A254575 - OEIS

%I #43 Feb 06 2023 06:49:42

%S 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,

%T 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,

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

%N 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).

%C 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.

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

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

%H Alois P. Heinz, <a href="/A254575/b254575.txt">Rows n = 1..30, flattened</a>

%H E. D. Demaine, M. L. Demaine, Y. N. Minsky, J. S. B. Mitchell, R. L. Rivest, M. Patrascu, <a href="https://arxiv.org/abs/1203.3602">Picture-Hanging Puzzles</a>, arXiv:1203.3602 [cs.DS], 2012-2014.

%e Triangle T(n,k) begins:

%e   1;

%e   1, 2, -1, -2;

%e   1, 2, -1, -2, 3, 2,  1, -2, -1, -3;

%e   1, 2, -1, -2, 3, 4, -3, -4,  2,  1, -2, -1, 4, 3, -4, -3;

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

%p T:= proc(n) option remember; `if`(n=1, 1, (m->

%p       ((x, y)-> [x[], y[], r(x), r(y)][])([T(m)],

%p        map(h-> h+sign(h)*m, [T(n-m)])))(iquo(n+1, 2)))

%p     end:

%p seq(T(n), n=1..7);

%t r[s_List] := -Reverse[s];

%t 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]]];

%t Table[T[n], {n, 1, 7}] // Flatten (* _Jean-François Alcover_, Nov 06 2017, after _Alois P. Heinz_ *)

%Y Row sums give A063524.

%Y Cf. A053644, A062383, A073121.

%K sign,tabf,look

%O 1,3

%A _Alois P. Heinz_, Feb 01 2015