

A227116


Given an equilateral triangular grid with side n, containing n(n+1)/2 points, a(n) is the minimal number of points to be removed from the grid, so that, if 3 of the remaining points are chosen, they do not form an equilateral triangle with sides parallel to the grid.


8



0, 1, 2, 4, 7, 9, 14, 18, 23, 29, 36, 44, 52, 61, 71
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OFFSET

1,3


COMMENTS

This is the complementary problem to A227308.
Numbers found by an exhaustive computational search for all solutions (see history).


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)
Ed Wynn, A comparison of encodings for cardinality constraints in a SAT solver, arXiv:1810.12975 [cs.LO], 2018.


FORMULA

a(n) + A227308(n) = n(n+1)/2.


EXAMPLE

n = 11: at least a(11) = 36 points (.) out of the 66 have to be removed, leaving 30 (X) behind:
.
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 .
There is no equilateral subtriangle with all vertices = X and sides parallel to the whole triangle.


CROSSREFS

Cf. A227308, A152125, A227133
Sequence in context: A090893 A100486 A139533 * A180742 A039904 A115162
Adjacent sequences: A227113 A227114 A227115 * A227117 A227118 A227119


KEYWORD

nonn,hard,more


AUTHOR

Heinrich Ludwig, Jul 01 2013


EXTENSIONS

Added a(12), a(13), Heinrich Ludwig, Sep 02 2013
Added a(14), Giovanni Resta, Sep 19 2013
a(15) from Heinrich Ludwig, Oct 27 2013


STATUS

approved



