r) = 112.
		

Links

• Clark Kimberling, Table of n, a(n) for n = 0..250
• Index entries for linear recurrences with constant coefficients, signature (5,-1,-3).

Crossrefs

Cf. A214992, A015530, A126473, A218986.

Programs

Mathematica

(See A218986.)

PARI

a(n) = round((-14+(217-83*sqrt(7))*(2-sqrt(7))^n+(2+sqrt(7))^n*(217+83*sqrt(7)))/84) \\ Colin Barker, Sep 02 2016

PARI

Vec((5-x-3*x^2)/((1-x)*(1-4*x-3*x^2)) + O(x^30)) \\ Colin Barker, Sep 02 2016

Formula

a(n) = ceiling(x*a(n-1)), where x=2+sqrt(7), a(0) = ceiling(x).

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

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

a(n) = (-14+(217-83*sqrt(7))*(2-sqrt(7))^n+(2+sqrt(7))^n*(217+83*sqrt(7)))/84. - Colin Barker, Sep 02 2016

E.g.f.: exp(x)*(exp(x)*(217*cosh(sqrt(7)*x) + 83*sqrt(7)*sinh(sqrt(7)*x)) - 7)/42. - Stefano Spezia, Aug 05 2025