A271915 Number of ways to choose three distinct points from a 5 X n grid so that they form an isosceles triangle.

0, 24, 108, 248, 444, 672, 932, 1204, 1512, 1836, 2188, 2548, 2936, 3332, 3756, 4192, 4656, 5128, 5628, 6136, 6672, 7216, 7788, 8368, 8976, 9592, 10236, 10888, 11568, 12256, 12972, 13696, 14448, 15208, 15996, 16792

Offset: 1

Author

N. J. A. Sloane, Apr 24 2016

Keywords

• nonn

Links

• 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).

Crossrefs

Row 5 of A271910.

Cf. A186434, A187452.

Programs

Mathematica

Join[{0, 24, 108, 248, 444, 672, 932, 1204, 1512, 1836, 2188, 2548, 2936, 3332}, LinearRecurrence[{2, 0, -2, 1}, {3756, 4192, 4656, 5128}, 20]] (* Jean-François Alcover, Sep 03 2018 *)

Formula

Conjectured g.f.: 4*x* (x^16-x^14+2*x^10+2*x^9-x^8-x^7 + 5*x^6+6*x^5+6*x^4+x^3-8*x^2-15*x-6) /((x+1)*(x-1)^3).

Conjectured recurrence: a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n > 18.

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