A296996 -id:A296996

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A296997

Number of 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.

+10
3

0, 4, 78, 544, 2260, 7068, 18298, 41472, 85032, 161300, 287430, 486624, 789308, 1234604, 1871730, 2761728, 3979088, 5613732, 7772862, 10583200, 14193060, 18774844, 24527338, 31678464, 40487800, 51249588, 64295478, 79997792, 98772492, 121082700, 147441890, 178417664

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

OFFSET

1,2

COMMENTS

Rotations and reflections of a placement are counted. If they are to be ignored, see A296996.

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

LINKS

Heinrich Ludwig, Table of n, a(n) for n = 1..256

Index entries for linear recurrences with constant coefficients, signature (5,-8,0,14,-14,0,8,-5,1).

FORMULA

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

a(n) = (n^6 - 3*n^4 - 3*n^3 + 8*n^2)/6 for n even,

a(n) = (n^6 - 3*n^4 - 3*n^3 + 8*n^2 - 3*n)/6 for n odd.

From Colin Barker, Dec 23 2017: (Start)

G.f.: 2*x^2*(2 + 29*x + 93*x^2 + 82*x^3 + 32*x^4 + x^5 + x^6) / ((1 - x)^7*(1 + x)^2).

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

(End)

MATHEMATICA

Array[(#^6 - 3 #^4 - 3 #^3 + 8 #^2)/6 - # Boole[OddQ@ #]/2 &, 32] (* Michael De Vlieger, Dec 23 2017 *)

CoefficientList[ Series[-2x (2 + 29x + 93x^2 + 82x^3 + 32x^4 + x^5 + x^6)/((x - 1)^7 (x + 1)^2), {x, 0, 31}], x] (* or *)

LinearRecurrence[{5, -8, 0, 14, -14, 0, 8, -5, 1}, {0, 4, 78, 544, 2260, 7068, 18298, 41472, 85032}, 32] (* Robert G. Wilson v, Jan 15 2018 *)

PROG

(PARI) concat(0, Vec(2*x^2*(2 + 29*x + 93*x^2 + 82*x^3 + 32*x^4 + x^5 + x^6) / ((1 - x)^7*(1 + x)^2) + O(x^40))) \\ Colin Barker, Dec 23 2017

CROSSREFS

Cf. A296468, A296996, A296998.

KEYWORD

nonn,easy

AUTHOR

Heinrich Ludwig, Dec 23 2017

STATUS

approved

A296999

Number of nonequivalent (mod D_8) ways to place 4 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.

+10
1

0, 1, 17, 226, 1550, 7221, 26120, 78484, 206242, 486640, 1056377, 2137506, 4085167, 7430276, 12964014, 21801632, 35520743, 56249658, 86880957, 131186720, 194133425, 282024809, 402949496, 566950056, 786640454, 1077397347, 1458190435, 1951789266, 2585856152, 3393157995

(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 A296998.

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,-4,4,-4,5,1,-5,6,-10,8,-8,10,-6,5,-1,-5,4,-4,4,1,-3,1).

FORMULA

a(n) = (n^8 - 6*n^6 - 12*n^5 + 64*n^4 + 8*n^3 - 136*n^2 + (n == 1 (mod 2))*(14*n^4 - 96*n^3 + 162*n^2 - 92*n + 93))/192 + (n == 2 (mod 6))*n/6 + (n == 2 (mod 4))*n/4 + (n == 5 (mod 6))*(n + 1)/6.

a(n) = (n^8 - 6*n^6 - 12*n^5)/192 + b(n) + c(n), where

b(n) = (64*n^4 + 8*n^3 - 136*n^2)/192 for n even,

b(n) = (78*n^4 - 88*n^3 + 26*n^2 - 92*n + 93)/192 for n odd,

c(n) = 0 for n == 0, 1, 3, 4, 7, 9 (mod 12),

c(n) = n/4 for n == 6, 10 (mod 12),

c(n) = n/6 for n == 8 (mod 12),

c(n) = 5/12*n for n == 2 (mod 12),

c(n) = (n + 1)/6 for n == 5, 11 (mod 12).

Conjectures from Colin Barker, Jan 21 2018: (Start)

G.f.: x^2*(1 + 14*x + 176*x^2 + 893*x^3 + 2861*x^4 + 6847*x^5 + 12704*x^6 + 20412*x^7 + 27052*x^8 + 33142*x^9 + 33910*x^10 + 33289*x^11 + 26586*x^12 + 20709*x^13 + 12212*x^14 + 7178*x^15 + 2639*x^16 + 1094*x^17 + 134*x^18 + 68*x^19 - 3*x^20 + 2*x^21) / ((1 - x)^9*(1 + x)^5*(1 - x + x^2)*(1 + x^2)^2*(1 + x + x^2)^2).

a(n) = 3*a(n-1) - a(n-2) - 4*a(n-3) + 4*a(n-4) - 4*a(n-5) + 5*a(n-6) + a(n-7) - 5*a(n-8) + 6*a(n-9) - 10*a(n-10) + 8*a(n-11) - 8*a(n-13) + 10*a(n-14) - 6*a(n-15) + 5*a(n-16) - a(n-17) - 5*a(n-18) + 4*a(n-19) - 4*a(n-20) + 4*a(n-21) + a(n-22) - 3*a(n-23) + a(n-24) for n>24.

(End)

MATHEMATICA

Array[(#^8 - 6 #^6 - 12 #^5 + 64 #^4 + 8 #^3 - 136 #^2 + Boole[OddQ@ #] (14 #^4 - 96 #^3 + 162 #^2 - 92 # + 93))/192 + Boole[Mod[#, 6] == 2] #/6 + Boole[Mod[#, 4] == 2] #/4 + Boole[Mod[#, 6] == 5] (# + 1)/6 &, 30] (* Michael De Vlieger, Jan 21 2018 *)