A231654 Number of non-equivalent ways to choose 5 points in an equilateral triangle grid of side n.
0, 0, 2, 48, 526, 3450, 16536, 63104, 204202, 580669, 1491096, 3520768, 7754502, 16098425, 31770760, 59998736, 109022244, 191454654, 326158974, 540703008, 874630262, 1383621756, 2144889472, 3263884272, 4882793214, 7190910467, 10437526372, 14947411024
Offset: 1
Links
- Heinrich Ludwig, Table of n, a(n) for n = 1..1000
- Index entries for linear recurrences with constant coefficients, signature (6,-10,-10,50,-34,-66,110,0,-110,66,34,-50,10,10,-6,1).
Programs
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Mathematica
Table[If[EvenQ[n], b = 0, b = 375*n^4 - 1170*n^3 + 210*n^2 - 405*n + 1035]; (n^10 + 5*n^9 - 10*n^8 - 70*n^7 + 25*n^6 + 584*n^5 - 420*n^4 - 480*n^3 - 1216*n^2 + 1536*n + b)/23040, {n, 30}] (* T. D. Noe, Nov 14 2013 *)
Formula
a(n) = (n^10 + 5*n^9 - 10*n^8 - 70*n^7 + 25*n^6 + 584*n^5 - 420*n^4 - 480*n^3 - 1216*n^2 + 1536*n + B)/23040 where B = 375*n^4 - 1170*n^3 + 210*n^2 - 405*n + 1035 if n odd, and B = 0 if n even.
G.f.: x^3*(x^11 -4*x^10 +14*x^9 -78*x^8 -189*x^7 -902*x^6 -1316*x^5 -1476*x^4 -794*x^3 -258*x^2 -36*x -2) / ((x -1)^11*(x +1)^5). - Colin Barker, Feb 15 2014