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A227327 Number of non-equivalent ways to choose two points in an equilateral triangle grid of side n. 15
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 (list; graph; refs; listen; history; text; internal format)
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
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).
Sequence in context: A155402 A155232 A188281 * A023609 A055364 A284870
KEYWORD
nonn,easy
AUTHOR
Heinrich Ludwig, Jul 07 2013
STATUS
approved

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Last modified May 26 13:39 EDT 2024. Contains 372826 sequences. (Running on oeis4.)