A056951 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].

-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, 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, 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

Offset: 1

Author

Henry Bottomley, Sep 05 2000

Keywords

• easy
• sign
• tabl

Examples

			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.
		

Crossrefs

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.

Cf. A130517 (unsigned version).

Programs

Mathematica

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 *)

Formula

T(n, k) = 2k - n - b with 1 <= k <= n (where b = 2 if 2k <= n + 1, b = 1 otherwise).