A184838 a(n) = n + floor(n*t) + floor(n*t^2) + floor(n*t^3) + floor(n/t), where t is the pentanacci constant.

12, 28, 42, 58, 72, 88, 103, 117, 132, 147, 162, 178, 192, 208, 221, 237, 252, 267, 282, 297, 312, 328, 341, 357, 371, 387, 402, 417, 432, 445, 460, 476, 490, 506, 520, 536, 551, 565, 580, 595, 610, 626, 640, 656, 669, 685, 700, 715, 730, 745, 760, 775, 789, 805, 819, 835, 850, 865, 880, 893, 909, 924, 939, 954, 969, 984, 999, 1013, 1029, 1043, 1059, 1074, 1089, 1104, 1118, 1133, 1149, 1163, 1179, 1193, 1209, 1223, 1238, 1253, 1268, 1283, 1299, 1313, 1327, 1341, 1357, 1372, 1387, 1402, 1417, 1432, 1447

Offset

1,1

Comments

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

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

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

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

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

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

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

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

Links

Table of n, a(n) for n=1..97.

Formula

Limit a(n)/n = t^4 = 14.937857588939362411757354...

a(n) = n + floor(n*p/q) + floor(n*r/q) + floor(n*s/q) + floor(n*u/q), where p=t, q=t^2, r=t^3, s=t^4, u=t^5, and t is the pentanacci constant.

Example

Given t = pentanacci constant, then t^4 = 1 + t + t^2 + t^3 + 1/t,
t = 1.965948236645..., t^2 = 3.864952469169..., t^3 = 7.598296491482..., t^4 = 14.93785758893..., t^5 = 29.36705478623...

Prog

(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)}

Crossrefs

Cf. A184835, A184836, A184837, A184839; A184812, A103814.

Sequence in context: A224613 A134618 A108405 * A044073 A044454 A098502

Adjacent sequences: A184835 A184836 A184837 * A184839 A184840 A184841

Keyword

nonn

Author

Paul D. Hanna, Jan 23 2011

Status

approved