cp's OEIS Frontend

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

%I A227327 #33 Nov 03 2024 04:54:15
%S A227327 0,1,4,10,22,41,72,116,180,265,380,526,714,945,1232,1576,1992,2481,
%T A227327 3060,3730,4510,5401,6424,7580,8892,10361,12012,13846,15890,18145,
%U A227327 20640,23376,26384,29665,33252,37146,41382,45961,50920,56260,62020,68201,74844
%N A227327 Number of non-equivalent ways to choose two points in an equilateral triangle grid of side n.
%C A227327 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).
%H A227327 Vincenzo Librandi, <a href="/A227327/b227327.txt">Table of n, a(n) for n = 1..1000</a>
%H A227327 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-5,5,1,-3,1).
%F A227327 a(n) = (n^4 + 2*n^3 + 8*n^2 - 8*n    )/48; if n even.
%F A227327 a(n) = (n^4 + 2*n^3 + 8*n^2 - 2*n - 9)/48; if n odd.
%F A227327 G.f.: -x^2*(x^3-x^2+x+1) / ((x-1)^5*(x+1)^2). - _Colin Barker_, Jul 12 2013
%e A227327 for n = 3 there are the following 4 choices of 2 points (X) (rotations and reflections being ignored):
%e A227327      X         X         X         .
%e A227327     X .       . .       . .       X X
%e A227327    . . .     X . .     . X .     . . .
%t A227327 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 *)
%t A227327 CoefficientList[Series[-x (x^3 - x^2 + x + 1) / ((x - 1)^5  (x + 1)^2), {x, 0, 50}], x] (* _Vincenzo Librandi_, Aug 02 2013 *)
%t A227327 LinearRecurrence[{3,-1,-5,5,1,-3,1},{0,1,4,10,22,41,72},50] (* _Harvey P. Dale_, May 11 2019 *)
%Y A227327 Corresponding questions about the number of ways in a square grid are treated by A083374 (2 points) and A178208 (3 points).
%Y A227327 Cf. A178073, A071244.
%K A227327 nonn,easy
%O A227327 1,3
%A A227327 _Heinrich Ludwig_, Jul 07 2013