A258756 a(n) = common difference giving the highest score in the game "Sequences" for a sequence with largest element n and as many elements as possible.

1, 1, 2, 2, 3, 3, 4, 3, 5, 5, 4, 4, 7, 3, 4, 4, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3, 4, 4, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset

3,3

Comments

The aim of the game is to find, in the given numbered tiles, arithmetic sequences of at least three elements. The score for a sequence is (max element)*(common difference ^ num elements). This sequence gives the common difference which produces the best score for a sequence with largest element n, and containing as many elements as possible.

This sequence converges to 3, which must have something to do with the fact that log(x)/x is maximized at x=e. Is n=45 the last time a(n) is not 3?

Links

Table of n, a(n) for n=3..132.

Christian Perfect, Sequences game

Prog

(Python)
def score(n, step):
...return n*(step**(n//step+(1 if n%step!=0 else 0)))
.
def best(n):
...return max(range(1, (n-1)//2+1), key=lambda step:score(n, step))

Crossrefs

Sequence in context: A375110 A283367 A322528 * A127431 A289777 A182921

Adjacent sequences: A258753 A258754 A258755 * A258757 A258758 A258759

Keyword

nonn,easy

Author

Christian Perfect, Jun 09 2015

Status

approved