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%I #60 Apr 17 2023 00:55:42
%S 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,
%T 23,22,29,30,31,32,33,34,35,36,45,44,43,42,41,40,39,38,37,46,47,48,49,
%U 50,51,52,53,54,55,66,65,64,63,62,61,60,59,58,57,56
%N 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.
%C 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.
%C Self-inverse permutation of the natural numbers.
%C 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
%C Mirror image of triangle in A056011. - _Philippe Deléham_, Apr 04 2009
%C From _Clark Kimberling_, Feb 03 2011: (Start)
%C When formatted as a rectangle R, for m > 1, the numbers n-1 and n+1 are neighbors (row, column, or diagonal) of R.
%C R(n,k) = n + (k+n-2)(k+n-1)/2 if n+k is odd;
%C R(n,k) = k + (n+k-2)(n+k-1)/2 if n+k is even.
%C Northwest corner:
%C 1, 2, 6, 7, 15, 16, 28
%C 3, 5, 8, 14, 17, 27, 30
%C 4, 9, 13, 18, 26, 31, 43
%C 10, 12, 19, 25, 32, 42, 49
%C 11, 20, 24, 33, 41, 50, 62
%C (End)
%C 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
%C For generalizations see A218890, A213927. - _Boris Putievskiy_, Mar 10 2013
%H Ivan Neretin, <a href="/A056023/b056023.txt">Table of n, a(n) for n = 1..5050</a>
%H Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations Integer Sequences And Pairing Functions</a>, arXiv:1212.2732 [math.CO], 2012.
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/PairingFunction.html">Pairing functions</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F T(n, k) = (n^2 - (n - 2*k)*(-1)^(n mod 2))/2 + n mod 2. - _Reinhard Zumkeller_, Apr 25 2004
%F 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
%e From _Philippe Deléham_, Apr 04 2009 (Start)
%e Triangle begins:
%e 1;
%e 2, 3;
%e 6, 5, 4;
%e 7, 8, 9, 10;
%e 15, 14, 13, 12, 11;
%e ...
%e (End)
%e Enumeration by boustrophedonic ("ox-plowing") diagonal method. - _Boris Putievskiy_, Dec 24 2012
%t (* As a rectangle: *)
%t r[n_, k_] := n + (k + n - 2) (k + n - 1)/2/; OddQ[n + k];
%t r[n_, k_] := k + (n + k - 2) (n + k - 1)/2/; EvenQ[n+k];
%t TableForm[Table[f[n, k], {n, 1, 10}, {k, 1, 15}]]
%t Table[r[n - k + 1, k], {n, 14}, {k, n, 1, -1}]//Flatten
%t (* _Clark Kimberling_, Feb 03 2011 *)
%t Module[{nn=15},If[OddQ[Length[#]],Reverse[#],#]&/@TakeList[Range[ (nn(nn+1))/2],Range[nn]]]//Flatten (* _Harvey P. Dale_, Feb 08 2022 *)
%Y Cf. A056011, A218890, A213927.
%K nonn,tabl,easy
%O 1,2
%A _Clark Kimberling_, Aug 01 2000
%E Name edited by _Andrey Zabolotskiy_, Apr 16 2023
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