A184839 a(n) = n + floor(n*t) + floor(n*t^2) + floor(n*t^3) + floor(n*t^4), where t is the pentanacci constant.

26, 56, 85, 115, 144, 174, 204, 232, 262, 291, 321, 351, 380, 410, 438, 468, 497, 526, 556, 585, 615, 645, 673, 703, 732, 762, 792, 821, 851, 878, 908, 938, 966, 996, 1025, 1055, 1085, 1113, 1143, 1172, 1202, 1232, 1261, 1291, 1319, 1349, 1379, 1408, 1437, 1466, 1496, 1525, 1554, 1584, 1613, 1643, 1673, 1702, 1731, 1759, 1789, 1819, 1848, 1878, 1906, 1936, 1965, 1994, 2024, 2053, 2083, 2113, 2142, 2172, 2200, 2230, 2260, 2289, 2319, 2348, 2377, 2406, 2435, 2465, 2494, 2524, 2554, 2583, 2611, 2640, 2670

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

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Formula

Limit a(n)/n = t^5 = 29.367054786236720687050865...

a(n) = n + floor(n*q/p) + floor(n*r/p) + floor(n*s/p) + floor(n*u/p), 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^5 = 1 + t + t^2 + t^3 + t^4,
t = 1.965948236645..., t^2 = 3.864952469169..., t^3 = 7.598296491482..., t^4 = 14.93785758893..., t^5 = 29.36705478623...

Mathematica

With[{t=Root[x^5-x^4-x^3-x^2-x-1, 1]}, Table[n+Total@@Through[ Floor[ n*t^Range[4]]], {n, 100}]] (* Harvey P. Dale, Dec 12 2019 *)

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^4)}

Crossrefs

Cf. A184835, A184836, A184837, A184838; A184812, A103814.

Sequence in context: A255027 A132767 A255020 * A038848 A245004 A161341

Adjacent sequences: A184836 A184837 A184838 * A184840 A184841 A184842

Keyword

nonn

Author

Paul D. Hanna, Jan 23 2011

Status

approved