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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.
1
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)
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
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