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A214997
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Power ceiling-floor sequence of 2+sqrt(2).
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4
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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
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OFFSET
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0,1
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COMMENTS
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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...
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LINKS
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Clark Kimberling, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (3,2,-2).
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FORMULA
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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
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EXAMPLE
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a(0) = ceiling(r) = 4, where r = 2+sqrt(2);
a(1) = floor(4*r) = 13; a(2) = ceiling(13*r) = 45.
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MATHEMATICA
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(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 *)
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PROG
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(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
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CROSSREFS
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Cf. A214992, A007052, A214996, A007070.
Sequence in context: A035356 A320652 A203573 * A189348 A165205 A149431
Adjacent sequences: A214994 A214995 A214996 * A214998 A214999 A215000
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KEYWORD
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nonn,easy
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AUTHOR
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Clark Kimberling, Nov 10 2012
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STATUS
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approved
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