A184823 - OEIS

%I #7 Mar 30 2012 18:37:25

%S 1,3,4,7,8,10,11,15,16,18,19,22,23,25,28,30,31,33,35,37,38,41,43,45,

%T 46,48,51,52,55,57,59,60,62,64,66,68,70,72,74,75,78,79,82,83,86,87,89,

%U 90,93,94,97,98,101,103,104,107,108,111,112,115,116,118,119,122,124,126,128,130,131,133,135,138,139,141,143,145,146,148,151,153,155,157,159,160,162,165,167,168,170,172,174,175,178,180,182,183,186,187,189,190,194,195,197,198,201,202,204,208,209,211,212,215,216,218,220,223,224

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

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

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

%C . A184823(n) = n + [n/t] + [n/t^2] + [n/t^3],

%C . A184824(n) = n + [n*t] + [n/t] + [n/t^2],

%C . A184825(n) = n + [n*t] + [n*t^2] + [n/t],

%C . A184826(n) = n + [n*t] + [n*t^2] + [n*t^3], where []=floor.

%C This is a special case of Clark Kimberling's results given in A184812.

%F Limit a(n)/n = t = 1.9275619754829253042619058...

%F a(n) = n + floor(n*p/s) + floor(n*q/s) + floor(n*r/s), where p=t, q=t^2, r=t^3, s=t^4, and t is the tetranacci constant.

%e Let t be the tetranacci constant, then t = 1 + 1/t + 1/t^2 + 1/t^3 and:

%e t = 1.92756197548..., t^2 = 3.71549516932..., t^3 = 7.16184720848..., t^4 = 13.8049043532...

%o (PARI) {a(n)=local(t=real(polroots(1+x+x^2+x^3-x^4)[2])); n+floor(n/t)+floor(n/t^2)+floor(n/t^3)}

%Y Cf. A184824, A184825, A184826; A184820, A184835, A184812, A086088.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jan 23 2011