OFFSET
1,2
COMMENTS
S must not contain 3 points A,B,C such that angle ABC = 90 degrees and |AB| = |BC|.
For example, this configuration is forbidden:
O B O O
O O O C
A O O O
O O O O
a(12) >= 29. - Robert Israel, Apr 22 2016
a(13) >= 32, a(14) >= 36, a(15) >= 38 (see pictures in Links). Note that a(14) >= 36 breaks the pattern of increasing by 3 at each step. - Giovanni Resta, Apr 23 2016
LINKS
Giovanni Resta, Illustration of a(3)-a(11)
Giovanni Resta, Illustration of lower bounds on a(12)-a(15)
EXAMPLE
Illustration for a(3) = 4:
X X X
O O O
O X O
Illustration for a(8) = 17:
O X O O O O O X
X O O O O O O X
O O O X O O O X
O O X O O O O X
O O O O O O O X
O O O O O O O X
O O O O O O X O
X X X X X X O O
MATHEMATICA
d[n_, a_, b_] := Block[{x1, y1, x2, y2}, x1 = Mod[a-1, n]; y1 = Floor[(a-1)/n]; x2 = Mod[b-1, n]; y2 = Floor[(b-1)/n]; (x1-x2)^2 + (y1-y2)^2]; isorQ[n_, a_, b_, c_] := Block[{k = Sort[{d[n, a, b], d[n, b, c], d[n, a, c]}]}, k[[1]] == k[[2]] && 2 k[[1]] == k[[3]]]; sol[n_] := sol[n] = Block[{m, L={}, nv=n^2, ne}, Do[If[ isorQ[n, x, y, z], AppendTo[L, {x, y, z}]], {x, n^2}, {y, x-1}, {z, y-1}]; ne = Length@L; m = Table[0, {ne}, {nv}]; Do[m[[i, L[[i]]]] = 1, {i, ne}]; Quiet@ LinearProgramming[ Table[-1, {nv}], m, Table[{2, -1}, {ne}], Table[{0, 1}, {nv}], Integers]]; a[n_] := Total[sol[n]]; Do[Print@ MatrixForm@ Partition[ sol@n, n], {n, 6}]; Array[a, 6]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Giovanni Resta and N. J. A. Sloane, Apr 22 2016
STATUS
approved