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

1,3

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)).

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).

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

Example

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 .     . . .

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 *)

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.

Sequence in context: A155402 A155232 A188281 * A023609 A055364 A284870

Adjacent sequences: A227324 A227325 A227326 * A227328 A227329 A227330

Keyword

nonn,easy

Author

Heinrich Ludwig, Jul 07 2013

Status

approved