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

1,2

Links

Table of n, a(n) for n=1..52.

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

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.

Example

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

Crossrefs

Cf. A088803.

Sequence in context: A312699 A131937 A183857 * A344012 A027924 A006578

Adjacent sequences: A088801 A088802 A088803 * A088805 A088806 A088807

Keyword

nonn

Author

Rob Arthan, Oct 17 2003

Status

approved