OFFSET
0,4
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).
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
MAPLE
MATHEMATICA
Table[n*Floor[(n-1)/2], {n, 0, 100}] (* Wesley Ivan Hurt, Nov 23 2013 *)
PROG
(Magma) [n*Floor((n-1)/2): n in [0..50]]; // Wesley Ivan Hurt, May 24 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Dec 11 1999
EXTENSIONS
Name change by Wesley Ivan Hurt, Nov 23 2013
STATUS
approved