A218992 Power ceiling sequence of 3+sqrt(10).

7, 44, 272, 1677, 10335, 63688, 392464, 2418473, 14903303, 91838292, 565933056, 3487436629, 21490552831, 132430753616, 816075074528, 5028881200785, 30989362279239, 190965054876220, 1176779691536560, 7251643204095581

Offset: 0

Author

Clark Kimberling, Nov 12 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 = 3+sqrt(10), and the limit p4(r) = 7.16724801485749657...

See A218991 for the power floor function, p1(x); for comparison of p1 and p4, we have limit(p4(r)/p1(r) = (3+sqrt(10))/5 = 1.23245553...

Examples

			a(0) = ceiling(r) = 7, where r = 3+sqrt(10);
a(1) = ceiling(7*r) = 44;
a(2) = ceiling(44*r) = 272.
		

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, A218991.

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

Programs

Magma

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

Mathematica

(See A218991.)
LinearRecurrence[{7,-5,-1},{7,44,272},20] (* Harvey P. Dale, Sep 22 2016 *)

Formula

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

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

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

a(n) = ((5+sqrt(10))*(3-sqrt(10))^(n+3)+(5-sqrt(10))*(3+sqrt(10))^(n+3)-10)/60. [Bruno Berselli, Nov 22 2012]