A327283 Irregular triangle T(n,k) read by rows: "residual summands" in reduced Collatz sequences (see Comments for definition and explanation).

1, 1, 5, 1, 5, 19, 73, 347, 1, 7, 29, 103, 373, 1631, 1, 5, 23, 133, 1, 11, 1, 5, 19, 65, 451, 1, 7, 53, 1, 5, 31, 125, 503, 2533, 1, 1, 5, 19, 185, 1, 7, 29, 151, 581, 2255, 10861, 1, 5, 23, 85, 287, 925

Offset

1,3

Comments

Let R_s be the reduced Collatz sequence (cf. A259663) starting with s and let R_s(k), k >= 0 be the k-th term in R_s. Then R_(2n-1)(k) = (3^k*(2n-1) + T(n,k))/2^j, where j is the total number of halving steps from R_(2n-1)(0) to R_(2n-1)(k). T(n,k) is defined here as the "residual summand".

The sequence without duplicates is a permutation of A116641.

Links

Table of n, a(n) for n=1..52.

Formula

T(n,k) = 2^j*R_(2n-1)(k) - 3^k*(2n-1), as defined in Comments.

T(n,1) = 1; for k>1: T(n,k) = 3*T(n,k-1) + 2^i, where i is the total number of halving steps from R_(2n-1)(0) to R_(2n-1)(k-1).

Example

Triangle starts:
  1;
  1, 5;
  1;
  1, 5, 19, 73,  347;
  1, 7, 29, 103, 373, 1631;
  1, 5, 23, 133;
  1, 11;
  1, 5, 19, 65,  451;
  1, 7, 53;
  1, 5, 31, 125, 503, 2533;
  1;
  1, 5, 19, 185;
  1, 7, 29, 151, 581, 2255, 10861;
  ...
T(5,4)=103 because R_9(4) = 13; the number of halving steps from R_9(0) to R_9(4) is 6, and 13 = (81*9 + 103)/64.

Crossrefs

Cf. A116623, A116641, A259663.

Sequence in context: A145825 A101692 A281105 * A105060 A229096 A290797

Adjacent sequences: A327280 A327281 A327282 * A327284 A327285 A327286

Keyword

nonn,tabf

Author

Bob Selcoe, Sep 15 2019

Status

approved