A097966 - OEIS

%I #14 Nov 09 2018 07:31:00

%S 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,

%T 14,6,11,15,7,12,17,9,16,22,10,17,23,11,19,26,14,24,33,15,26,36,17,29,

%U 40,21,37,51,23,40,55,26,45,62,33,57,77,35,61,83

%N 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.

%C From _Franck Maminirina Ramaharo_, Nov 08 2018: (Start)

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

%C n\k |    1    2    3

%C ----+---------------

%C   1 |   16   32   36

%C   2 |   17   30   44

%C   3 |   24   36   51

%C   4 |   68  120  176

%C   5 |  105  170  233

%C   6 |   99  186  240

%C   7 |  420  680  932

%C   8 |  470  848 1129

%C   9 |  519  870 1227. (End)

%F From _Franck Maminirina Ramaharo_, Nov 08 2018: (Start)

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

%F       0, 4, 0

%F   M = 1, 1, 3

%F       3, 3, 0

%F and

%F       0, 1, 1

%F   A = 1, 1, 2

%F       1, 2, 2.

%F 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)

%e Triangle begins:

%e   1,  2,  3;

%e   1,  2,  4;

%e   2,  3,  4;

%e   1,  3,  4;

%e   2,  4,  6;

%e   2,  5,  6;

%e   3,  5,  7;

%e   3,  7,  9;

%e   4,  7, 10;

%e   4,  8, 11;

%e   6, 10, 14;

%e   6, 11, 15;

%e    ... - _Franck Maminirina Ramaharo_, Nov 08 2018

%t 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))];

%t A[n_] := M.A[n - 1]; A[0] := {{0, 1, 1}, {1, 1, 2}, {1, 2, 2}};

%t Table[Floor[M.A[n]], {n, 1, 12}]//Flatten

%Y Cf. A097964.

%K nonn,tabf,less

%O 1,2

%A _Roger L. Bagula_, Sep 06 2004

%E Edited, new name, and offset corrected by _Franck Maminirina Ramaharo_, Nov 08 2018