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A300001 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. 2

%I #16 Oct 24 2020 13:59:25

%S 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,

%T 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,

%U 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

%N 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.

%C No solutions exist for n = 2, 3 and 5.

%C a(n) = A290820(n) for n <= 8. It is conjectured that a(n) < A290820(n) for all n > 12.

%C 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

%H Ales Drapal, Carlo Hamalainen, <a href="http://arxiv.org/abs/0910.5199">An enumeration of equilateral triangle dissections</a>, arXiv:0910.5199 [math.CO], 2009-2010.

%F a(n^2) = n for all n>=1, a(n^2-3) = n for all n>=3. - Corrected by _Peter Munn_, Feb 24 2018

%F 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

%e a(9)=3 a(10)=4 a(11)=5

%e * * *

%e / \ / \ / \

%e *---* *---* + +

%e / \ / \ / \ / \ / \

%e *---*---* *---*---* + +

%e / \ / \ / \ / \ / \ / \ / \

%e *---*---*---* + *---* + *---+---+---*

%e / \ / \ / \ / \ / \

%e *---+---*---+---* *---*---* + +

%e / \ / \ / \ / \

%e *---*---*---*---+---*

%e .

%e a(12)=6 a(13)=4 a(14)=5

%e * * *

%e / \ / \ / \

%e *---* *---* + +

%e / \ / \ / \ / \ / \

%e *---*---* *---*---* + +

%e / \ / \ / \ / \ / \ / \ / \

%e *---*---*---* *---* *---* *---+---+---*

%e / \ / \ / \ / \ / \ / \ / \ / \ / \

%e * + + + *---*---*---*---* *---*---*---* +

%e / \ / \ / \ / \ / \ / \

%e + + + + *---*---*---*---+---*

%e / \ / \

%e *---+---+---*---+---+---*

%e .

%e a(15)=6 a(16)=4 a(17)=5

%e * * *

%e / \ / \ / \

%e + + *---* + +

%e / \ / \ / \ / \

%e + + *---*---* + +

%e / \ / \ / \ / \ / \

%e + + *---*---*---* *---*---*---*

%e / \ / \ / \ / \ / \ / \ / \ / \ / \

%e *---*---*---*---* *---*---*---*---* *---*---*---*---*

%e / \ / \ / \ / \ / \ / \ / \ / \

%e *---* *---* *---* *---*---*---*---*---*

%e / \ / \ / \ / \ / \ / \

%e *---*---*---*---*---*---*

%e .

%e a(18)=6 a(19)=5 a(20)=6

%e * * *

%e / \ / \ / \

%e + + + + *---*

%e / \ / \ / \ / \

%e + + *---*---* *---*---*

%e / \ / \ / \ / \ / \ / \

%e + + *---* *---* *---*---*---*

%e / \ / \ / \ / \ / \ / \ / \ / \ / \

%e *---*---*---*---* *---*---*---*---* + *---*---* +

%e / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \

%e *---*---* *---*---* *---*---*---*---*---* + *---* +

%e / \ / \ / \ / \ / \ / \ / \

%e *---*---*---+---*---*---* *---+---+---*---+---+---*

%Y Cf. A068527, A123120, A290820, A299705.

%K nonn

%O 1,4

%A _Hugo Pfoertner_, Feb 20 2018

%E a(21)-a(100) from _Peter Munn_, Feb 24 2018

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Last modified March 28 08:22 EDT 2024. Contains 371236 sequences. (Running on oeis4.)