A271913 Number of ways to choose three distinct points from a 4 X n grid so that they form an isosceles triangle.
0, 16, 68, 148, 248, 360, 488, 620, 768, 924, 1096, 1272, 1464, 1660, 1872, 2088, 2320, 2556, 2808, 3064, 3336, 3612, 3904, 4200, 4512, 4828, 5160, 5496, 5848, 6204, 6576, 6952, 7344, 7740, 8152, 8568, 9000, 9436, 9888, 10344, 10816, 11292, 11784, 12280, 12792, 13308, 13840, 14376, 14928, 15484
Offset: 1
Keywords
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).
Programs
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Mathematica
Join[{0, 16, 68, 148, 248, 360, 488, 620}, LinearRecurrence[{2, 0, -2, 1}, {768, 924, 1096, 1272}, 42]] (* Jean-François Alcover, Sep 03 2018 *)
Formula
Conjectured g.f.: 4*x*(x^10-x^8+2*x^6+x^5+4*x^4+4*x^3-3*x^2-9*x-4)/((x+1)*(x-1)^3).
Conjectured recurrence: a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n > 12.
Conjectures from Colin Barker, Apr 25 2016: (Start)
a(n) = -3/2*(143+(-1)^n)+64*n+5*n^2 for n>8.
a(n) = 5*n^2+64*n-216 for n>8 and even.
a(n) = 5*n^2+64*n-213 for n>8 and odd.
(End)
The conjectured g.f. and recurrence are true. See paper in links. - Chai Wah Wu, May 07 2016
Extensions
More terms from Jean-François Alcover, Sep 03 2018