cp's OEIS Frontend

A330285 The maximum number of arithmetic progressions in a sequence of length n.

%I A330285 #48 Feb 16 2025 08:33:59
%S A330285 0,0,1,3,7,12,20,29,41,55,72,90,113,137,164,194,228,263,303,344,390,
%T A330285 439,491,544,604,666,731,799,872,946,1027,1109,1196,1286,1379,1475,
%U A330285 1579,1684,1792,1903,2021,2140,2266,2393,2525,2662,2802,2943,3093,3245,3402,3562,3727
%N A330285 The maximum number of arithmetic progressions in a sequence of length n.
%C A330285 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 A330285 Encyclopedia of Mathematics, <a href="http://www.encyclopediaofmath.org/index.php?title=Density_of_a_sequence&amp;oldid=34860">Density of a sequence</a>
%H A330285 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ArithmeticProgression.html">Arithmetic Progression</a>
%F A330285 a(n) = Sum_{i=1..n} Sum_{j=1..i} floor((i - 1)/(j + 1)).
%o A330285 (PARI) a(n) = sum(i=1, n, sum(j=1, i, floor((i - 1)/(j + 1))))
%Y A330285 Cf. A049988, A130518, A104429, A065825, A092482.
%Y A330285 Partial sums of A002541.
%K A330285 nonn
%O A330285 1,4
%A A330285 _Joseph Wheat_, Dec 21 2019