A254575 Triangle T(n,k) in which the n-th row encodes how to hang a picture by wrapping rope around n nails using a polynomial number of twists, such that removing one nail causes the picture to fall; n>=1, 1<=k<=A073121(n).

1, 1, 2, -1, -2, 1, 2, -1, -2, 3, 2, 1, -2, -1, -3, 1, 2, -1, -2, 3, 4, -3, -4, 2, 1, -2, -1, 4, 3, -4, -3, 1, 2, -1, -2, 3, 2, 1, -2, -1, -3, 4, 5, -4, -5, 3, 1, 2, -1, -2, -3, 2, 1, -2, -1, 5, 4, -5, -4, 1, 2, -1, -2, 3, 2, 1, -2, -1, -3, 4, 5, -4, -5, 6, 5

Offset: 1

Author

Alois P. Heinz, Feb 01 2015

Keywords

• sign
• tabf
• look

Comments

In step k the rope has to be wrapped around nail |T(n,k)| clockwise if T(n,k)>0 and counterclockwise if T(n,k)<0.

1 or (-1) appears A062383(n-1) times in row n.

n or (-n) appears A053644(n) times in row n.

Examples

			Triangle T(n,k) begins:
  1;
  1, 2, -1, -2;
  1, 2, -1, -2, 3, 2,  1, -2, -1, -3;
  1, 2, -1, -2, 3, 4, -3, -4,  2,  1, -2, -1, 4, 3, -4, -3;
		

Links

• Alois P. Heinz, Rows n = 1..30, flattened
• E. D. Demaine, M. L. Demaine, Y. N. Minsky, J. S. B. Mitchell, R. L. Rivest, M. Patrascu, Picture-Hanging Puzzles, arXiv:1203.3602 [cs.DS], 2012-2014.

Crossrefs

Row sums give A063524.

Cf. A053644, A062383, A073121.

Programs

Maple

r:= s-> seq(-s[-k], k=1..nops(s)):
T:= proc(n) option remember; `if`(n=1, 1, (m->
      ((x, y)-> [x[], y[], r(x), r(y)][])([T(m)],
       map(h-> h+sign(h)*m, [T(n-m)])))(iquo(n+1, 2)))
    end:
seq(T(n), n=1..7);

Mathematica

r[s_List] := -Reverse[s];
T[1] = {1}; T[n_] := T[n] = Module[{ m = Quotient[n+1, 2]}, Function[{x, y}, {x, y, r[x], r[y]} // Flatten][T[m], Function[h, h + Sign[h]*m] /@ T[n - m]]];
Table[T[n], {n, 1, 7}] // Flatten (* Jean-François Alcover, Nov 06 2017, after Alois P. Heinz *)