A184835 - OEIS

%I #14 Feb 15 2014 15:45:41

%S 1,3,4,7,8,10,11,15,16,18,19,22,23,25,27,31,32,34,35,38,39,41,43,46,

%T 47,49,50,53,54,57,60,62,63,65,67,69,70,73,75,77,78,80,82,84,86,89,91,

%U 93,94,96,98,100,101,104,106,108,109,112,114,116,119,121,123,124,126,128,130,131,134,136,138,139,141,143,146,148,150,152,154,155,157,159,161,163,165,167,169,170,173,175,177,179,182,183,185,186,189,190,193,194,197,198,200,201,205,206,209,210,213,214,216,217,220,222,224,227,228

%N a(n) = n + floor(n/t) + floor(n/t^2) + floor(n/t^3) + floor(n/t^4), where t is the pentanacci constant.

%C This is one of five sequences that partition the positive integers.

%C Given t is the pentanacci constant, then the following sequences are disjoint:

%C . A184835(n) = n + [n/t] + [n/t^2] + [n/t^3] + [n/t^4],

%C . A184836(n) = n + [n*t] + [n/t] + [n/t^2] + [n/t^3],

%C . A184837(n) = n + [n*t] + [n*t^2] + [n/t] + [n/t^2],

%C . A184838(n) = n + [n*t] + [n*t^2] + [n*t^3] + [n/t],

%C . A184839(n) = n + [n*t] + [n*t^2] + [n*t^3] + [n*t^4], where []=floor.

%C This is a special case of _Clark Kimberling_'s results given in A184812.

%F Limit a(n)/n = t = 1.9659482366454853371899373...

%F 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.

%e Given t = pentanacci constant, then t = 1 + 1/t + 1/t^2 + 1/t^3 + 1/t^4,

%e t = 1.965948236645..., t^2 = 3.864952469169..., t^3 = 7.598296491482..., t^4 = 14.93785758893..., t^5 = 29.36705478623...

%t 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 *)

%o (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)}

%Y Cf. A184836, A184837, A184838, A184839; A184820, A184823, A184812, A103814.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jan 23 2011