A218985 Power ceiling sequence of 2+sqrt(6).

5, 23, 103, 459, 2043, 9091, 40451, 179987, 800851, 3563379, 15855219, 70547635, 313900979, 1396699187, 6214598707, 27651793203, 123036370227, 547449067315, 2435869009715, 10838374173491, 48225234713395, 214577687200563, 954761218229043, 4248200247317299

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(6), and the limit p4(r) = 5.2127890589687233047437696796862841514303439...

See A218984 for the power floor function, p1(x). For comparison of p4 and p1, limit(p4(r)/p1(r)) = 2*(1+sqrt(6))/5 = 1.379795897113271239278913629882356556786378...

Links

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

Index entries for linear recurrences with constant coefficients, signature (5,-2,-2).

Formula

a(n) = [x*a(n-1)], where x=2+sqrt(6), a(0) = [x].

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

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

a(n) = (1/15)*(-3 + (39-16*sqrt(6))*(2-sqrt(6))^n + (2+sqrt(6))^n*(39+16*sqrt(6))). - Colin Barker, Nov 13 2017

Example

a(0) = ceiling(r) = 5, where r = 2+sqrt(6).
a(1) = ceiling(5*r) = 23; a(2) = ceiling(23*r) = 103.

Mathematica

(See A218984.)

Prog

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

Crossrefs

Cf. A214992, A090017, A123347, A218984.

Sequence in context: A054441 A289803 A102285 * A129162 A167660 A290924

Adjacent sequences: A218982 A218983 A218984 * A218986 A218987 A218988

Keyword

nonn,easy

Author

Clark Kimberling, Nov 11 2012

Status

approved