OFFSET

1,1

COMMENTS

a(1) = A078441(9).

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.

A majority of numbers the sequence generate trivial 9-tuples (m, m, m, m, m, m, m, m, m).

For a(n) < 200000, the following sets have been identified:

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, ...

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, ...

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, ...

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, ...

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, ...

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, ...

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, ...

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, ...

EXAMPLE

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.

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).

MATHEMATICA

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

CROSSREFS

KEYWORD

nonn

AUTHOR

Michel Lagneau, Jan 30 2016

STATUS

approved