A184820 - OEIS

%I #12 Mar 30 2012 18:37:25

%S 1,3,4,7,8,10,12,14,15,17,19,21,23,25,27,28,31,32,34,35,38,39,41,44,

%T 45,47,48,51,52,54,56,58,59,62,64,65,67,69,71,72,75,76,78,80,82,84,85,

%U 88,89,91,93,95,96,98,100,102,103,106,108,109,112,113,115,116,119,120,122,124,126,128,129,132,133,135,137,139,140,143,144,146,148,150,152,153,156,157,159,161,163,164,166,169,170,172,174,176,177,179,181,183,184,187,188,190,193,194,196,197,200,201,203,205,207,208,210,213,214,216,218,220,221,224,225,227,228,231,233,234,237,238,240,242,244,245,247

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

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

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

%C . A184820(n) = n + [n/t] + [n/t^2],

%C . A184821(n) = n + [n*t] + [n/t],

%C . A184822(n) = n + [n*t] + [n*t^2], where []=floor.

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

%F Limit a(n)/n = t = 1.8392867552141611325518525646532866...

%F a(n) = n + floor(n*p/r) + floor(n*q/r), where p=t, q=t^2, r=t^3, and t is the tribonacci constant (see Clark Kimberling's formula in A184812).

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

%e t = 1.8392867552..., t^2 = 3.3829757679..., t^3 = 6.2222625231...

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

%Y Cf. A184821, A184822, A184812, A058265; A184823, A184835.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jan 22 2011