%I #2 Mar 30 2012 17:34:40

%S 3,8,24,64,7776,20736,13824,36864,45349632,120932352,80621568,

%T 214990848,322486272,859963392,573308928,1528823808,1028294561267712,

%U 2742118830047232,1828079220031488,4874877920083968,7312316880125952

%N A coding sequence of binary based integers using powers of {2,3} for {0,1}.

%C Based on a 2 for zero and three for one substitution, this apparently unique

%C Goedelization of the binary numbers uses the Fibonacci sequence to make

%C the digit ordering unique.

%F The sum of digits is weighted by the Fibonacci sequence to give:

%F a(n)=6^(sum[n]*Fibonacci[n])*2^(Count[n,2]+PosititionSum[n,2])*3^(Count[n,3]+PosititionSum[n,3])/3

%t Table[6^(Sum[Table[((Reverse[IntegerDigits[n, 2]]) /. 0 -> 2) /. 1 -> 3, {n, 0, 50}][[n]][[ m]]*Fibonacci[m], {m, 1, Length[ Table[((Reverse[IntegerDigits[n, 2]]) /. 0 -> 2) /. 1 -> 3, {n, 0, 50}][[n]]]}])*2^(-(Count[ Table[((IntegerDigits[n, 2]) /. 0 -> 2) /. 1 -> 3, {n, 0, 50}][[n]], 2] + Apply[Plus, Flatten[ Position[Table[((Reverse[IntegerDigits[n, 2]]) /. 0 -> 2) /. 1 -> 3, {n, 0, 50}][[n]], 2]]]))*3^(-( Count[Table[((IntegerDigits[n, 2]) /. 0 -> 2) /. 1 -> 3, {n, 0, 50}][[ n]], 3] + Apply[Plus, Flatten[Position[Table[(( Reverse[IntegerDigits[n, 2]]) /. 0 -> 2) /. 1 -> 3, { n, 0, 50}][[n]], 3]]]))/3, {n, 1, 51}]

%K nonn,uned

%O 0,1

%A _Roger L. Bagula_, Apr 28 2010