A097966 Rectangular array read by rows (n > 0, 1 <= k <= 3): T(n,k) = floor(b(n,k)/6^(2*(A002264(n) + 1)/3)), where b(n,k) = b(n-3,k) + 13*b (n-6,k) + 36*b(n-9,k), with initial values given in comments.

1, 2, 3, 1, 2, 4, 2, 3, 4, 1, 3, 4, 2, 4, 6, 2, 5, 6, 3, 5, 7, 3, 7, 9, 4, 7, 10, 4, 8, 11, 6, 10, 14, 6, 11, 15, 7, 12, 17, 9, 16, 22, 10, 17, 23, 11, 19, 26, 14, 24, 33, 15, 26, 36, 17, 29, 40, 21, 37, 51, 23, 40, 55, 26, 45, 62, 33, 57, 77, 35, 61, 83

Offset

1,2

Comments

From Franck Maminirina Ramaharo, Nov 08 2018: (Start)

The initial values for b(n,k), 1 <= n <= 9, 1 <= k <= 3, are

n\k | 1 2 3

----+---------------

1 | 16 32 36

2 | 17 30 44

3 | 24 36 51

4 | 68 120 176

5 | 105 170 233

6 | 99 186 240

7 | 420 680 932

8 | 470 848 1129

9 | 519 870 1227. (End)

Links

Table of n, a(n) for n=1..72.

Formula

From Franck Maminirina Ramaharo, Nov 08 2018: (Start)

Let M and A denote the following 3 X 3 matrices:

0, 4, 0

M = 1, 1, 3

3, 3, 0

and

0, 1, 1

A = 1, 1, 2

1, 2, 2.

Then applying floor() to the entries in (h*M)^(n + 1)*A, where h = 1/(6^(2/3)), yields row 3*n - 2 to 3*n. (End)

Example

Triangle begins:
  1,  2,  3;
  1,  2,  4;
  2,  3,  4;
  1,  3,  4;
  2,  4,  6;
  2,  5,  6;
  3,  5,  7;
  3,  7,  9;
  4,  7, 10;
  4,  8, 11;
  6, 10, 14;
  6, 11, 15;
   ... - Franck Maminirina Ramaharo, Nov 08 2018

Mathematica

M = N[(16/9)^(1/3)*({{0, 1, 0}, {1, 1, 0}, {0, 0, 0}}*(1/4) + {{0, 1, 0}, {0, 0, 1}, {1, 1, 0}}*(3/4))];
A[n_] := M.A[n - 1]; A[0] := {{0, 1, 1}, {1, 1, 2}, {1, 2, 2}};
Table[Floor[M.A[n]], {n, 1, 12}]//Flatten

Crossrefs

Cf. A097964.

Sequence in context: A072193 A233359 A279345 * A177993 A256990 A071503

Adjacent sequences: A097963 A097964 A097965 * A097967 A097968 A097969

Keyword

nonn,tabf,less

Author

Roger L. Bagula, Sep 06 2004

Extensions

Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Nov 08 2018

Status

approved