OFFSET
1,1
COMMENTS
Or numbers k such that h(k) + h(k+6) = h(k+1) + h(k+5) and h(k+3) = (h(k) + h(k+6))/2, where h(k) is the length of k, f(k), f(f(k)), ..., 1 in the Collatz (or 3x + 1) problem.
a(1) = A078441(7).
The symmetry can be seen from the differences between consecutive h(k) (see the example).
The 7-tuple of consecutive h(k) are symmetric about the central value h(k+3) which are averages of both their immediate neighbors, their second neighbors and their third neighbors.
A majority of numbers of the sequence generate trivial 7-tuples {m, m, m, m, m, m, m}.
The 7-tuples {h(k)} of the form {m, p, p, p, p, p, q} are generated by the numbers of the sequence 1377, 4897, ...
The 7-tuples {h(k)} of the form {m, m, p, m, q, m, m} are generated by the numbers of the sequence 5511, 58757, ...
The 7-tuples {h(k)} of the form {m, p, m, m, m, q, p} are generated by the numbers of the sequence 9514, ...
The 7-tuples {h(k)} of the form {m, m, p, p, p, q, q} are generated by the numbers of the sequence 21442, 25666, ...
The 7-tuples {h(k)} of the form {m, m, m, p, q, q, q} are generated by the numbers of the sequence 55108, 55293, ...
EXAMPLE
In 7-tuple of consecutive {h(k)}: {h(9514),h(9515),h(9516),h(9517),h(9518),h(9519),h(9520)} = {78,52,78,78,78,104,78}, the central value is 78, and 78+78 = 52+104 = 2*78. Hence, 9514 is in the sequence.
Alternatively, the symmetry can be seen from the differences between consecutive {h(k)}. For {78,52,78,78,78,104,78}, the differences {h(k+1)-h(k)} are {-26,26,0,0,26,-26}.
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+6]==f[m+1]+f[m+5]&&f[m+2]+f[m+4]==f[m]+f[m+6]&& f[m]+f[m+6]==f[m+2]+f[m+4]&&f[m+3]==(f[m]+f[m+6])/2, AppendTo[lst, m]], {m, 1, 6000}]; lst
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, Jan 29 2016
STATUS
approved