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A214993 Power floor sequence of (golden ratio)^5. 4
11, 121, 1341, 14871, 164921, 1829001, 20283931, 224952241, 2494758581, 27667296631, 306835021521, 3402852533361, 37738212888491, 418523194306761, 4641493350262861, 51474950047198231, 570865943869443401, 6331000332611075641, 70211869602591275451 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

See A214992 for a discussion of power floor sequence and also the power floor function, p1(x) = limit of a(n,x)/x^n.  The present sequence is a(n,r), where r = (golden ratio)^5, and the limit p1(r) = (3/22)*(3+2*sqrt(5)).

LINKS

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

Index entries for linear recurrences with constant coefficients, signature (12,-10,-1).

FORMULA

a(n) = [x*a(n-1)], where x=((1+sqrt(5))/2)^5, a(0) = [x].

a(n) = 1 (mod 10).

a(n) = 12*a(n-1) - 10*a(n-2) - a(n-3).

G.f.: (11 - 11*x - x^2)/(1 - 12*x + 10*x^2 + x^3).

a(n) = (1/55)*(5 + (300-134*sqrt(5))*((11-5*sqrt(5))/2)^n + 2*(11/2+(5*sqrt(5))/2)^n*(150+67*sqrt(5))). - Colin Barker, Nov 13 2017

EXAMPLE

a(0) = [r] = [11.0902] = 11, where r = (1+sqrt(5))^5.

a(1) = [11*r] = 121; a(2) = [121*r] = 1341.

MATHEMATICA

x = GoldenRatio^5; z = 30; (* z = # terms in sequences *)

z1 = 100; (* z1 = # digits in approximations *)

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]]

Table[p1[n], {n, 0, z}]  (* A214993 *)

Table[p2[n], {n, 0, z}]  (* A049666 *)

Table[p3[n], {n, 0, z}]  (* A015457 *)

Table[p4[n], {n, 0, z}]  (* A214994 *)

LinearRecurrence[{12, -10, -1}, {11, 121, 1341}, 30] (* G. C. Greubel, Feb 01 2018 *)

PROG

(PARI) Vec((11 - 11*x - x^2) / ((1 - x)*(1 - 11*x - x^2)) + O(x^20)) \\ Colin Barker, Nov 13 2017

(MAGMA) I:=[11, 121, 1341]; [n le 3 select I[n] else 12*Self(n-1)-10*Self(n-2)-Self(n-3): n in [1..30]]; // (* G. C. Greubel, Feb 01 2018 *)

CROSSREFS

Cf. A214992, A214994, A049666, A015457.

Sequence in context: A003590 A072051 A092846 * A263608 A088760 A080486

Adjacent sequences:  A214990 A214991 A214992 * A214994 A214995 A214996

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Nov 09 2012

STATUS

approved

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Last modified February 25 20:31 EST 2018. Contains 299655 sequences. (Running on oeis4.)