OFFSET
1,2
LINKS
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
KEYWORD
nonn,more
AUTHOR
N. J. A. Sloane, Apr 24 2016
EXTENSIONS
More terms from Harvey P. Dale, May 10 2018
STATUS
approved