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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 22 10:55 EDT 2018. Contains 316436 sequences. (Running on oeis4.)