A348844 Irregular triangle T(n,k) read by rows: row n gives the pairs of odd and even number of moves for the Juniper Green game JG(n) with n cards, for n >= 2, if the first card taken away is labeled K, for K = 2, 4, ..., 2*floor(n/2).

0, 1, 1, 0, 2, 1, 2, 1, 3, 2, 3, 2, 6, 6, 7, 4, 6, 6, 8, 8, 9, 6, 8, 8, 14, 19, 18, 21, 19, 17, 18, 21, 25, 24, 31, 27, 30, 21, 31, 27, 36, 36, 51, 52, 49, 39, 51, 52, 43, 43, 41, 41, 59, 59, 54, 44, 59, 59, 48, 48, 189, 190, 286, 283, 253, 268, 307, 309, 266, 262, 222, 220, 209

Offset: 2

Author

Wolfdieter Lang, Jan 02 2022

Keywords

• nonn
• tabf

Comments

The Ian Stewart links for the Juniper Green game are given in A348842.

The length of row n is 2*A009619(n-2), for n >= 2.

The sum of row n is A348842(n).

In the irregular triangle A348843 the numbers of the pairs have been summed.

Examples

			The irregular triangle T(n,k) begins:
n\ k   1    2    3    4    5    6    7    8    9   10   11   12   13   14 ...
   K        2         4         6         8        10        12        14 ...
-----------------------------------------------------------------------------
2:     0    1
3:     1    0
4:     2    1    2    1
5:     3    2    3    2
6:     6    6    7    4    6    6
7:     8    8    9    6    8    8
8:    14   19   18   21   19   17   18   21
9:    25   24   31   27   30   21   31   27
10:   36   36   51   52   49   39   51   52   43   43
11:   41   41   59   59   54   44   59   59   48   48
12:  189  190  286  283  253  268  307  309  266  262  222  220
13:  209  211  315  313  282  296  340  342  287  282  245  243
14:  257  257  462  459  433  448  489  488  394  391  372  367  394  391
15:  542  550  996  990  843  910 1019 1083  992 1044  757  800  824  810
...
-------------------------------------------------------------------------------
n = 2: The 1 = A348842(2) game JG(2) is [2, 1], with an even number of moves (B wins); hence row n = 2 is 0, 1, because there is no game with an odd number of moves. Thus JG(2) is called secondary.
n = 4: The 6 games JG(4) are: [2, 1, 3], [2, 1, 4]  and [2, 4, 1, 3] for K = 2, and [4, 1, 2], [4, 1, 3] and [4, 2, 1, 3], for K = 4; hence row n = 4 gives 2, 1, for K = 2 as well as for K = 4. This means that in these six games A wins four times and B twice. But B can always win by reacting on 2 with 4, and on 4 with 2, leading to [2, 4, 1, 3] and  [4, 2, 1, 3]. Thus the game JG(4) is called secondary.
n = 6: There are 35 games, A wins 19 times and B 16 times. For K = 2 and K = 6  6 times A, 6 times B, and for K = 4 4 times A and 7 times B. Again B is a safe winner reacting to K = 2 with 4 ([2, 4, 1, 5] or [2, 4, 1, 3]), to K = 4 with 2, then 5  ([4, 2, 1, 5]), and to K = 6 with 3 then 5 ([6, 3, 1, 5]). Thus JG(6) is also secondary.
n = 9: There are 216 games, A wins 117 times and B 99 times. There is a strategy for B, and JG(9) is secondary.
		

Crossrefs

Cf. A009619, A348842, A348843.