A271911 Number of ways to choose three distinct points from a 2 X n grid so that they form an isosceles triangle.

0, 4, 10, 16, 24, 32, 42, 52, 64, 76, 90, 104, 120, 136, 154, 172, 192, 212, 234, 256, 280, 304, 330, 356, 384, 412, 442, 472, 504, 536, 570, 604, 640, 676, 714, 752, 792, 832, 874, 916, 960, 1004, 1050, 1096, 1144, 1192, 1242, 1292, 1344, 1396, 1450, 1504

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.

Index entries for linear recurrences with constant coefficients, signature (2, 0, -2, 1).

Formula

Conjectured g.f.: 2*x*(2*x^2-x-2)/((x+1)*(x-1)^3). It would be nice to have a proof!

Conjectures from Colin Barker, Apr 24 2016: (Start)

a(n) = (-1+(-1)^n+16*n+2*n^2)/4, or equivalently, a(n) = (n^2+8*n)/2 if n even, (n^2+8*n-1)/2 if n odd.

a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n>4.

(End)

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

Example

n=3: Label the points
1 2 3
4 5 6
There are 8 small isosceles triangles like 124 plus 135 and 246, so a(3) = 10.

Mathematica

LinearRecurrence[{2, 0, -2, 1}, {0, 4, 10, 16}, 60] (* Harvey P. Dale, May 10 2018 *)

Crossrefs

Row 2 of A271910.

Cf. A186434, A187452.

Same start as, but totally different from, 2*A213707.

Sequence in context: A049881 A341064 A366661 * A322948 A277368 A067274

Adjacent sequences: A271908 A271909 A271910 * A271912 A271913 A271914

Keyword

nonn,more

Author

N. J. A. Sloane, Apr 24 2016

Extensions

More terms from Harvey P. Dale, May 10 2018

Status

approved