A184824 - OEIS

%I #9 Dec 16 2024 22:19:16

%S 2,6,9,14,17,21,24,29,32,36,39,44,47,50,54,58,61,65,69,73,76,80,84,88,

%T 91,95,100,102,106,110,114,117,121,125,129,132,136,140,144,147,152,

%U 154,158,161,166,169,173,176,181,184,188,191,196,200,203,207,210,214,217,222,225,229,232,237,240,244,248,252,255,258,262,266,269,273,277,281,284,288,292,296,300,304,307,310,314,318,322,325,329,333,337,340,345,348,352,355,359,362,366,369,374,377,381,384,389,392,396,401,404,408

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

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

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

%C . A184823(n) = n + [n/t] + [n/t^2] + [n/t^3],

%C . A184824(n) = n + [n*t] + [n/t] + [n/t^2],

%C . A184825(n) = n + [n*t] + [n*t^2] + [n/t],

%C . A184826(n) = n + [n*t] + [n*t^2] + [n*t^3], where []=floor.

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

%F Limit a(n)/n = t^2 = 3.7154951693276375317543272...

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

%e Let t be the tetranacci constant, then t^2 = 1 + t + 1/t + 1/t^2 and:

%e t = 1.92756197548..., t^2 = 3.71549516932..., t^3 = 7.16184720848..., t^4 = 13.8049043532...

%o (PARI) {a(n)=local(t=real(polroots(1+x+x^2+x^3-x^4)[2])); n+floor(n*t)+floor(n/t)+floor(n/t^2)}

%Y Cf. A184823, A184825, A184826; A184812, A086088.

%K nonn

%O 1,1

%A _Paul D. Hanna_, Jan 23 2011