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A258756
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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.
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0
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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
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OFFSET
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3,3
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COMMENTS
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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?
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LINKS
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PROG
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(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))
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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