A184826 a(n) = n + floor(n*t) + floor(n*t^2) + floor(n*t^3) where t is the tetranacci constant.

12, 26, 40, 53, 67, 81, 96, 109, 123, 137, 150, 164, 179, 192, 205, 219, 233, 246, 261, 275, 289, 302, 316, 330, 344, 358, 372, 385, 398, 412, 427, 440, 454, 468, 482, 495, 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

Offset: 1

Author

Paul D. Hanna, Jan 23 2011

Keywords

• nonn

Comments

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

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

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

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

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

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

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

Examples

			Let t be the tetranacci constant, then t^4 = 1 + t + t^2 + t^3 and:
t = 1.92756197548..., t^2 = 3.71549516932..., t^3 = 7.16184720848..., t^4 = 13.8049043532...
		

Crossrefs

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

Programs

Mathematica

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 *)

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)}

Formula

Limit a(n)/n = t^4 = 13.804904353297009893939920...

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.