A271912 Number of ways to choose three distinct points from a 3 X n grid so that they form an isosceles triangle.

0, 10, 36, 68, 108, 150, 200, 252, 312, 374, 444, 516, 596, 678, 768, 860, 960, 1062, 1172, 1284, 1404, 1526, 1656, 1788, 1928, 2070, 2220, 2372, 2532, 2694, 2864, 3036, 3216, 3398, 3588, 3780, 3980, 4182, 4392, 4604, 4824, 5046, 5276, 5508, 5748, 5990, 6240, 6492, 6752, 7014, 7284, 7556

Offset

1,2

Links

Table of n, a(n) for n=1..52.

Chai Wah Wu, Counting the number of isosceles triangles in rectangular regular grids, arXiv:1605.00180 [math.CO], 2016.

Index entries for linear recurrences with constant coefficients, signature (2, 0, -2, 1).

Formula

Conjectured g.f.: 2*x*(2*x^4+4*x^3+2*x^2-8*x-5)/((x+1)*(x-1)^3).

Conjectured recurrence: a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n > 6.

Conjectures from Colin Barker, Apr 25 2016: (Start)

a(n) = (-3*(47+(-1)^n)+64*n+10*n^2)/4 for n>2.

a(n) = (5*n^2+32*n-72)/2 for n>2 and even.

a(n) = (5*n^2+32*n-69)/2 for n>2 and odd.

(End)

The conjectured g.f. and recurrence are true. See paper in links. - Chai Wah Wu, May 07 2016

Example

n=2: Label the points
1 2 3
4 5 6
There are 8 small isosceles triangles like 124 plus 135 and 246, so a(2) = 10.

Mathematica

Join[{0, 10}, LinearRecurrence[{2, 0, -2, 1}, {36, 68, 108, 150}, 50]] (* Jean-François Alcover, Oct 10 2018 *)

Crossrefs

Row 3 of A271910.

Cf. A186434, A187452.

Sequence in context: A359959 A309783 A072517 * A288947 A328146 A033585

Adjacent sequences: A271909 A271910 A271911 * A271913 A271914 A271915

Keyword

nonn

Author

N. J. A. Sloane, Apr 24 2016

Extensions

More terms from Jean-François Alcover, Oct 10 2018

Status

approved