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%I #47 Jul 16 2022 22:51:35
%S 0,0,1,3,7,12,20,29,41,55,72,90,113,137,164,194,228,263,303,344,390,
%T 439,491,544,604,666,731,799,872,946,1027,1109,1196,1286,1379,1475,
%U 1579,1684,1792,1903,2021,2140,2266,2393,2525,2662,2802,2943,3093,3245,3402,3562,3727
%N The maximum number of arithmetic progressions in a sequence of length n.
%C 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.
%H Encyclopedia of Mathematics, <a href="http://www.encyclopediaofmath.org/index.php?title=Density_of_a_sequence&oldid=34860">Density of a sequence</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ArithmeticProgression.html">Arithmetic Progression</a>
%F a(n) = Sum_{i=1..n} Sum_{j=1..i} floor((i - 1)/(j + 1)).
%o (PARI) a(n) = sum(i=1, n, sum(j=1, i, floor((i - 1)/(j + 1))))
%Y Cf. A049988, A130518, A104429, A065825, A092482.
%Y Partial sums of A002541.
%K nonn
%O 1,4
%A _Joseph Wheat_, Dec 21 2019
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