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A248864 Minimal perimeter of an n-dollar construction consisting of 3-dollar triangles and 4-dollar squares. 0
4, 5, 6, 5, 6, 7, 6, 7, 8, 7, 8, 7, 6, 9, 8, 7, 8, 8, 8, 9, 8, 9 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn,more
AUTHOR
Gordon Hamilton, Mar 03 2015
STATUS
approved

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)