A296996 Number of nonequivalent (mod D_8) ways to place 3 points on an n X n point grid so that no point is equally distant from two other points on the same row or the same column.

0, 1, 14, 75, 310, 911, 2373, 5254, 10824, 20305, 36300, 61081, 99294, 154735, 234955, 345836, 498848, 702609, 973674, 1324135, 1776950, 2348511, 3069649, 3961970, 5065800, 6408961, 8043048, 10003189, 12354174, 15139615, 18439575, 22307416, 26840704, 32103905, 38214470

Offset

1,3

Comments

Rotations and reflections of placements are not counted. If they are to be counted see A296997.

The condition of placements is also known as "no 3-term arithmetic progressions".

Links

Heinrich Ludwig, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (3,1,-11,6,14,-14,-6,11,-1,-3,1).

Formula

a(n) = (n^6 -3*n^4 +5*n^3 -4*n^2 +4n)/48 + (n == 1 mod 2)*(8*n^3 -18n^2 +7*n)/48.

From Colin Barker, Jan 12 2018: (Start)

G.f.: x^2*(1 + 11*x + 32*x^2 + 82*x^3 + 54*x^4 + 57*x^5 + 2*x^6 + 2*x^7 - x^8) / ((1 - x)^7*(1 + x)^4).

a(n) = (n^6 - 3*n^4 + 5*n^3 - 4*n^2 + 4*n) / 48 for n even.

a(n) = (n^6 - 3*n^4 + 13*n^3 - 22*n^2 + 11*n) / 48 for n odd.

a(n) = 3*a(n-1) + a(n-2) - 11*a(n-3) + 6*a(n-4) + 14*a(n-5) - 14*a(n-6) - 6*a(n-7) + 11*a(n-8) - a(n-9) - 3*a(n-10) + a(n-11) for n>11.

(End)

Mathematica

Array[(#^6 - 3 #^4 + 5 #^3 - 4 #^2 + 4 #)/48 + Boole[OddQ@ #] (8 #^3 - 18 #^2 + 7 #)/48 &, 35] (* or *)
Rest@ CoefficientList[Series[x^2*(1 + 11 x + 32 x^2 + 82 x^3 + 54 x^4 + 57 x^5 + 2 x^6 + 2 x^7 - x^8)/((1 - x)^7*(1 + x)^4), {x, 0, 35}], x] (* Michael De Vlieger, Jan 12 2018 *)

Prog

(PARI) concat(0, Vec(x^2*(1 + 11*x + 32*x^2 + 82*x^3 + 54*x^4 + 57*x^5 + 2*x^6 + 2*x^7 - x^8) / ((1 - x)^7*(1 + x)^4) + O(x^40))) \\ Colin Barker, Jan 12 2018

Crossrefs

Cf. A296997.

Sequence in context: A196411 A108650 A093567 * A270704 A200554 A152100

Adjacent sequences: A296993 A296994 A296995 * A296997 A296998 A296999

Keyword

nonn,easy

Author

Heinrich Ludwig, Jan 12 2018

Status

approved