A123120 -id:A123120 - cp's OEIS frontend

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.

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

Offset: 1

Hugo Pfoertner, Feb 20 2018

nonn

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

			            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
              *                       *                       *
             / \                     / \                     / \
            +   +                   +   +                   *---*
           /     \                 /     \                 / \ / \
          +       +               *---*---*               *---*---*
         /         \             / \     / \             / \ / \ / \
        +           +           *---*   *---*           *---*---*---*
       /             \         / \ / \ / \ / \         / \ / \ / \ / \
      *---*---*---*---*       *---*---*---*---*       +   *---*---*   +
     / \ / \ / \ / \ / \     / \ / \ / \ / \ / \     /     \ / \ /     \
    *---*---*   *---*---*   *---*---*---*---*---*   +       *---*       +
   / \ / \ /     \ / \ / \                         /         \ /         \
  *---*---*---+---*---*---*                       *---+---+---*---+---+---*

Ales Drapal, Carlo Hamalainen, An enumeration of equilateral triangle dissections, arXiv:0910.5199 [math.CO], 2009-2010.

Cf. A068527, A123120, A290820, A299705.

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

a(21)-a(100) from Peter Munn, Feb 24 2018

A179101 Numbers which are not the sum of exactly one positive square and a sum of squares-minus-1.

Original entry on oeis.org

2, 3, 5, 6, 8, 11, 14

Offset: 1

Benoit Jubin, Jun 29 2010

Sequence motivated by the study of certain replicate tilings, where each tile can be replaced by a square number of tiles.

Adding multiples of 3=2^2-1 to the numbers 1=1^2, 9=3^2 and 17=3^2+(3^2-1), one obtains all the integers not in the sequence.

Cf. A000290 (squares), A005563 (squares-minus-1), A123120, A078135.

A223729 Numbers appearing in a theorem on the representation of numbers as sums of five non-vanishing squares.

Original entry on oeis.org

1, 2, 4, 5, 7, 10, 13, 28

Offset: 1

Wolfdieter Lang, Mar 27 2013

See A047701 for the positive numbers that are not the sum of five nonzero squares, which are precisely 1, 2, 3, 4 and 5 + a(n), n = 1,...,8. This appears as Theorem 2. in Grosswald's book on p. 73-4 with references to E. Dubouis (1911) and some others.

E. Dubouis, Solution of a problem of J. Tannery, L'Intermédiaire Math. 18 (1911) 55-56.
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 73-74.

Cf. A123120, A047701.

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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.

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A179101 Numbers which are not the sum of exactly one positive square and a sum of squares-minus-1.

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A223729 Numbers appearing in a theorem on the representation of numbers as sums of five non-vanishing squares.

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