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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. 5
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified November 27 03:56 EST 2021. Contains 349345 sequences. (Running on oeis4.)