OFFSET
0,1
COMMENTS
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...
LINKS
Clark Kimberling, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (17,17,-1).
FORMULA
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).
a(n) = 17*a(n-1) + 17*a(n-2) - a(n-3).
G.f.: (18 + 16*x - x^2)/((1 + x)*(1 - 18*x + x^2)).
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
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
EXAMPLE
a(0) = ceiling(r) = [17.9] = 18 , where r=(1+sqrt(5))^6;
a(1) = floor(18*r) = 322; a(2) = ceiling(322*r ) = 5779.
MATHEMATICA
x = GoldenRatio^6; 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}] (* A007805 *)
Table[p2[n], {n, 0, z}] (* A156085 *)
Table[p3[n], {n, 0, z}] (* A214995 *)
Table[p4[n], {n, 0, z}] (* A049660 *)
Table[p4[n] - p1[n], {n, 0, z}] (* A049660 *)
Table[p3[n] - p2[n], {n, 0, z}] (* A099279 *)
LinearRecurrence[{17, 17, -1}, {18, 322, 5779}, 30] (* Harvey P. Dale, Feb 25 2013 *)
PROG
(PARI) Vec((18+16*x-x^2)/((1+x)*(1-18*x+x^2)) + O(x^20)) \\ Colin Barker, Mar 04 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Nov 09 2012
STATUS
approved