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A271912 Number of ways to choose three distinct points from a 3 X n grid so that they form an isosceles triangle. 1
0, 10, 36, 68, 108, 150, 200, 252, 312, 374, 444, 516, 596, 678, 768, 860, 960, 1062, 1172, 1284, 1404, 1526, 1656, 1788, 1928, 2070, 2220, 2372, 2532, 2694, 2864, 3036, 3216, 3398, 3588, 3780, 3980, 4182, 4392, 4604, 4824, 5046, 5276, 5508, 5748, 5990, 6240, 6492, 6752, 7014, 7284, 7556 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Table of n, a(n) for n=1..52.

Chai Wah Wu, Counting the number of isosceles triangles in rectangular regular grids, arXiv:1605.00180 [math.CO], 2016.

FORMULA

Conjectured g.f.: 2*x*(2*x^4+4*x^3+2*x^2-8*x-5)/((x+1)*(x-1)^3).

Conjectured recurrence: a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n > 6.

Conjectures from Colin Barker, Apr 25 2016: (Start)

a(n) = (-3*(47+(-1)^n)+64*n+10*n^2)/4 for n>2.

a(n) = (5*n^2+32*n-72)/2 for n>2 and even.

a(n) = (5*n^2+32*n-69)/2 for n>2 and odd.

(End)

The conjectured g.f. and recurrence are true. See paper in links. - Chai Wah Wu, May 07 2016

EXAMPLE

n=2: Label the points

1 2 3

4 5 6

There are 8 small isosceles triangles like 124 plus 135 and 246, so a(2) = 10.

MATHEMATICA

Join[{0, 10}, LinearRecurrence[{2, 0, -2, 1}, {36, 68, 108, 150}, 50]] (* Jean-François Alcover, Oct 10 2018 *)

CROSSREFS

Row 3 of A271910.

Cf. A186434, A187452.

Sequence in context: A117404 A309783 A072517 * A288947 A328146 A033585

Adjacent sequences:  A271909 A271910 A271911 * A271913 A271914 A271915

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Apr 24 2016

EXTENSIONS

More terms from Jean-François Alcover, Oct 10 2018

STATUS

approved

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Last modified September 29 12:15 EDT 2020. Contains 337431 sequences. (Running on oeis4.)