

A271913


Number of ways to choose three distinct points from a 4 X n grid so that they form an isosceles triangle.


4



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
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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.


FORMULA

Conjectured g.f.: 4*x*(x^10x^8+2*x^6+x^5+4*x^4+4*x^33*x^29*x4)/((x+1)*(x1)^3).
Conjectured recurrence: a(n) = 2*a(n1)2*a(n3)+a(n4) 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*n216 for n>8 and even.
a(n) = 5*n^2+64*n213 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]] (* JeanFrançois Alcover, Sep 03 2018 *)


CROSSREFS

Row 4 of A271910.
Cf. A186434, A187452.
Sequence in context: A216306 A321180 A100186 * 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 JeanFrançois Alcover, Sep 03 2018


STATUS

approved



