A348843 Irregular triangle read by rows: row n gives for the Julian Green game with n cards labeled 1, 2, ..., n, for n >= 2, the number of games if the first removed card has label K = 2*k, for k = 1, 2, ..., floor(n/2).

1, 1, 3, 3, 5, 5, 12, 11, 12, 16, 15, 16, 33, 39, 36, 39, 49, 58, 51, 58, 72, 103, 88, 103, 86, 82, 118, 98, 118, 96, 379, 569, 521, 616, 528, 442, 420, 628, 578, 682, 569, 488, 514, 921, 881, 977, 785, 739, 785, 1092, 1986, 1753, 2102, 2036, 1557, 1634

Offset: 2

Author

Wolfdieter Lang, Dec 23 2021

Keywords

• nonn
• tabf

Comments

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

The length of row n is A008619(n-2), for n >= 2.

The sum of row n gives A348842(n).

See A348844 for the table where the present entries are split into the number of games with odd and even number of moves.

T(n, K=2*k) is determined from the sequence of removed numbers in the Julian Green game with cards labeled 1, 2, ..., n, named JG(n), if the first removed card number is K. Such a sequence is obtained from row n of table A348390 which provides the possible card numbers (the reservoir) for the move following the sequence entry with card number q, for q from {1, 2, ..., n} (q is here the number k of A348390, giving the sequence of proper divisors of d(n,k) followed by the sequence of multiples m(n, k) of k which are > k and <= n). However, numbers of the reservoir only qualify if they have not yet been removed in the game. E.g., a JG(5) game starting with {2}. The reservoir for the next entry (move) is {1, 4}, q = k = 2 from row n = 5 of A348390. If the game continues as {2, 1}, the next move comes form the reservoir for q = 1, that is {2, 3, 4, 5}, but 2 has already been removed, that is, the game continues as {2, 1, 3} or {2, 1, 4} or {2, 1, 5}. The next move comes from reservoir {1} or {2,4} or {1} from q = 3 or 4, or 5, respectively, but these numbers have all to be omitted. Thus the three games have an odd number of moves (namely 3) and the player who starts (player A) wins. The game starting with {2, 4} continues as {2, 4, 1} (from q = 4 with reservoir {1, 2}, but 2 has to be omitted). The next move uses either number 3 or 5 (q = 1, {2, 3, 4, 5}, omitting 2 and 4). Then the game finishes either as {2, 4, 1, 3} or as {2, 4, 1, 5} because the reservoir {1} for q = 3 and also for q = 5 cannot be used. In these two cases the second player (B) wins. Thus there are 3 + 2 = 5 games for start number K = 2 if n = 5 (see also row 5, K = 2 in table A348844).

Examples

			The irregular triangle T(n, k) begins:
n\ k    1     2     3     4     5     6     7     8     9    10 ...
   K    2     4     6     8    10    12    14    16    18    29 ...
-------------------------------------------------------------------
2:      1
3:      1
4:      3     3
5:      5     5
6:     12    11    12
7:     16    15    16
8:     33    39    36    39
9:     49    58    51    58
10:    72   103    88   103    86
11:    82   118    98   118    96
12:   379   569   521   616   528   442
13:   420   628   578   682   569   488
14:   514   921   881   977   785   739   785
15:  1092  1986  1753  2102  2036  1557  1634
16:  2382  4594  4569  5666  5214  3933  3927  5666
17:  2525  4864  4835  6024  5474  4143  4070  6024
18:  7430 17220 16208 21258 18760 15158 13053 21258 17544
19:  7811 18087 16995 22418 19553 15860 13434 22418 18422
20: 28538 57489 74728 90526 86225 70069 54336 90526 80493 70164
...
-------------------------------------------------------------------
n = 5:  The 5 games starting with card labeled 2 are [2, 1, 3], [2, 1, 4],
[2, 1, 5], [2, 4, 1, 3], [2, 4, 1, 5], and the 5 games starting with number 4 are: [4, 1, 2], [4, 1, 3], [4, 1, 5], [4, 2, 1, 3], [4, 2, 1, 5]. ------------------------------------------------------------------
		

Links

• See A348842.

Crossrefs

Cf. A008619, A348842, A348844.