OFFSET
0,1
COMMENTS
See A214992 for a discussion of power floor sequence and the power floor function, p1(x) = lim_{n->oo} a(n,x)/x^n. The present sequence is a(n,r), where r = 3+sqrt(10), and the limit p1(r) = 5.815421188487681054332319082...
See A218992 for the power floor function, p4. For comparison with p1, we have lim_{r->oo} p4(r)/p1(r) = (3+sqrt(10))/5 = 1.23245553....
LINKS
Clark Kimberling, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (7,-5,-1).
FORMULA
a(n) = floor(r*a(n-1)), where r=3+sqrt(10), a(0) = floor(r).
a(n) = 7*a(n-1) - 5*a(n-2) - a(n-3).
G.f.: (6 - 6*x - x^2)/(1 - 7*x + 5*x^2 + x^3).
a(n) = ((5+sqrt(10))*(3-sqrt(10))^(n+2) + (5-sqrt(10))*(3+sqrt(10))^(n+2)+2)/12. - Bruno Berselli, Nov 22 2012
EXAMPLE
a(0) = floor(r) = 6, where r = 3+sqrt(10);
a(1) = floor(6*r) = 36;
a(2) = floor(36*r) = 221.
MATHEMATICA
x = 3 + Sqrt[10]; z = 30; (* z = # terms in sequences *)
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]]
t1 = Table[p1[n], {n, 0, z}] (* A218991 *)
t2 = Table[p2[n], {n, 0, z}] (* A005668 *)
t3 = Table[p3[n], {n, 0, z}] (* A015451 *)
t4 = Table[p4[n], {n, 0, z}] (* A218992 *)
PROG
(Magma) [IsZero(n) select Floor(r) else Floor(r*Self(n)) where r is 3+Sqrt(10): n in [0..20]]; // Bruno Berselli, Nov 22 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Nov 12 2012
STATUS
approved