OFFSET
1,2
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
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
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
EXAMPLE
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
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 *)
CROSSREFS
KEYWORD
AUTHOR
Clark Kimberling, Aug 01 2000
EXTENSIONS
Name edited by Andrey Zabolotskiy, Apr 16 2023
STATUS
approved