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

4,3

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

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)

Crossrefs

Cf. A002623, A300760.

Sequence in context: A056278 A161625 A234848 * A069712 A076971 A103529

Adjacent sequences: A300758 A300759 A300760 * A300762 A300763 A300764

Keyword

nonn,easy

Author

Heinrich Ludwig, Mar 15 2018

Status

approved