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

1,1

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)

Links

Table of n, a(n) for n=1..63.

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)

Example

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

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

Crossrefs

Cf. A097966.

Sequence in context: A361128 A309200 A361170 * A133133 A024710 A140264

Adjacent sequences: A097961 A097962 A097963 * A097965 A097966 A097967

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