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%I #18 Jan 20 2022 16:32:50
%S 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,
%T 21,25,24,31,27,30,21,31,27,36,36,51,52,49,39,51,52,43,43,41,41,59,59,
%U 54,44,59,59,48,48,189,190,286,283,253,268,307,309,266,262,222,220,209
%N 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).
%C The Ian Stewart links for the Juniper Green game are given in A348842.
%C The length of row n is 2*A009619(n-2), for n >= 2.
%C The sum of row n is A348842(n).
%C In the irregular triangle A348843 the numbers of the pairs have been summed.
%e The irregular triangle T(n,k) begins:
%e n\ k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...
%e K 2 4 6 8 10 12 14 ...
%e -----------------------------------------------------------------------------
%e 2: 0 1
%e 3: 1 0
%e 4: 2 1 2 1
%e 5: 3 2 3 2
%e 6: 6 6 7 4 6 6
%e 7: 8 8 9 6 8 8
%e 8: 14 19 18 21 19 17 18 21
%e 9: 25 24 31 27 30 21 31 27
%e 10: 36 36 51 52 49 39 51 52 43 43
%e 11: 41 41 59 59 54 44 59 59 48 48
%e 12: 189 190 286 283 253 268 307 309 266 262 222 220
%e 13: 209 211 315 313 282 296 340 342 287 282 245 243
%e 14: 257 257 462 459 433 448 489 488 394 391 372 367 394 391
%e 15: 542 550 996 990 843 910 1019 1083 992 1044 757 800 824 810
%e ...
%e -------------------------------------------------------------------------------
%e 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.
%e 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.
%e 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.
%e 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.
%Y Cf. A009619, A348842, A348843.
%K nonn,tabf
%O 2,5
%A _Wolfdieter Lang_, Jan 02 2022
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