A348842 Number of Juniper Green games with n cards.

0, 1, 1, 6, 10, 35, 47, 147, 216, 452, 512, 3055, 3365, 5602, 12160, 35951, 37959, 147889, 154998, 703094, 1178850, 1467813

Offset: 1

Author

Wolfdieter Lang, Dec 23 2021

Keywords

• nonn
• more

Comments

For the rules of this two person game with cards labeled from 1 to n, for n >= 1, called JG(n), see the Ian Stewart links.

It is reported (see the FEEDBACK and the German version), that E. P. Wigner used this game in some lecture in the thirties. There the prime factorization of n! into prime powers, with the number of odd or even (>= 2) exponents, seems to have played a role (see A055460(n) and A348841(n) for the number of primes with these exponents in the factorization of n!, respectively).

The repertoire of card numbers for JG(n) that can be chosen if the latest removed card had label k is shown in A348390. Of course, only those card numbers not yet removed in earlier moves qualify. E.g., n = 4, k = 2: repertoire 1, 4.

The total number of games JG(n), for n >= 2, if the first removed card has label K = 2*k, for k = 1, 2, ... ,floor(n/2), is given in A348843.

For the irregular table which gives in row n the odd and even number of moves in the a(n) JG(n) games see A348844. This gives the number of times Alice (the first mover), respectively Bob wins.

Links

• Ian Stewart, Juniper Green, Scientific American No. 3 March 1997, pp. 118-120.
• Ian Stewart, FEEDBACK, Scientific American No. 5 November 1997 p. 112.
• The author's email is given, juniper green Online game with 100 cards.

Crossrefs

Cf. A055460, A348390, A348843, A348844.

Formula

a(n) = Sum_{k=1..floor(n/2)} A348843(n, k) = Sum_{k=1..2*floor(n/2)} A348844(n, k), for n >= 2.