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

1,1

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.

Links

Table of n, a(n) for n=1..98.

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.

Example

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...

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 *)

Prog

(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)}

Crossrefs

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

Sequence in context: A025138 A075689 A054303 * A119412 A105814 A297427

Adjacent sequences: A184823 A184824 A184825 * A184827 A184828 A184829

Keyword

nonn

Author

Paul D. Hanna, Jan 23 2011

Status

approved