0,2

For small n, this is equal to the binomial coefficient C(n,19). We have eleven possibilities for the first ball thrown in the first frame (gutter, 1, 2, ..., 9, strike) and the same possibilities occur for each of the other nine frames. So without even considering the second ball in each frame, at a minimum we have 11^10 possibilities. In fact, the true number of games is much larger due to the effect of the second ball in each frame. It's easy to show that the total number of possible games is: 66^9 x 241 = 5,726,805,883,325,784,576. The number of possible games with scores n = 0, 1, 2, 3, 4, 5, are: 1, 20, 210, 1540, 8855, 42504... The maximum number of possible games is for the score n = 77, namely 172542309343732000. Then there are a declining number of ways to get very high scores. For n = 288, 289, ..., 300 the totals are: 12, 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1. The distribution of number of possible games as a function of n is not precisely symmetric about its maximum point. - Jonathan Vos Post (jvospost3(AT)gmail.com), Jul 15 2005

Cooper, C. N. and Kennedy, R. E. "A Generating Function for the Distribution of the Scores of All Possible Bowling Games." In The Lighter Side of Mathematics (Ed. R. K. Guy and R. E. Woodrow). Washington, DC: Math. Assoc. Amer., 1994.

Cooper, C. N. and Kennedy, R. E. "Is the Mean Bowling Score Awful?" In The Lighter Side of Mathematics (Ed. R. K. Guy and R. E. Woodrow). Washington, DC: Math. Assoc. Amer., 1994.

Curtis Cooper and Robert E. Kennedy, Mathematics Magazine, Vol. 63, No. 4, pp. 239-243.

Balmoral Software, All About Bowling Scores, 2005,

Eric Weisstein's World of Mathematics, Bowling.

The final terms are a(290) = 11, a(291) = ... = a(300) = 1.

Cf. A010972.

Adjacent sequences: A060850 A060851 A060852 this_sequence A060854 A060855 A060856

Sequence in context: A074668 A139620 A094311 this_sequence A010972 A126905 A022585

nonn,fini,easy

Micah Friese (friesem(AT)stolaf.edu), May 03 2001