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A184838 a(n) = n + floor(n*t) + floor(n*t^2) + floor(n*t^3) + floor(n/t), where t is the pentanacci constant. 5

%I #7 Mar 30 2012 18:37:25

%S 12,28,42,58,72,88,103,117,132,147,162,178,192,208,221,237,252,267,

%T 282,297,312,328,341,357,371,387,402,417,432,445,460,476,490,506,520,

%U 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

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

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

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

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

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

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

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

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

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

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

%F 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.

%e Given t = pentanacci constant, then t^4 = 1 + t + t^2 + t^3 + 1/t,

%e t = 1.965948236645..., t^2 = 3.864952469169..., t^3 = 7.598296491482..., t^4 = 14.93785758893..., t^5 = 29.36705478623...

%o (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)}

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

%K nonn

%O 1,1

%A _Paul D. Hanna_, Jan 23 2011

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