A261393 - OEIS

%I #17 Aug 26 2015 12:55:53

%S 0,1,2,4,4,5,8,13,8,7,10,14,16,17,26,40,16,11,14,16,20,19,28,41,32,25,

%T 34,44,52,53,80,121,32,19,22,20,28,23,32,43,40,29,38,46,56,55,82,122,

%U 64,41,50,52,68,61,88,125,104,79,106,134

%N Additive terms of the rational Collatz tree.

%C A full binary tree can be constructed by inverting the Collatz function and removing the even and odd branch conditions. The rational numbers formed by this tree have the form c = ((2^x + a(n))/(3^y)) where (y = A000120(n)) and (x >= A029837(c+1)-y).

%H Charles R Greathouse IV, <a href="/A261393/b261393.txt">Table of n, a(n) for n = 0..10000</a>

%F a(0) = 0, a(2*n) = 2*a(n), and a(2*n+1) = a(n) + 3^A000120(n).

%e a(0) = 0.

%e a(1) = a(2*0+1) = a(0) + 3^A000120(0) = 0 + 3^0 = 1.

%e a(2) = a(2*1) = 2*a(1) = 2*1 = 2.

%e a(3) = a(2*1+1) = a(1) + 3^A000120(1) = 1 + 3^1 = 4.

%e a(4) = a(2*2) = 2*a(2) = 2*2 = 4.

%e a(5) = a(2*2+1) = a(2) + 3^A000120(2) = 2 + 3^1 = 5.

%e a(6) = a(2*3) = 2*a(3) = 2*4 = 8.

%e a(7) = a(2*3+1) = a(3) + 3^A000120(3) = 4 + 3^2 = 13.

%e a(8) = a(2*4) = 2*a(4) = 2*4 = 8.

%o (Python)

%o def h(n):

%o   if(n == 0):     return 0

%o   elif(n%2 == 0): return h(n/2)

%o   else:           return h((n-1)/2) + 1

%o def a(n):

%o   if(n == 0):     return 0

%o   elif(n%2 == 0): return 2*a(n/2)

%o   else:           return a((n-1)/2) + 3**h((n-1)/2)

%o (PARI) h(n)=if(n,n>>=valuation(n,2);h(n\2)+1,0)

%o a(n)=if(n,my(k=valuation(n,2));n>>=k+1;(a(n)+3^h(n))<<k,0) \\ _Charles R Greathouse IV_, Aug 18 2015

%Y Cf. A000120, A029837.

%K nonn,easy

%O 0,3

%A _Jared Deckard_, Aug 17 2015