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A218986 Power floor sequence of 2+sqrt(7). 3
4, 18, 83, 385, 1788, 8306, 38587, 179265, 832820, 3869074, 17974755, 83506241, 387949228, 1802315634, 8373110219, 38899387777, 180716881764, 839565690386, 3900413406835, 18120350698497, 84182643014492, 391091624153458, 1816914425657307, 8440932575089601 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

See A214992 for a discussion of power floor sequence and the power floor function, p1(x) = limit of a(n,x)/x^n.  The present sequence is a(n,r), where r = 2+sqrt(7), and the limit p1(r) = 3.83798607113023840500712572585708...

See A218987 for the power floor function, p4.  For comparison with p1, limit(p4(r)/p1(r) = 4-sqrt(7).

LINKS

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

Index entries for linear recurrences with constant coefficients, signature (5,-1,-3).

FORMULA

a(n) = [x*a(n-1)], where x=2+sqrt(7), a(0) = [x].

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

G.f.:  (4 - 2*x - 3*x^2)/(1 - 5*x + x^2 + 3*x^3).

a(n) = (14+(161-61*sqrt(7))*(2-sqrt(7))^n+(2+sqrt(7))^n*(161+61*sqrt(7)))/84. - Colin Barker, Sep 02 2016

EXAMPLE

a(0) = [r] = 4, where r = 2+sqrt(7);

a(1) = [4*r] = 18; a(2) = [18*r] = 83.

MATHEMATICA

x = 2 + Sqrt[7]; z = 30; (* z = # terms in sequences *)

f[x_] := Floor[x]; c[x_] := Ceiling[x];

p1[0] = f[x]; p2[0] = f[x]; p3[0] = c[x]; p4[0] = c[x];

p1[n_] := f[x*p1[n - 1]]

p2[n_] := If[Mod[n, 2] == 1, c[x*p2[n - 1]], f[x*p2[n - 1]]]

p3[n_] := If[Mod[n, 2] == 1, f[x*p3[n - 1]], c[x*p3[n - 1]]]

p4[n_] := c[x*p4[n - 1]]

t1 = Table[p1[n], {n, 0, z}]  (* A218986 *)

t2 = Table[p2[n], {n, 0, z}]  (* A015530 *)

t3 = Table[p3[n], {n, 0, z}]  (* A126473 *)

t4 = Table[p4[n], {n, 0, z}]  (* A218987 *)

LinearRecurrence[{5, -1, -3}, {4, 18, 83}, 30] (* Harvey P. Dale, Jun 18 2014 *)

PROG

(PARI) a(n) = round((14+(161-61*sqrt(7))*(2-sqrt(7))^n+(2+sqrt(7))^n*(161+61*sqrt(7)))/84) \\ Colin Barker, Sep 02 2016

(PARI) Vec((4-2*x-3*x^2)/((1-x)*(1-4*x-3*x^2)) + O(x^30)) \\ Colin Barker, Sep 02 2016

CROSSREFS

Cf. A214992, A015530, A126473, A218987.

Sequence in context: A279285 A129160 A187077 * A143646 A290916 A014348

Adjacent sequences:  A218983 A218984 A218985 * A218987 A218988 A218989

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Nov 11 2012

STATUS

approved

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Last modified February 19 17:00 EST 2019. Contains 320311 sequences. (Running on oeis4.)