## S000260

Irregular triangle of a conjectured periodicity in the Collatz (3x+1) iteration.

2, 4, 16, 12, 20, 8, 44, 76, 40, 16, 28, 52, 24, 20, 76, 60, 20, 56, 84, 68, 8, 152, 80, 108, 52, 24, 312, 152, 80, 152, 264, 56, 84, 68, 144, 204, 212, 56, 84, 36, 32, 176, 172, 36, 112, 236, 4, 316, 68, 144, 24, 152, 264, 116, 236, 152, 264

1

The length of the patterns is given in A186009.

T. D. Noe, Plot of 13 rows of irregular triangle

T. D. Noe, Table of 13 rows of irregular triangle

The first seven irregular rows are {2}, {4}, {16}, {12, 20}, {8, 44, 76}, {40, 16, 28, 52, 24, 20, 76}, and {60, 20, 56, 84, 68, 8, 152, 80, 108, 52, 24, 312}. Note that the sum of the numbers in row r is 2^s, where s = A020914(r).

(Mma) (* the range of the Do loop is in A020914 *) t = {}; Do[found = 0; lim = Infinity; seq = {}; n = 0; While[n < lim, n++; cnt = 1; m = n; While[If[EvenQ[m], m = m/2, m = (3*m + 1)/2]; m > n, cnt++]; If[cnt == nn, AppendTo[seq, n]; found++; If[found == 1, lim = n + 2^nn]]]; AppendTo[t, Differences[seq]], {nn, {1, 2, 4, 5, 7, 8, 10, 12, 13}}]; Flatten[t]

Cf.

nonn,tabf,nice

T. D. Noe, Sep 17 2014