A355850 Number of monotonic lattice paths of length n which do not pass above the line y = x/(log_2(3)-1).

1, 1, 2, 3, 6, 12, 22, 44, 88, 169, 338, 646, 1292, 2584, 5055, 10110, 20220, 39915, 79830, 157008, 314016, 628032, 1244631, 2489262, 4978524, 9899008, 19798016, 39596032, 78879609, 157759218, 313777086, 627554172, 1255108344, 2502016784, 5004033568, 10008067136, 19971766007

Offset: 0

Author

C. Krishna, Jul 18 2022

Keywords

• nonn

Comments

A monotonic lattice path is one which starts at (0,0) consists entirely of steps of length 1, moving only rightwards (x,y) -> (x+1,y) or upwards (x,y) -> (x,y+1).

a(n+1) <= 2*a(n).

Consider a stock with an initial price of x. At every time step it either gains or loses 50% of its value with equal probability (x -> 0.5*x or x-> 1.5*x). An investor buys this stock and sells it if its current price exceeds its initial price and holds onto it otherwise. What is the probability the stock is not sold after n time steps?

The answer to this is a(n) / 2^n. Limit_{n->oo} a(n) / 2^n = c ~ 0.2863153965300.

Notice that the expected value of the stock remains constant after each time step ((0.5*x + 1.5*x)/2 = x), but the expected log of the stock price is constantly decreasing, at a rate of (1/2) * log(3/4), each time step.

Using the central limit theorem, there is a 100% likelihood that the stock price falls below any arbitrarily small positive value, p > 0 in the long run.

Since there is a probability 1-c that the investment yields a profit, a probability c that the stock is never sold, and the stock maintains a constant expected value, the expected profit if sold is cx/(1-c) ~ 0.401179169535*x.

Examples

			n, a(n), list of paths
n = 0: 1 {(0,0)}
n = 1: 1 {(0,0) -> (1,0)}
n = 2: 2 {(0,0) -> (1,0) -> (2,0), (0,0) -> (1,0) -> (1,1)}
n = 3: 3 {(0,0) -> (1,0) -> (2,0) -> (3,0), (0,0) -> (1,0) -> (2,0) -> (2,1) (0,0) -> (1,0) -> (1,1) -> (2,1)}
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..3323

Crossrefs

Cf. A001405, A020857, A126042.