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 A300761 Number of non-equivalent ways (mod D_2) to select 4 points from n equidistant points on a straight line so that no selected point is equally distant from two other selected points. 2

%I

%S 0,1,3,6,15,28,53,87,140,210,310,434,600,803,1061,1368,1747,2190,2723,

%T 3337,4060,4884,5840,6916,8148,9525,11083,12810,14747,16880,19253,

%U 21851,24720,27846,31278,34998,39060,43447,48213,53340,58887,64834,71243,78093,85448

%N Number of non-equivalent ways (mod D_2) to select 4 points from n equidistant points on a straight line so that no selected point is equally distant from two other selected points.

%C The condition of the selection is also known as "no 3-term arithmetic progressions".

%C A reflection of a selection is not counted. If reflections are to be counted see A300760.

%H Heinrich Ludwig, <a href="/A300761/b300761.txt">Table of n, a(n) for n = 4..1000</a>

%F a(n) = (n^4 - 12*n^3 + 54*n^2 - 88*n)/48 + (n == 1 (mod 2))*(-4*n + 19)/16 + (n == 5 (mod 6))/3 + (n == 2 (mod 6))/3 + (n == 2 (mod 4))/2.

%F a(n) = (n^4 - 12*n^3 + 54*n^2 - 88*n)/48 + b(n) + c(n), where

%F b(n) = 0 for n even

%F b(n) = (-4*n + 19)/16 for n odd

%F c(n) = 0 for n == 0,1,3,4,7,9 (mod 12)

%F c(n) = 1/3 for n == 5,8,11 (mod 12)

%F c(n) = 1/2 for n == 6,10 (mod 12)

%F c(n) = 5/6 for n == 2 (mod 12).

%F From _Colin Barker_, Mar 15 2018: (Start)

%F G.f.: x^5*(1 + x + 4*x^3 + x^4 + 5*x^5) / ((1 - x)^5*(1 + x)^2*(1 + x^2)*(1 + x + x^2)).

%F a(n) = 2*a(n-1) - a(n-3) - 2*a(n-5) + 2*a(n-6) + a(n-8) - 2*a(n-10) + a(n-11) for n>14.

%F (End)

%Y Cf. A002623, A300760.

%K nonn,easy

%O 4,3

%A _Heinrich Ludwig_, Mar 15 2018

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Last modified August 5 19:30 EDT 2020. Contains 336213 sequences. (Running on oeis4.)