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A214997 Power ceiling-floor sequence of 2+sqrt(2). 4
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 (list; graph; refs; listen; history; text; internal format)
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
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
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
Sequence in context: A035356 A320652 A203573 * A189348 A165205 A149431
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Nov 10 2012
STATUS
approved

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Last modified April 16 00:00 EDT 2024. Contains 371696 sequences. (Running on oeis4.)