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A346965 a(n) is the number of ascending subsequences in reducing n to 1 using the Collatz reduction, or -1 if n refutes the Collatz conjecture.

%I A346965 #65 Mar 27 2025 10:14:58
%S A346965 0,0,1,0,1,1,3,0,4,1,3,1,2,3,2,0,3,4,4,1,1,3,2,1,5,2,17,3,4,2,16,0,6,
%T A346965 3,2,4,4,4,6,1,17,1,6,3,4,2,16,1,5,5,5,2,2,17,17,3,7,4,6,2,3,16,15,0,
%U A346965 6,6,5,3,3,2,16,4,18,4,2,4,5,6,6,1,4,17,17
%N A346965 a(n) is the number of ascending subsequences in reducing n to 1 using the Collatz reduction, or -1 if n refutes the Collatz conjecture.
%C A346965 In this sequence, a subsequence is considered ascending for as long as a (3*n + 1) / 2 step is required.
%H A346965 Douglas Boffey, <a href="/A346965/b346965.txt">Table of n, a(n) for n = 1..20000</a>
%H A346965 Douglas Boffey, <a href="/A346965/a346965.c.txt">Code used for generating b-file</a>
%F A346965 a(2^n) = 0.
%F A346965 a((2^n*(2*x+1)-1) * 2^y) = a(3^n*(2*x+1)-1) + 1, where x, y >= 0.
%F A346965 a(n) = a(A085062(n)) + (n mod 2) for n >= 2. - _Alan Michael Gómez Calderón_, Feb 09 2025
%F A346965 a(n) = A160541(A000265(n)). - _Alan Michael Gómez Calderón_, Mar 19 2025
%e A346965 a(9) = 4, viz.
%e A346965   9->14;
%e A346965   14->7->11->17->26;
%e A346965   26->13->20;
%e A346965   20->10->5->8.
%o A346965 (C) /* A007814 */
%o A346965 int num_clear_bits(unsigned n) {
%o A346965   if (n == 0)
%o A346965     return -1;
%o A346965   return log2(n & -n);
%o A346965 }
%o A346965 int A346965(unsigned n) {
%o A346965   int x;
%o A346965   int result = 0;
%o A346965   n >>= num_clear_bits(n);
%o A346965   while (n > 1) {
%o A346965     x = num_clear_bits(n + 1);
%o A346965     n = ((n >> x) + 1) * pow(3, x) - 1;
%o A346965     n >>= num_clear_bits(n);
%o A346965     ++result;
%o A346965   }
%o A346965   return result;
%o A346965 }
%Y A346965 Cf. A000265, A070168, A078719, A085062, A160541, A221469.
%K A346965 nonn
%O A346965 1,7
%A A346965 _Douglas Boffey_, Aug 09 2021