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A300001
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Side length of the smallest equilateral triangle that can be dissected into n equilateral triangles with integer sides, or 0 if no such triangle exists.
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2
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1, 0, 0, 2, 0, 3, 4, 4, 3, 4, 5, 6, 4, 5, 6, 4, 5, 6, 5, 6, 6, 5, 7, 6, 5, 7, 6, 6, 7, 6, 7, 7, 6, 7, 7, 6, 7, 7, 8, 7, 7, 8, 7, 8, 8, 7, 8, 8, 7, 8, 9, 8, 8, 9, 8, 8, 9, 8, 9, 9, 8, 9, 9, 8, 9, 9, 9, 10, 9, 9, 10, 9, 9, 10, 9, 10, 10, 9, 10, 10, 9, 10, 10, 10, 10, 10, 11, 10, 10, 11, 10, 10, 11, 10, 11, 11, 10, 11, 11, 10
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OFFSET
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1,4
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COMMENTS
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No solutions exist for n = 2, 3 and 5.
a(n) = A290820(n) for n <= 8. It is conjectured that a(n) < A290820(n) for all n > 12.
The seven numbers mentioned by Peter Munn in the Formula section [1, 2, 4, 5, 7, 10, 13] coincide with the seven terms of A123120. - M. F. Hasler and Omar E. Pol, Feb 23 2018
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LINKS
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Table of n, a(n) for n=1..100.
Ales Drapal, Carlo Hamalainen, An enumeration of equilateral triangle dissections, arXiv:0910.5199 [math.CO], 2009-2010.
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FORMULA
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a(n^2) = n for all n>=1, a(n^2-3) = n for all n>=3. - Corrected by Peter Munn, Feb 24 2018
For n > 23, if A068527(n) = 1, 2, 4, 5, 7, 10 or 13 then a(n) = ceiling(sqrt(n)) + 1 else a(n) = ceiling(sqrt(n)). - Peter Munn, Feb 23 2018
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EXAMPLE
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a(9)=3 a(10)=4 a(11)=5
* * *
/ \ / \ / \
*---* *---* + +
/ \ / \ / \ / \ / \
*---*---* *---*---* + +
/ \ / \ / \ / \ / \ / \ / \
*---*---*---* + *---* + *---+---+---*
/ \ / \ / \ / \ / \
*---+---*---+---* *---*---* + +
/ \ / \ / \ / \
*---*---*---*---+---*
.
a(12)=6 a(13)=4 a(14)=5
* * *
/ \ / \ / \
*---* *---* + +
/ \ / \ / \ / \ / \
*---*---* *---*---* + +
/ \ / \ / \ / \ / \ / \ / \
*---*---*---* *---* *---* *---+---+---*
/ \ / \ / \ / \ / \ / \ / \ / \ / \
* + + + *---*---*---*---* *---*---*---* +
/ \ / \ / \ / \ / \ / \
+ + + + *---*---*---*---+---*
/ \ / \
*---+---+---*---+---+---*
.
a(15)=6 a(16)=4 a(17)=5
* * *
/ \ / \ / \
+ + *---* + +
/ \ / \ / \ / \
+ + *---*---* + +
/ \ / \ / \ / \ / \
+ + *---*---*---* *---*---*---*
/ \ / \ / \ / \ / \ / \ / \ / \ / \
*---*---*---*---* *---*---*---*---* *---*---*---*---*
/ \ / \ / \ / \ / \ / \ / \ / \
*---* *---* *---* *---*---*---*---*---*
/ \ / \ / \ / \ / \ / \
*---*---*---*---*---*---*
.
a(18)=6 a(19)=5 a(20)=6
* * *
/ \ / \ / \
+ + + + *---*
/ \ / \ / \ / \
+ + *---*---* *---*---*
/ \ / \ / \ / \ / \ / \
+ + *---* *---* *---*---*---*
/ \ / \ / \ / \ / \ / \ / \ / \ / \
*---*---*---*---* *---*---*---*---* + *---*---* +
/ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \
*---*---* *---*---* *---*---*---*---*---* + *---* +
/ \ / \ / \ / \ / \ / \ / \
*---*---*---+---*---*---* *---+---+---*---+---+---*
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CROSSREFS
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Cf. A068527, A123120, A290820, A299705.
Sequence in context: A330492 A350534 A101336 * A137218 A306765 A087819
Adjacent sequences: A299998 A299999 A300000 * A300002 A300003 A300004
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KEYWORD
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nonn
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AUTHOR
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Hugo Pfoertner, Feb 20 2018
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EXTENSIONS
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a(21)-a(100) from Peter Munn, Feb 24 2018
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STATUS
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approved
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