A218989 Power ceiling sequence of 2+sqrt(8).

5, 25, 121, 585, 2825, 13641, 65865, 318025, 1535561, 7414345, 35799625, 172855881, 834622025, 4029911625, 19458134601, 93952184905, 453641278025, 2190373851721, 10576060518985, 51065737482825, 246567192007241, 1190531717960265, 5748395639870025

Offset

0,1

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(8), and the limit p4(r) = (18 + 13*sqrt(2))/2 = 5.1978251872643193763459933449608678602008191971286...

See A218988 for the power floor function, p1(x); for comparison of p1 and p4, we have limit(p4(r)/p1(r) = 4 - sqrt(7).

Links

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

Index entries for linear recurrences with constant coefficients, signature (5,0,-4).

Formula

a(n) = ceiling(x*a(n-1)), where x=2+sqrt(8), a(0) = ceiling(x).

a(n) = 5*a(n-1) - 4*a(n-3).

G.f.: (5 - 4*x^2) / ((1 - x)*(1 - 4*x - 4*x^2)). Corrected by Colin Barker, Nov 13 2017

a(n) = (1/7)*(-1 + (18-13*sqrt(2))*(2-2*sqrt(2))^n + (2*(1+sqrt(2)))^n*(18+13*sqrt(2))). - Colin Barker, Nov 13 2017

Example

a(0) = ceiling(r) = 5, where r = 2+sqrt(8);
a(1) = ceiling(5*r) = 25; a(2) = ceiling(25*r) = 121.

Mathematica

(See A218988.)

Prog

(PARI) Vec((5 - 4*x^2) / ((1 - x)*(1 - 4*x - 4*x^2)) + O(x^40)) \\ Colin Barker, Nov 13 2017

Crossrefs

Cf. A214992, A057087, A086347, A218988.

Sequence in context: A123871 A268453 A343801 * A078717 A275906 A249454

Adjacent sequences: A218986 A218987 A218988 * A218990 A218991 A218992

Keyword

nonn,easy

Author

Clark Kimberling, Nov 11 2012

Status

approved