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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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,9

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.

LINKS

Table of n, a(n) for n=0..81.

EXAMPLE

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.

PROG

(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

CROSSREFS

Cf. A256598, A259663.

Sequence in context: A127324 A083917 A117974 * A193426 A156062 A156064

Adjacent sequences:  A328034 A328035 A328036 * A328038 A328039 A328040

KEYWORD

nonn,tabf

AUTHOR

Bob Selcoe, Oct 03 2019

STATUS

approved

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Last modified October 1 15:55 EDT 2020. Contains 337443 sequences. (Running on oeis4.)