A227327 Number of non-equivalent ways to choose two points in an equilateral triangle grid of side n.

0, 1, 4, 10, 22, 41, 72, 116, 180, 265, 380, 526, 714, 945, 1232, 1576, 1992, 2481, 3060, 3730, 4510, 5401, 6424, 7580, 8892, 10361, 12012, 13846, 15890, 18145, 20640, 23376, 26384, 29665, 33252, 37146, 41382, 45961, 50920, 56260, 62020, 68201, 74844

Offset: 1

Author

Heinrich Ludwig, Jul 07 2013

Keywords

• nonn
• easy

Comments

The sequence is an alternating composition of A178073 and A071244: a(n) = 2*A071244((n+1)/2) if n is odd, otherwise a(n) = A178073(n/2).

Examples

			for n = 3 there are the following 4 choices of 2 points (X) (rotations and reflections being ignored):
     X         X         X         .
    X .       . .       . .       X X
   . . .     X . .     . X .     . . .
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

Crossrefs

Corresponding questions about the number of ways in a square grid are treated by A083374 (2 points) and A178208 (3 points).

Cf. A178073, A071244.

Programs

Mathematica

Table[b = n^4 + 2*n^3 + 8*n^2; If[EvenQ[n], c = b - 8*n, c = b - 2*n - 9]; c/48, {n, 43}] (* T. D. Noe, Jul 09 2013 *)
CoefficientList[Series[-x (x^3 - x^2 + x + 1) / ((x - 1)^5  (x + 1)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 02 2013 *)
LinearRecurrence[{3,-1,-5,5,1,-3,1},{0,1,4,10,22,41,72},50] (* Harvey P. Dale, May 11 2019 *)

Formula

a(n) = (n^4 + 2*n^3 + 8*n^2 - 8*n )/48; if n even.

a(n) = (n^4 + 2*n^3 + 8*n^2 - 2*n - 9)/48; if n odd.

G.f.: -x^2*(x^3-x^2+x+1) / ((x-1)^5*(x+1)^2). - Colin Barker, Jul 12 2013