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%I #10 Jun 04 2017 02:42:36
%S 1680,1991,2987,2988,2989,2990,2991,2992,3982,3983,3984,3985,3986,
%T 4722,4723,5313,5314,5315,5316,5317,6576,6577,6578,7083,7084,7085,
%U 7086,7087,7088,7089,7090,7091,7794,7795,7976,7977,7978,7979,7980,7981,8769,8770,8771
%N a(n) begins the first chain of 9 consecutive positive integers of h-values with symmetrical gaps about the center, where h(k) is the length of the finite sequence k, f(k), f(f(k)), ...., 1 in the Collatz (or 3x + 1) problem.
%C a(1) = A078441(9).
%C The 9-tuple of consecutive h(k) are symmetric about the central value h(k+4) which are averages of both their immediate neighbors, their second neighbors, their third neighbors and their fourth neighbors.
%C A majority of numbers the sequence generate trivial 9-tuples (m, m, m, m, m, m, m, m, m).
%C For a(n) < 200000, the following sets have been identified:
%C The 9-tuples {h(k)} of the form {m, p, p, p, p, p, p, p, q} are generated by the numbers of the sequence 12608, 16915, 39169, ...
%C The 9-tuples {h(k)} of the form {m, p, q, q, q, q, q, m, p} are generated by the numbers of the sequence 40553, ...
%C The 9-tuples {h(k)} of the form {m, p, p, p, q, m, m, m, p} are generated by the numbers of the sequence 55107, 124739, ...
%C The 9-tuples {h(k)} of the form {m, m, m, m, p, q, q, q, q} are generated by the numbers of the sequence 55292, 90396, 118109, ...
%C The 9-tuples {h(k)} of the form {m, m, m, p, m, q, m, m, m} are generated by the numbers of the sequence 58756, 71236, 79428, ...
%C The 9-tuples {h(k)} of the form {m, m, p, m, m, m, q, m, m} are generated by the numbers of the sequence 78021, ...
%C The 9-tuples {h(k)} of the form {m, p, m, m, m, m, m, q, m} are generated by the numbers of the sequence 93600, 124768, ...
%C The 9-tuples {h(k)} of the form {m, m, m, p, p, p, q, q, q} are generated by the numbers of the sequence 160705, ...
%e In 9-tuple of consecutive h(k): {h(55107),h(55108),...,h(55115)} = {184,60,60,60,122,184,184,184,60}, the central value is 122, and 184+60 = 2*122. Hence, 55107 is in the sequence.
%e Alternatively, the symmetry can be seen from the differences between consecutive h(k). For {184,60,60,60,122,184,184,184,60}, the differences h(k+1)-h(k) are (-124,0,0,62,62,0,0,-124).
%t lst={};f[n_]:=Module[{a=n,k=0},While[a!=1,k++;If[EvenQ[a],a=a/2,a=a*3+1]];k];Do[If[f[m]+f[m+8]==f[m+1]+f[m+7]&&f[m+2]+f[m+6]==f[m+3]+f[m+5]&& f[m]+f[m+8]==f[m+3]+f[m+5]&&f[m+4]==(f[m]+f[m+8])/2,AppendTo[lst,m]],{m,1,6000}];lst
%Y Cf. A006577, A078441, A268177, A268253.
%K nonn
%O 1,1
%A _Michel Lagneau_, Jan 30 2016
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