A024610 -id:A024610 - cp's OEIS frontend

A328787 Number of distinct sizes of equilateral triangles with vertices in an equilateral triangular array of points with n rows.

1, 2, 4, 6, 8, 11, 14, 17, 21, 25, 28, 33, 38, 43, 49, 54, 59, 66, 73, 80, 87, 93, 100, 109, 116, 124, 133, 142, 150, 161, 172, 181, 191, 201, 211, 224, 234, 246, 258, 271, 282, 295, 307, 319, 333, 346, 359, 375, 389, 403, 420, 435, 448, 465, 482, 499, 514

Offset: 2

Peter Kagey, Oct 27 2019

nonn

a(n) <= A024610(n-1) - 1.

			For n = 4, the a(4) = 4 sizes of equilateral triangles are 1, sqrt(3), 2, and 3. For example,
     *         o         *              *
    * *       * o       o o            o o
   o o o     o o *     * o *          o o o
  o o o o,  o * o o,  o o o o,  and  * o o *.
For n = 7, the a(7) = 11 equilateral triangles have side lengths 1, sqrt(3), 2, sqrt(7), 3, sqrt(12), sqrt(13), 4, sqrt(21), 5, and 6.

Peter Kagey, Table of n, a(n) for n = 2..10000

Cf. A000332, A003136, A024610.

A108279 is the analog for squares on square grids.

A328793 Least k such that there exists an equilateral triangle of side length sqrt(A003136(n)) with vertices in an equilateral triangular array of points with k rows.

Original entry on oeis.org

2, 4, 3, 5, 4, 7, 6, 5, 8, 7, 6, 10, 9, 8, 7, 11, 10, 9, 13, 8, 11, 10, 14, 13, 9, 12, 11, 16, 15, 14, 10, 13, 12, 16, 15, 11, 14, 19, 18, 13, 17, 16, 12, 15, 20, 19, 14, 17, 13, 16, 21, 20, 19, 15, 18, 14, 22, 17, 21, 20, 16, 19, 25, 24, 15, 18, 22, 21, 17

Offset: 1

Peter Kagey, Oct 27 2019

a(A024610(n) - 1) = n + 1 for all n > 0, and

a(A024610(n) + k) > n + 1 for all n > 0, k >= 0.

			For n = 2, there exists an equilateral triangle with side length sqrt(A003136(2)) = sqrt(3) and vertices on equilateral triangular array with a(2) = 4 rows:
     o
    * o
   o o *
  o * o o.
However there is no equilateral triangle of side length sqrt(3) with vertices on a smaller triangular array with three or fewer rows.

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A003136, A024610, A328787.

cp's OEIS Frontend

A328787 Number of distinct sizes of equilateral triangles with vertices in an equilateral triangular array of points with n rows.

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