A184826 - OEIS

%I #7 Oct 18 2012 16:20:47

%S 12,26,40,53,67,81,96,109,123,137,150,164,179,192,205,219,233,246,261,

%T 275,289,302,316,330,344,358,372,385,398,412,427,440,454,468,482,495,

%U 509,524,537,551,565,578,591,606,620,633,647,661,675,689,703,717,730,744,758,772,785,799,813,826,840,855,869,882,896,910,923,938,952,965,978,992,1006,1019,1034,1048,1062,1075,1089,1103,1117,1131,1144,1158,1171,1185,1200,1213,1227,1241,1255,1268,1283,1297,1310,1324,1337,1351

%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^4 = 13.804904353297009893939920...

%F a(n) = n + floor(n*q/p) + floor(n*r/p) + floor(n*s/p), 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^4 = 1 + t + t^2 + t^3 and:

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

%t Module[{t=x/.FindRoot[x^4-x^3-x^2-x-1==0,{x,2},WorkingPrecision->200], t2,t3},t2=t^2;t3=t^3;Table[n+Floor[t*n]+Floor[t2*n]+Floor[t3*n], {n,100}]] (* _Harvey P. Dale_, Oct 18 2012 *)

%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. A184823, A184824, A184825; A184812, A086088.

%K nonn

%O 1,1

%A _Paul D. Hanna_, Jan 23 2011