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

1,2

Links

Table of n, a(n) for n=1..36.

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.: 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

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 *)

Crossrefs

Row 5 of A271910.

Cf. A186434, A187452.

Sequence in context: A100150 A305950 A060334 * A187163 A211577 A101862

Adjacent sequences: A271912 A271913 A271914 * A271916 A271917 A271918

Keyword

nonn

Author

N. J. A. Sloane, Apr 24 2016

Status

approved