A248864 Minimal perimeter of an n-dollar construction consisting of 3-dollar triangles and 4-dollar squares.

4, 5, 6, 5, 6, 7, 6, 7, 8, 7, 8, 7, 6, 9, 8, 7, 8, 8, 8, 9, 8, 9

Offset

6,1

Comments

The squares and equilateral triangles have edge lengths of 1.

What ratios of triangle cost to square cost make large minimal perimeter shapes dominated by squares? What ratios make them dominated by triangles? - Gordon Hamilton, Mar 17 2015

a(19) - a(27) have not been proved to be optimal. - Gordon Hamilton, Mar 17 2015

Links

Table of n, a(n) for n=6..27.

Example

a(6) = 4 is created by gluing two equilateral triangles along an edge to make a rhombus with perimeter 4:
.
                          /\
                         /  \
                        /____\
                        \    /
                         \  /
                          \/
.
a(16) = 8 because 4($4) = $16 and the four squares can be arranged so the shape has perimeter 8:
.
                    +------+------+
                    |      |      |
                    |      |      |
                    |      |      |
                    +------+------+
                    |      |      |
                    |      |      |
                    |      |      |
                    +------+------+
.
a(17) = 7 because 3($3) + 2($4) = $17 and three triangles can be built on top of two squares to create a shape with perimeter 7:
                        _______
                       /\     /\
                      /  \   /  \
                     /    \ /    \
                    +------+------+
                    |      |      |
                    |      |      |
                    |______|______|
.
a(18) = 6 because 6($3) = $18 and the six triangles can be built into a hexagon of perimeter 6.
                         ______
                        /\    /\
                       /  \  /  \
                      /____\/____\
                      \    /\    /
                       \  /  \  /
                        \/____\/
.
a(19) = 9 because 5($3) + 1($4) = $19 and this is one of the minimal perimeter shapes:
                         ______
                        /\    /
                       /  \  /
                      /____\/____
                      \    /\    /
                       \  /  \  /
                        \/____\/
                         |    |
                         |    |
                         |____|
.
a(20) = 8 because 4($3) + 2($4) = $20 and four triangles can be built on top of two squares to create this minimal perimeter shape:
                           +
                          / \
                         /   \
                        /_____\
                       /\     /\
                      /  \   /  \
                     /    \ /    \
                    +------+------+
                    |      |      |
                    |      |      |
                    |______|______|
.
a(21) = 7 because 7($3) = $21 and the seven triangles can be built into this minimal perimeter shape.
                         ______
                        /\    /\
                       /  \  /  \
                      /____\/____\
                      \    /\    /\
                       \  /  \  /  \
                        \/____\/____\
.
a(26) = 8 because 6($3) + 2($4) = $26 and the following shape minimizes the perimeter:
                        _______
                       /\     /\
                      /  \   /  \
                     /    \ /    \
                    +------+------+
                    |      |      |
                    |      |      |
                    |      |      |
                    +------+------+
                     \    / \    /
                      \  /   \  /
                       \/_____\/
.
a(33) = 9 because 7($3) + 3($4) = $33 and the following construction works: Take a triangle. Glue the three squares to its three edges. Use the remaining 6 triangles to make the convex shape of perimeter 9.

Crossrefs

Sequence in context: A002129 A113184 A136004 * A134299 A112780 A021223

Adjacent sequences: A248861 A248862 A248863 * A248865 A248866 A248867

Keyword

nonn,more

Author

Gordon Hamilton, Mar 03 2015

Status

approved