A088804 a(n) gives the number of steps taken in a process which manipulates piles of tokens arranged in a line. There are 2n (or 2n+1) tokens in all. Initially they are all in one pile. At each step, from each pile with more than 1 token, one token is taken and added to the pile on its left and one is taken and added to the pile on its right. The redistributions in each step are done in parallel.

1, 4, 8, 14, 21, 29, 39, 51, 63, 77, 93, 110, 128, 148, 170, 192, 216, 242, 268, 296, 326, 358, 390, 424, 460, 496, 534, 574, 615, 657, 701, 747, 793, 841, 891, 941, 993, 1047, 1103, 1159, 1217, 1277, 1337, 1399, 1463, 1529, 1595, 1663, 1733, 1803, 1875, 1949

Offset: 1

Author

Rob Arthan, Oct 17 2003

Keywords

• nonn

Examples

			E.g., a(2) = 4 because there are 4 steps in the process beginning with 4 tokens:
0 0 4 0 0
0 1 2 1 0
0 2 0 2 0
1 0 2 0 1
1 1 0 1 1
		

Links

• R. Anderson, L. Lovasz, P. Shor, J. Spencer, E. Tardos, S. Winograd, Disks, balls and walls: analysis of a combinatorial game, Amer. Math. Monthly, 6, 96, pp. 481-493, 1989.
• Anders Björner, László Lovászb, Peter W. Shor, Chip-firing games on graphs, European Journal of Combinatorics 12, pp. 283-291, 1991.
• Mikkel Thorup, Firing Games

Crossrefs

Cf. A088803.

Formula

The sequence is asymptotically quadratic with a(n) ~= c*n^2, where c is between 0.33 and 1, with estimate 0.7078 for n = 1, 000.