A328037 Irregular triangle T(n,k) read by rows: "quotient trajectories" in reduced Collatz sequences; i.e., T(n,k) = q-value(A256598(n,k)) where q-value(z) = (z - A259663(m,j))/2^(m+j) and (m,j) is the unique pair such that z == A259663(m,j) (mod 2^(m+j)). (See Comments for definitions.)

0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 1, 0, 2, 0, 0, 0, 1, 5, 0, 0, 2, 6, 20, 15, 5, 17, 3, 9, 29, 2, 8, 24, 74, 27, 5, 15, 47, 17, 53

Offset: 0

Author

Bob Selcoe, Oct 03 2019

Keywords

• nonn
• tabf

Comments

Coefficients T(m,j) in the array A259663 are least residues in congruence classes T(m,j) mod 2^(m+j). T"(m,j) denotes all members of that class.

Reduced Collatz sequences (i.e., reduced sequences) are standard Collatz sequences excluding even terms. Row n in triangle A256598 shows the reduced sequence starting with 2n+1.

Let every positive odd number z = T"(m,j)_q, where q is the quotient of z in T"(m,j). For example, T(2,3) = 7 in A259663, so T"(2,3) contains all numbers == 7 (mod 32). So z = T"(2,3)_0 = 7, z = T"(2,3)_1 = 39, z = T"(2,3)_2 = 71, etc.

T(n,k) is the q-value of A256598(n,k) (see Example below). Thus, each row n is defined here as a "quotient trajectory" for the reduced sequence with starting term 2n+1.

Examples

			Triangle starts:
  0;
  0, 0, 0;
  0, 0;
  0, 0, 2, 0, 0, 0;
  1, 0, 0, 2, 0, 0, 0;
  0, 2, 0, 0, 0;
  0, 0, 0;
  0, 0, 0, 0, 0, 0;
  2, 0, 0, 0;
  0, 1, 0, 2, 0, 0, 0;
  0, 0;
  0, 0, 0, 0, 0;
  3, 0, 1, 0, 2, 0, 0, 0;
  ...
n=13 starts with 27 = T"(2,2)_1 and takes 41 steps: 1, 5, 0, 0, 2, 6, 20, 15, 5, ..., 0, 0, 0.
Row n=12 maps to the reduced sequence n=12 in A256598: 25 -> 19 -> 29 -> 11 -> 17 -> 13 -> 5 -> 1, which is T"(2,1)_3 -> T"(3,2)_0 -> T"(3,1)_1 -> T"(2,2)_0 -> T"(2,1)_2 -> T"(3,1)_0 -> T"(4,1)_0 -> T"(2,1)_0.
		

Crossrefs

Cf. A256598, A259663.

Programs

PARI

Tdt(n, k) = if (n==2, if (k%2, 2^k-1, 3*2^k-1), if (n==3, if (k%2, 7*2^k-1, 5*2^k-1), mj = 2^(n-3) % 2^(n-2); mk = k % 2^(n-2); (2^k*3^(mj-mk) - 1) % 2^(n+k))); \\ A259663
qvalue(m) = {my(line = 2, i, md); while (1, i = line; for (j=1, line-1, md = Tdt(i, j); if (m % (2^(i+j)) == md % (2^(i+j)), return((m-md)/2^(i+j))); i--;); line ++;);}
row(n) = {my(oddn = 2*n+1, vl = List(oddn), x); while (oddn != 1, x = 3*oddn+1; oddn = x >> valuation(x, 2); listput(vl, oddn)); my(v = Vec(vl)); for (i=1, #v, v[i] = qvalue(v[i]);); v;} \\ A256598
tabf(nn) = {for (n=0, nn, my(rown = row(n)); for (k=1, #rown, print1(rown[k], ", ")); print;);} \\ Michel Marcus, Oct 04 2019