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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
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS 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 LINKS Ales Drapal, Carlo Hamalainen, An enumeration of equilateral triangle dissections, arXiv:0910.5199 [math.CO], 2009-2010. FORMULA 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 EXAMPLE 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 * * * / \ / \ / \ + + + + *---* / \ / \ / \ / \ + + *---*---* *---*---* / \ / \ / \ / \ / \ / \ + + *---* *---* *---*---*---* / \ / \ / \ / \ / \ / \ / \ / \ / \ *---*---*---*---* *---*---*---*---* + *---*---* + / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ *---*---* *---*---* *---*---*---*---*---* + *---* + / \ / \ / \ / \ / \ / \ / \ *---*---*---+---*---*---* *---+---+---*---+---+---* CROSSREFS Cf. A068527, A123120, A290820, A299705. Sequence in context: A330492 A350534 A101336 * A137218 A306765 A087819 Adjacent sequences: A299998 A299999 A300000 * A300002 A300003 A300004 KEYWORD nonn AUTHOR Hugo Pfoertner, Feb 20 2018 EXTENSIONS a(21)-a(100) from Peter Munn, Feb 24 2018 STATUS approved

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Last modified March 28 03:48 EDT 2023. Contains 361577 sequences. (Running on oeis4.)