0,3

The (4,2)-case of the Motzkin Tennis Ball Problem is a variation of the Tennis Ball Problem that generates a(n). On each turn, i, four balls labeled i are placed in the bucket and then any two are removed and placed on the lawn. We consider all possible combinations of balls on the lawn after n turns.

The number of ways to choose n numbers, ranging from 0 to 4, so that their sum is 2n and so that when you take k numbers from the left, the sum of these numbers is <= 2k (e.g. the combination of {141} is impossible, for 1+4 > 2k). Thus a(1) = {2}; a(2) = {04}, {13} and {22}; a(3) = {024}, {033}, {042}, {114}, {123}, {132}, {204}, {213} and {222} - Joost Vermeij (joost_vermeij(AT)hotmail.com), Jun 12 2005

D. Merlini, R. Sprugnoli, M.C. Verri, The Tennis Ball Problem, Journal of Combinatorial Theory, Series A, Vol. 99 (2002), pp. 307-344.

G.f. in Mathematica notation (1/(4x))(1 + x + Sqrt[(1 - 6 x + 5 x^2)] - Sqrt[2] Sqrt[1 + Sqrt[(1 - 6 x + 5 x^2)] + x (-2 - 5 x + Sqrt[(1 - 6 x + 5 x^2)])]). - Nour-Eddine Fahssi (fahssin(AT)yahoo.fr), Jan 10 2008

a(3)=9, since the possible combinations of balls on the lawn after 3 turns is 111122, 111123, 111133, 111222, 111223, 111233, 112222, 112223, 112233, if on each turn there are 4 identically labeled balls received and 2 selected.

Cf. A066357, A001006.

Sequence in context: A122452 A091841 A063020 this_sequence A039628 A005964 A129416

Adjacent sequences: A104181 A104182 A104183 this_sequence A104185 A104186 A104187

nonn

Nicholas Billler (billern(AT)gmail.com), Mar 11 2005