A280471 Solutions to the Gamow-Stern Elevator Problem, a(n) = ceiling((log_10 5)/(log_10 (1+2(n-2)))) for integer n >= 3.

2, 3, 4, 4, 5, 6, 7, 8, 9, 9, 10, 11, 12, 13, 13, 14, 15, 16, 17, 17, 18, 19, 20, 21, 21, 22, 23, 24, 25, 25, 26, 27, 28, 29, 29, 30, 31, 32, 33, 33, 34, 35, 36, 37, 38, 38, 39, 40, 41, 42, 42, 43, 44, 45, 46, 47, 48, 49, 50, 50, 51, 52, 53, 54, 54, 55, 56, 57

Offset

3,1

Comments

Gives the number a(n) of elevators in an ideal system needed for a building with n floors to have a probability greater than or equal to 0.4 that the first elevator arriving to a given floor will be on its way down.

Also the least positive integer n such that 1/2 + (1/2)(1-2p)|1-2p|^(n-1) >= 0.4.

1/2 + (1/2)(1-2p)|1-2p|^(n-1) approaches 1/2 as n goes to infinity.

References

George Gamow and Marvin Stern, Puzzle-Math, (New York: Viking Press, 1958).

Donald E. Knuth, Fundamental Algorithms, Volume 1 of The Art of Computer Programming (Reading, Massachusetts: Addison-Wesley, 1968).

Donald E. Knuth, Selected Papers on Fun and Games, (Stanford, California: CSLI Publications, 2011), pages 79-86.

Links

Table of n, a(n) for n=3..70.

Formula

a(n) = ceiling((log_10 5)/(log_10 (1+2/(n-2)))) for integer n >= 3.

Example

For n=20, a(20) = ceiling((log_10 5)/(log_10 (1+2/(20-2))))= ceiling((log_10 5)/(log_10 (10/9))) = 16.

Mathematica

a[n_] := Print[Ceiling[(Log10[5])/(Log10[1+2/(n-2)])]]
For[i = 3, True, i++, a[i]]

Crossrefs

Sequence in context: A187737 A109401 A307294 * A172266 A362949 A056847

Adjacent sequences: A280468 A280469 A280470 * A280472 A280473 A280474

Keyword

easy,nonn

Author

Luke Botta, Jan 03 2017

Status

approved