Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #12 Jan 31 2022 10:52:21
%S 1,1,3,3,5,5,12,11,12,16,15,16,33,39,36,39,49,58,51,58,72,103,88,103,
%T 86,82,118,98,118,96,379,569,521,616,528,442,420,628,578,682,569,488,
%U 514,921,881,977,785,739,785,1092,1986,1753,2102,2036,1557,1634
%N 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).
%C The Ian Stewart links for the Juniper Green game are given in A348842.
%C The length of row n is A008619(n-2), for n >= 2.
%C The sum of row n gives A348842(n).
%C See A348844 for the table where the present entries are split into the number of games with odd and even number of moves.
%C 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).
%H See A348842.
%e The irregular triangle T(n, k) begins:
%e n\ k 1 2 3 4 5 6 7 8 9 10 ...
%e K 2 4 6 8 10 12 14 16 18 29 ...
%e -------------------------------------------------------------------
%e 2: 1
%e 3: 1
%e 4: 3 3
%e 5: 5 5
%e 6: 12 11 12
%e 7: 16 15 16
%e 8: 33 39 36 39
%e 9: 49 58 51 58
%e 10: 72 103 88 103 86
%e 11: 82 118 98 118 96
%e 12: 379 569 521 616 528 442
%e 13: 420 628 578 682 569 488
%e 14: 514 921 881 977 785 739 785
%e 15: 1092 1986 1753 2102 2036 1557 1634
%e 16: 2382 4594 4569 5666 5214 3933 3927 5666
%e 17: 2525 4864 4835 6024 5474 4143 4070 6024
%e 18: 7430 17220 16208 21258 18760 15158 13053 21258 17544
%e 19: 7811 18087 16995 22418 19553 15860 13434 22418 18422
%e 20: 28538 57489 74728 90526 86225 70069 54336 90526 80493 70164
%e ...
%e -------------------------------------------------------------------
%e n = 5: The 5 games starting with card labeled 2 are [2, 1, 3], [2, 1, 4],
%e [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]. ------------------------------------------------------------------
%Y Cf. A008619, A348842, A348844.
%K nonn,tabf
%O 2,3
%A _Wolfdieter Lang_, Dec 23 2021