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.

0, 1, 3, 6, 15, 28, 53, 87, 140, 210, 310, 434, 600, 803, 1061, 1368, 1747, 2190, 2723, 3337, 4060, 4884, 5840, 6916, 8148, 9525, 11083, 12810, 14747, 16880, 19253, 21851, 24720, 27846, 31278, 34998, 39060, 43447, 48213, 53340, 58887, 64834, 71243, 78093, 85448

Offset: 4

Author

Heinrich Ludwig, Mar 15 2018

Keywords

• nonn
• easy

Comments

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

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

Links

• Heinrich Ludwig, Table of n, a(n) for n = 4..1000

Crossrefs

Cf. A002623, A300760.

Formula

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.

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

b(n) = 0 for n even

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

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

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

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

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

From Colin Barker, Mar 15 2018: (Start)

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)).

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.

(End)