A214995 - OEIS

%I #21 Oct 31 2024 01:37:03

%S 18,322,5779,103699,1860804,33390772,599173093,10751724901,

%T 192931875126,3462022027366,62123464617463,1114760341086967,

%U 20003562674947944,358949367807976024,6441085057868620489,115580581673827192777,2074009385071020849498

%N Power ceiling-floor sequence of (golden ratio)^6.

%C See A214992 for a discussion of power ceiling-floor sequence and the power ceiling-floor function, p3(x) = limit of a(n,x)/x^n.  The present sequence is a(n,r), where r = (golden ratio)^6, and the limit p3(r) = 17.94722275971790615684809...

%H Clark Kimberling, <a href="/A214995/b214995.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 (17,17,-1).

%F a(n) = floor(x*a(n-1)) if n is odd, a(n) = ceiling(x*a(n-1)) if n is even, where x=((1+sqrt(5))/2)^6 and a(0) = ceiling(x).

%F a(n) = 17*a(n-1) + 17*a(n-2) - a(n-3).

%F G.f.: (18 + 16*x - x^2)/((1 + x)*(1 - 18*x + x^2)).

%F a(n) = (4*(-1)^n+(718-321*sqrt(5))*(9+4*sqrt(5))^(-n)+(9+4*sqrt(5))^n*(718+321*sqrt(5)))/80. - _Colin Barker_, Mar 04 2016

%F E.g.f.: exp(-x)*(2 + exp(10*x)*(718*cosh(4*sqrt(5)*x) + 321*sqrt(5)*sinh(4*sqrt(5)*x)))/40. - _Stefano Spezia_, Oct 28 2024

%e a(0) = ceiling(r) = [17.9] = 18 , where r=(1+sqrt(5))^6;

%e a(1) = floor(18*r) = 322; a(2) = ceiling(322*r ) = 5779.

%t x = GoldenRatio^6; 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}]  (* A007805 *)

%t Table[p2[n], {n, 0, z}]  (* A156085 *)

%t Table[p3[n], {n, 0, z}]  (* A214995 *)

%t Table[p4[n], {n, 0, z}]  (* A049660 *)

%t Table[p4[n] - p1[n], {n, 0, z}]  (* A049660 *)

%t Table[p3[n] - p2[n], {n, 0, z}]  (* A099279 *)

%t LinearRecurrence[{17,17,-1},{18,322,5779},30] (* _Harvey P. Dale_, Feb 25 2013 *)

%o (PARI) Vec((18+16*x-x^2)/((1+x)*(1-18*x+x^2)) + O(x^20)) \\ _Colin Barker_, Mar 04 2016

%Y Cf. A214992, A007805, A156085, A049660.

%K nonn,easy

%O 0,1

%A _Clark Kimberling_, Nov 09 2012