OFFSET
1,2
COMMENTS
This is one of five sequences that partition the positive integers.
Given t is the pentanacci constant, then the following sequences are disjoint:
. A184835(n) = n + [n/t] + [n/t^2] + [n/t^3] + [n/t^4],
. A184836(n) = n + [n*t] + [n/t] + [n/t^2] + [n/t^3],
. A184837(n) = n + [n*t] + [n*t^2] + [n/t] + [n/t^2],
. A184838(n) = n + [n*t] + [n*t^2] + [n*t^3] + [n/t],
. A184839(n) = n + [n*t] + [n*t^2] + [n*t^3] + [n*t^4], where []=floor.
This is a special case of Clark Kimberling's results given in A184812.
FORMULA
Limit a(n)/n = t = 1.9659482366454853371899373...
a(n) = n + floor(n*p/u) + floor(n*q/u) + floor(n*r/u) + floor(n*s/u), where p=t, q=t^2, r=t^3, s=t^4, u=t^5, and t is the pentanacci constant.
EXAMPLE
Given t = pentanacci constant, then t = 1 + 1/t + 1/t^2 + 1/t^3 + 1/t^4,
t = 1.965948236645..., t^2 = 3.864952469169..., t^3 = 7.598296491482..., t^4 = 14.93785758893..., t^5 = 29.36705478623...
MATHEMATICA
With[{pc=x/.FindRoot[x^5-x^4-x^3-x^2-x-1==0, {x, 1.96}, WorkingPrecision-> 100]}, Table[n+Total[Table[Floor[n/pc^i], {i, 4}]], {n, 150}]] (* Harvey P. Dale, Jun 21 2011 *)
PROG
(PARI) {a(n)=local(t=real(polroots(1+x+x^2+x^3+x^4-x^5)[1])); n+floor(n/t)+floor(n/t^2)+floor(n/t^3)+floor(n/t^4)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 23 2011
STATUS
approved