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%I #43 Sep 08 2022 08:44:58
%S 0,0,0,3,4,10,12,21,24,36,40,55,60,78,84,105,112,136,144,171,180,210,
%T 220,253,264,300,312,351,364,406,420,465,480,528,544,595,612,666,684,
%U 741,760,820,840,903,924,990,1012,1081,1104,1176
%N a(n) = n * floor((n-1)/2).
%C T(n,2), array T as in A050186; a count of aperiodic binary words.
%C The Row2 triangle sums A159797 lead to the sequence given above for n >= 1 with a(1)=0. For the definitions of the Row2 and other triangle sums see A180662. - _Johannes W. Meijer_, May 20 2011
%C The number of chords joining n equally distributed points on a circle with a length less than the diameter. - _Wesley Ivan Hurt_, Nov 23 2013
%C a(n) is the maximum possible length of a circuit in the complete graph on n vertices. - _Geoffrey Critzer_, May 23 2014
%C For n > 0, a(n) is half the sum of the perimeters of the distinct rectangles that can be made with positive integer sides such that L + W = n, W < L. For example, a(14) = 84; the rectangles are 1 X 13, 2 X 12, 3 X 11, 4 X 10, 5 X 9, 6 X 8 (the 7 X 7 rectangle is not considered since we have W < L). The sum of the perimeters gives 28 + 28 + 28 + 28 + 28 + 28 = 168, half of which is 84. - _Wesley Ivan Hurt_, Nov 23 2017
%C Sum of the middle side lengths of all integer-sided triangles with perimeter 3n whose side lengths are in arithmetic progression (For example, when n=5 there are two triangles with perimeter 3(5) = 15 whose side lengths are in arithmetic progression: [3,5,7] and [4,5,6]; thus a(5) = 5+5 = 10). - _Wesley Ivan Hurt_, Nov 01 2020
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F a(n) = n * floor((n-1)/2).
%F From _R. J. Mathar_, Aug 08 2009: (Start)
%F a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
%F G.f.: x^3*(3+x) / ((1+x)^2*(1-x)^3). (End)
%F a(n) = binomial(n,2) - (n/2) * ((n+1) mod 2). - _Wesley Ivan Hurt_, Nov 23 2013
%F E.g.f.: x*(x*cosh(x) + sinh(x)*(x - 1))/2. - _Stefano Spezia_, Nov 02 2020
%p A050187:=n->n*floor((n-1)/2); seq(A050187(n), n=0..100); # _Wesley Ivan Hurt_, Nov 23 2013
%t Table[n*Floor[(n-1)/2], {n,0,100}] (* _Wesley Ivan Hurt_, Nov 23 2013 *)
%o (Magma) [n*Floor((n-1)/2): n in [0..50]]; // _Wesley Ivan Hurt_, May 24 2014
%Y Cf. A050186, A159797, A180662.
%K nonn,easy
%O 0,4
%A _Clark Kimberling_, Dec 11 1999
%E Name change by _Wesley Ivan Hurt_, Nov 23 2013
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