A214997 Power ceiling-floor sequence of 2+sqrt(2).

4, 13, 45, 153, 523, 1785, 6095, 20809, 71047, 242569, 828183, 2827593, 9654007, 32960841, 112535351, 384219721, 1311808183, 4478793289, 15291556791, 52208640585, 178251448759, 608588513865, 2077851157943, 7094227604041, 24221208100279, 82696377193033

Offset

0,1

Comments

See A214992 for a discussion of power ceiling-floor sequence and power ceiling-floor function, p3(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = 2+sqrt(2), and the limit p3(r) = 3.8478612632206289...

a(n) is the number of words over {0,1,2,3} of length n+1 that avoid 23, 32, and 33. As an example, a(2)=45 corresponds to the 45 such words of length 3; these are all 64 words except for the 19 prohibited cases which are 320, 321, 322, 323, 230, 231, 232, 233, 330, 331, 332, 333, 023, 123, 223, 032, 132, 033, 133. - Greg Dresden and Mina BH Arsanious, Aug 09 2023

Links

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

Index entries for linear recurrences with constant coefficients, signature (3,2,-2).

Formula

a(n) = floor(x*a(n-1)) if n is odd, a(n) = ceiling(x*a(n-1)) if n is even, where x = 2+sqrt(2) and a(0) = ceiling(x).

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

G.f.: (4 + x - 2*x^2)/(1 - 3*x - 2*x^2 + 2*x^3).

a(n) = (1/14)*(2*(-1)^n + (27-19*sqrt(2))*(2-sqrt(2))^n + (2+sqrt(2))^n*(27+19*sqrt(2))). - Colin Barker, Nov 13 2017

a(n) = p(n)+q(n)+r(n)+s(n) = p(n+1) where p,q,r,s are defined by A(n) = the column vector (p(n),q(n),r(n),s(n)) = M^n * A(0), where A(0) = (1,1,1,1) and M is the 4 X 4 matrix = [[1,1,1,1], [1,1,1,1], [1,1,1,0], [1,1,0,0]]. - Mina BH Arsanious, Jan 18 2025

Sum_{k=0..n} a(k) = (r(n-2)-3)/2 where r(n) is defined in the previous formula. - Mina BH Arsanious, May 21 2025

Sum_{k=0..n} a(k) = (a(n+1) - 2*a(n-1) - 3)/2. - Mina BH Arsanious, Mar 19 2026

Example

a(0) = ceiling(r) = 4, where r = 2+sqrt(2);
a(1) = floor(4*r) = 13; a(2) = ceiling(13*r) = 45.

Mathematica

(* See A214996 *)
CoefficientList[Series[(4+x-2*x^2)/(1-3*x-2*x^2+2*x^3), {x, 0, 50}], x] (* G. C. Greubel, Feb 01 2018 *)

Prog

(PARI) Vec((4 + x - 2*x^2) / ((1 + x)*(1 - 4*x + 2*x^2)) + O(x^40)) \\ Colin Barker, Nov 13 2017
(Magma) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!((4 +x-2*x^2)/(1-3*x-2*x^2+2*x^3))); // G. C. Greubel, Feb 01 2018

Crossrefs

Cf. A214992, A007052, A214996, A007070.

Sequence in context: A035356 A320652 A203573 * A189348 A165205 A149431

Adjacent sequences: A214994 A214995 A214996 * A214998 A214999 A215000

Keyword

nonn,easy

Author

Clark Kimberling, Nov 10 2012

Status

approved