

A330285


The maximum number of arithmetic progressions for a sequence of length n.


0



0, 0, 1, 3, 7, 12, 20, 29, 41, 55, 72, 90, 113, 137, 164, 194, 228, 263, 303, 344, 390, 439, 491, 544, 604, 666, 731, 799, 872, 946, 1027, 1109, 1196, 1286, 1379, 1475, 1579, 1684, 1792, 1903, 2021, 2140, 2266, 2393, 2525, 2662, 2802, 2943, 3093, 3245, 3402, 3562, 3727
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OFFSET

1,4


COMMENTS

The partial arithmetic density D_n(A) up to n is merely the number of arithmetic progressions, A(s(n)), divided by the total number of nonempty subsets of {s(1), s(2), ... , s(n)}, i.e. A(s(n))/(2^n  1). As n approaches infinity, D_n(A) converges to zero. Furthermore, the infinite sum of the partial densities for any sequence always converges to the total density D(A). Every infinite arithmetic progression has the same total density, Sum_{n >= 1} a(n)/(2^n  1) = alpha ~ 1.25568880818612911696845537; sequences with a finite number of progressions have D(A) < alpha; and sequences without any arithmetic progressions have D(A) = 0.


LINKS

Table of n, a(n) for n=1..53.
Encyclopedia of Mathematics, Density of a sequence
Eric Weisstein's World of Mathematics, Arithmetic Progression


FORMULA

a(n) = Sum_{i=1..n} Sum_{j=1..i} floor((i  1)/(j + 1)).


PROG

(PARI) a(n) = sum(i=1, n, sum(j=1, i, floor((i  1)/(j + 1))))


CROSSREFS

Cf. A049988, A130518, A104429, A065825, A092482.
Sequence in context: A132273 A130050 A173256 * A002049 A025582 A247556
Adjacent sequences: A330282 A330283 A330284 * A330286 A330287 A330288


KEYWORD

nonn


AUTHOR

Joseph Wheat, Dec 21 2019


STATUS

approved



