A214995 Power ceiling-floor sequence of (golden ratio)^6.
18, 322, 5779, 103699, 1860804, 33390772, 599173093, 10751724901, 192931875126, 3462022027366, 62123464617463, 1114760341086967, 20003562674947944, 358949367807976024, 6441085057868620489, 115580581673827192777, 2074009385071020849498
Offset: 0
Examples
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.
Links
- Clark Kimberling, Table of n, a(n) for n = 0..250
- Index entries for linear recurrences with constant coefficients, signature (17,17,-1).
Programs
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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 *) -
PARI
Vec((18+16*x-x^2)/((1+x)*(1-18*x+x^2)) + O(x^20)) \\ Colin Barker, Mar 04 2016
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
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