|
%I #18 Feb 04 2020 06:37:45
%S -1,-2,1,-3,-1,2,-4,-2,1,3,-5,-3,-1,2,4,-6,-4,-2,1,3,5,-7,-5,-3,-1,2,
%T 4,6,-8,-6,-4,-2,1,3,5,7,-9,-7,-5,-3,-1,2,4,6,8,-10,-8,-6,-4,-2,1,3,5,
%U 7,9,-11,-9,-7,-5,-3,-1,2,4,6,8,10,-12,-10,-8,-6,-4,-2,1,3,5,7,9,11,-13,-11,-9,-7,-5,-3,-1,2,4,6,8,10,12,-14,-12,-10
%N Triangle whose rows show the result of flipping the first, first two, ... and finally first n coins when starting with the stack (1,2,3,4,...,n) [starting with all heads up, where signs show whether particular coins end up heads or tails].
%F T(n, k) = 2k - n - b with 1 <= k <= n (where b = 2 if 2k <= n + 1, b = 1 otherwise).
%e Third row is constructed by starting from (1, 2, 3), going to (-1, 2, 3), then going to (-2, 1, 3) and finally going to (-3, -1, 2). Rows are: (-1), (-2, 1), (-3, -1, 2), (-4, -2, 1, 3) etc. as each row is reverse of previous row, with signs changed and -n added as the first term in the row.
%t t[n_, 1] := -n; t[n_, n_] := n - 1; t[n_, k_] := 2 * k - n - If[2 * k <= n + 1, 2, 1]; Table[t[n, k], {n, 14}, {k, n}] // Flatten (* _Jean-François Alcover_, Oct 03 2013 *)
%Y A003558 is the number of times the operation needs to be repeated to return to the starting point, taking no account of heads/tails (i.e., signs). A002326 is the number required if heads/tails (i.e., signs) are also required to return to their original position.
%Y Cf. A130517 (unsigned version).
%K easy,sign,tabl
%O 1,2
%A _Henry Bottomley_, Sep 05 2000
|
