A097964 Rectangular array read by rows (n > 0, 1 <= k <= 3): T(n,k) = floor(b(n,k)/2^((A002264(n) + 1)/3)), where b(n,k) = b(n-3,k) + 3*b (n-6,k) + 2*b(n-9,k), with initial values given in comments.

2, 5, 7, 3, 5, 8, 2, 3, 6, 5, 8, 13, 6, 11, 17, 4, 8, 12, 10, 17, 27, 12, 21, 34, 9, 15, 24, 20, 34, 54, 25, 42, 68, 18, 30, 49, 40, 68, 108, 50, 85, 136, 36, 61, 97, 80, 135, 216, 101, 170, 271, 72, 121, 194, 160, 270, 430, 201, 339, 541, 144, 242, 387

Offset: 1

Author

Roger L. Bagula, Sep 06 2004

Keywords

• nonn
• tabf
• less

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 | 4 8 12

2 | 5 8 13

3 | 4 6 10

4 | 10 16 26

5 | 13 22 35

6 | 9 16 25

7 | 26 44 70

8 | 32 54 86

9 | 23 38 61. (End)

Examples

			Array begins:
   2,  5,  7;
   3,  5,  8;
   2,  3,  6;
   5,  8, 13;
   6, 11, 17;
   4,  8, 12;
  10, 17, 27;
  12, 21, 34;
   9, 15, 24;
  20, 34, 54;
  25, 42, 68;
  18, 30, 49;
   ... - _Franck Maminirina Ramaharo_, Nov 08 2018
		

Crossrefs

Cf. A097966.

Programs

Mathematica

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

Formula

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

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

0, 2, 0

M = 1, 1, 1

1, 1, 0

and

0, 1, 1

A = 1, 1, 2

1, 2, 3.

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

Extensions

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