A050187 a(n) = n * floor((n-1)/2).

0, 0, 0, 3, 4, 10, 12, 21, 24, 36, 40, 55, 60, 78, 84, 105, 112, 136, 144, 171, 180, 210, 220, 253, 264, 300, 312, 351, 364, 406, 420, 465, 480, 528, 544, 595, 612, 666, 684, 741, 760, 820, 840, 903, 924, 990, 1012, 1081, 1104, 1176

Offset: 0

Author

Clark Kimberling, Dec 11 1999

Keywords

• nonn
• easy

Comments

T(n,2), array T as in A050186; a count of aperiodic binary words.

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

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

a(n) is the maximum possible length of a circuit in the complete graph on n vertices. - Geoffrey Critzer, May 23 2014

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

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

Links

• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A050186, A159797, A180662.

Programs

Magma

[n*Floor((n-1)/2): n in [0..50]]; // Wesley Ivan Hurt, May 24 2014

Maple

A050187:=n->n*floor((n-1)/2); seq(A050187(n), n=0..100); # Wesley Ivan Hurt, Nov 23 2013

Mathematica

Table[n*Floor[(n-1)/2], {n,0,100}] (* Wesley Ivan Hurt, Nov 23 2013 *)

Formula

a(n) = n * floor((n-1)/2).

From R. J. Mathar, Aug 08 2009: (Start)

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).

G.f.: x^3*(3+x) / ((1+x)^2*(1-x)^3). (End)

a(n) = binomial(n,2) - (n/2) * ((n+1) mod 2). - Wesley Ivan Hurt, Nov 23 2013

E.g.f.: x*(x*cosh(x) + sinh(x)*(x - 1))/2. - Stefano Spezia, Nov 02 2020

Extensions

Name change by Wesley Ivan Hurt, Nov 23 2013