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%I #25 Mar 18 2024 12:00:34
%S 11,121,1341,14871,164921,1829001,20283931,224952241,2494758581,
%T 27667296631,306835021521,3402852533361,37738212888491,
%U 418523194306761,4641493350262861,51474950047198231,570865943869443401,6331000332611075641,70211869602591275451
%N Power floor sequence of (golden ratio)^5.
%C 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)).
%H Clark Kimberling, <a href="/A214993/b214993.txt">Table of n, a(n) for n = 0..250</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (12,-10,-1).
%F a(n) = [x*a(n-1)], where x=((1+sqrt(5))/2)^5, a(0) = [x].
%F a(n) = 1 (mod 10).
%F a(n) = 12*a(n-1) - 10*a(n-2) - a(n-3).
%F G.f.: (11 - 11*x - x^2)/(1 - 12*x + 10*x^2 + x^3).
%F 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
%e a(0) = [r] = [11.0902] = 11, where r = (1+sqrt(5))^5.
%e a(1) = [11*r] = 121; a(2) = [121*r] = 1341.
%t x = GoldenRatio^5; z = 30; (* z = # terms in sequences *)
%t z1 = 100; (* z1 = # digits in approximations *)
%t f[x_] := Floor[x]; c[x_] := Ceiling[x];
%t p1[0] = f[x]; p2[0] = f[x]; p3[0] = c[x]; p4[0] = c[x];
%t p1[n_] := f[x*p1[n - 1]]
%t p2[n_] := If[Mod[n, 2] == 1, c[x*p2[n - 1]], f[x*p2[n - 1]]]
%t p3[n_] := If[Mod[n, 2] == 1, f[x*p3[n - 1]], c[x*p3[n - 1]]]
%t p4[n_] := c[x*p4[n - 1]]
%t Table[p1[n], {n, 0, z}] (* A214993 *)
%t Table[p2[n], {n, 0, z}] (* A049666 *)
%t Table[p3[n], {n, 0, z}] (* A015457 *)
%t Table[p4[n], {n, 0, z}] (* A214994 *)
%t LinearRecurrence[{12,-10,-1}, {11,121,1341}, 30] (* _G. C. Greubel_, Feb 01 2018 *)
%o (PARI) Vec((11 - 11*x - x^2) / ((1 - x)*(1 - 11*x - x^2)) + O(x^20)) \\ _Colin Barker_, Nov 13 2017
%o (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
%Y Cf. A214992, A214994, A049666, A015457.
%K nonn,easy
%O 0,1
%A _Clark Kimberling_, Nov 09 2012
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