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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 (list; graph; refs; listen; history; text; internal format)
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^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: 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 Jean-François Alcover, Sep 03 2018

STATUS

approved

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Last modified January 18 16:27 EST 2020. Contains 331011 sequences. (Running on oeis4.)