

A227308


Given an equilateral triangular grid with side n consisting of n(n+1)/2 points, a(n) is the maximum number of points that can be painted so that, if any 3 of the painted ones are chosen, they do not form an equilateral triangle with sides parallel to the grid.


8



1, 2, 4, 6, 8, 12, 14, 18, 22, 26, 30, 34, 39, 44, 49
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OFFSET

1,2


COMMENTS

Numbers found by an exhaustive computational search for all solutions. This sequence is complementary to A227116: A227116(n) + A227308(n) = n(n+1)/2.
Up to n=12 there is always a symmetric maximal solution. For n=13 and n=15 symmetric solutions contain at most a(n)1 painted points.  Heinrich Ludwig, Oct 26 2013


REFERENCES

Mohammad K. Azarian, Equating Distances and Altitude in an Equilateral Triangle, Problem 316, Mathematics and Computer Education, Vol. 28, No. 3, Fall 1994, p. 337. Solution published in Vol. 29, No. 3, Fall 1995, pp. 324325.
Mohammad K. Azarian, A Trigonometric Characterization of Equilateral Triangle, Problem 336, Mathematics and Computer Education, Vol. 31, No. 1, Winter 1997, p. 96. Solution published in Vol. 32, No. 1, Winter 1998, pp. 8485.


LINKS

Table of n, a(n) for n=1..15.
Heinrich Ludwig, Illustration of a(2)..a(15)
Giovanni Resta, Illustration of a(3)a(14)


EXAMPLE

n = 11. At most a(11) = 30 points (X) of 66 can be painted, while 36 (.) must remain unpainted.
.
X X
X . X
X . . X
X . . . X
X . . . . X
. X X . X X .
. X . X X . X .
. . X X . X X . .
X . . . . . . . . X
. X X X . . . X X X .
In this pattern there is no equilateral subtriangle with all vertices = X and sides parallel to the whole triangle.


MATHEMATICA

ivar[r_, c_] := r*(r1)/2 + c; a[n_] := Block[{m, qq, nv = n*(n+1)/2, ne}, qq = Union[ Flatten[Table[{ivar[r, c], ivar[rj, c], ivar[r, c+j]}, {r, 2, n}, {c, r  1}, {j, Min[r  1, r  c]}], 2], Flatten[Table[{ivar[r, c], ivar[r + j, c], ivar[r, c  j]}, {r, 2, n}, {c, 2, r}, {j, Min[c  1, n  r]}], 2]]; ne = Length@qq; m = Table[0, {ne}, {nv}]; Do[m[[i, qq[[i]]]] = 1, {i, ne}]; Total@ Quiet@ LinearProgramming[ Table[1, {nv}], m, Table[{2, 1}, {ne}], Table[{0, 1}, {nv}], Integers]]; Array[a, 9] (* Giovanni Resta, Sep 19 2013 *)


CROSSREFS

Cf. A227116 (the complementary problem), A152125, A227133, A002717.
Sequence in context: A260652 A089623 A089681 * A214294 A233578 A057220
Adjacent sequences: A227305 A227306 A227307 * A227309 A227310 A227311


KEYWORD

nonn,hard,more


AUTHOR

Heinrich Ludwig, Jul 06 2013


EXTENSIONS

a(12), a(13) from Heinrich Ludwig, Sep 02 2013
a(14) from Giovanni Resta, Sep 19 2013
a(15) from Heinrich Ludwig, Oct 26 2013


STATUS

approved



