[seqfan] Re: Fibonaroot

[Hans Havermann]

Eric Angelini:

> If we call S(k) a sequence with the seed [1,k],... is there a S(k) sequence which doesn't (end in a loop)?


For k<10^7, everything evolves into one of three loops:

{1,1,2,3,5,8,13,21,34,1,8,9,17,8,25,6,31,37,68,6,74,8,1,9}, from smallest k = 1.

{1,7,8,6,14,20,34,9,43,52,95,147,8,155,163,318,4,322,326,9,335,344,4,6}, from smallest k = 3. [Your S(3) contained an error.]

{1,6,7,13,20,6,8,14,4,9,13,4,17,21,38,59,97,156,1,157,158,9,167,176}, from smallest k = 4.

Allowing seeds [j,k], j>1, the latter two loops may be generated from 'smallest' [2,1] and [3,1], respectively. I found two additional loops:

{6,6,12,9,21,30,6,9}, from 'smallest' [3,3].

{9}, from 'smallest' [9,9].