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

1,2

Links

Table of n, a(n) for n=1..50.

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^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

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 *)

Crossrefs

Row 4 of A271910.

Cf. A186434, A187452.

Sequence in context: A321180 A100186 A344600 * A178574 A005906 A247663

Adjacent sequences: A271910 A271911 A271912 * A271914 A271915 A271916

Keyword

nonn

Author

N. J. A. Sloane, Apr 24 2016

Extensions

More terms from Jean-François Alcover, Sep 03 2018

Status

approved