A056023 The positive integers written as a triangle, where row n is written from right to left if n is odd and otherwise from left to right.

1, 2, 3, 6, 5, 4, 7, 8, 9, 10, 15, 14, 13, 12, 11, 16, 17, 18, 19, 20, 21, 28, 27, 26, 25, 24, 23, 22, 29, 30, 31, 32, 33, 34, 35, 36, 45, 44, 43, 42, 41, 40, 39, 38, 37, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56

Offset: 1

Author

Clark Kimberling, Aug 01 2000

Keywords

• nonn
• tabl
• easy

Comments

A triangle such that (1) every positive integer occurs exactly once; (2) row n consists of n consecutive numbers; (3) odd-numbered rows are decreasing; (4) even-numbered rows are increasing; and (5) column 1 is increasing.

Self-inverse permutation of the natural numbers.

T(2*n-1,1) = A000217(2*n-1) = T(2*n,1) - 1; T(2*n,4*n) = A000217(2*n) = T(2*n+1,4*n+1) - 1. - Reinhard Zumkeller, Apr 25 2004

Mirror image of triangle in A056011. - Philippe Deléham, Apr 04 2009

From Clark Kimberling, Feb 03 2011: (Start)

When formatted as a rectangle R, for m > 1, the numbers n-1 and n+1 are neighbors (row, column, or diagonal) of R.

R(n,k) = n + (k+n-2)(k+n-1)/2 if n+k is odd;

R(n,k) = k + (n+k-2)(n+k-1)/2 if n+k is even.

Northwest corner:

1, 2, 6, 7, 15, 16, 28

3, 5, 8, 14, 17, 27, 30

4, 9, 13, 18, 26, 31, 43

10, 12, 19, 25, 32, 42, 49

11, 20, 24, 33, 41, 50, 62

(End)

a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers. - Boris Putievskiy, Dec 24 2012

For generalizations see A218890, A213927. - Boris Putievskiy, Mar 10 2013

Examples

			From _Philippe Deléham_, Apr 04 2009 (Start)
Triangle begins:
  1;
  2,   3;
  6,   5,  4;
  7,   8,  9, 10;
  15, 14, 13, 12, 11;
  ...
(End)
Enumeration by boustrophedonic ("ox-plowing") diagonal method. - _Boris Putievskiy_, Dec 24 2012
		

Links

• Ivan Neretin, Table of n, a(n) for n = 1..5050
• Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.
• Eric Weisstein's MathWorld, Pairing functions
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A056011, A218890, A213927.

Programs

Mathematica

(* As a rectangle: *)
r[n_, k_] := n + (k + n - 2) (k + n - 1)/2/; OddQ[n + k];
r[n_, k_] := k + (n + k - 2) (n + k - 1)/2/; EvenQ[n+k];
TableForm[Table[f[n, k], {n, 1, 10}, {k, 1, 15}]]
Table[r[n - k + 1, k], {n, 14}, {k, n, 1, -1}]//Flatten
(* Clark Kimberling, Feb 03 2011 *)
Module[{nn=15},If[OddQ[Length[#]],Reverse[#],#]&/@TakeList[Range[ (nn(nn+1))/2],Range[nn]]]//Flatten (* Harvey P. Dale, Feb 08 2022 *)

Formula

T(n, k) = (n^2 - (n - 2*k)*(-1)^(n mod 2))/2 + n mod 2. - Reinhard Zumkeller, Apr 25 2004

a(n) = ((i + j - 1)*(i + j - 2) + ((-1)^t + 1)*j - ((-1)^t - 1)*i)/2, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n and t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 24 2012

Extensions

Name edited by Andrey Zabolotskiy, Apr 16 2023