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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). 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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.

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.

EXAMPLE

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;

MAPLE

r:= s-> seq(-s[-k], k=1..nops(s)):

T:= proc(n) option remember; local m; m:= iquo(n+1, 2);

      `if`(n=1, 1, ((x, y)-> [x[], y[], r(x), r(y)][])(

       [T(m)], map(h-> h+sign(h)*m, [T(n-m)])))

    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-Fran├žois Alcover, Nov 06 2017, after Alois P. Heinz *)

CROSSREFS

Cf. A073121.

Sequence in context: A304536 A272863 A112632 * A275344 A206826 A175835

Adjacent sequences:  A254572 A254573 A254574 * A254576 A254577 A254578

KEYWORD

sign,tabf,look

AUTHOR

Alois P. Heinz, Feb 01 2015

STATUS

approved

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Last modified February 21 06:40 EST 2019. Contains 320371 sequences. (Running on oeis4.)