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A213927 T(n,k) = (z*(z-1)-(-1+(-1)^(z^2 mod 3))*n+(1+(-1)^(z^2 mod 3))*k)/2, where z=n+k-1; n, k > 0, read by antidiagonals.

%I A213927 #35 Feb 16 2025 08:33:17
%S A213927 1,2,3,6,5,4,7,8,9,10,11,12,13,14,15,21,20,19,18,17,16,22,23,24,25,26,
%T A213927 27,28,29,30,31,32,33,34,35,36,45,44,43,42,41,40,39,38,37,46,47,48,49,
%U A213927 50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,78
%N A213927 T(n,k) = (z*(z-1)-(-1+(-1)^(z^2 mod 3))*n+(1+(-1)^(z^2 mod 3))*k)/2, where z=n+k-1; n, k > 0, read by antidiagonals.
%C A213927 Self-inverse permutation of the natural numbers.
%C A213927 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.
%C A213927 In general, let b(z) be a sequence of integers and denote number of antidiagonal table T(n,k) by z=n+k-1. Natural numbers placed in table T(n,k) by antidiagonals. The order of placement - by  antidiagonal downwards, if b(z) is odd; by  antidiagonal upwards, if b(z) is even. T(n,k) read by antidiagonals downwards. For A218890 -- the order of placement -- at the beginning m antidiagonals downwards, next m antidiagonals upwards and so on - b(z)=floor((z+m-1)/m). For this sequence b(z)=z^2 mod 3. (This comment should be edited for clarity, _Joerg Arndt_, Dec 11 2014)
%H A213927 Boris Putievskiy, <a href="/A213927/b213927.txt">Rows n = 1..140 of triangle, flattened</a>
%H A213927 Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations [of] Integer Sequences And Pairing Functions</a> arXiv:1212.2732 [math.CO], 2012.
%H A213927 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/PairingFunction.html">MathWorld: Pairing functions</a>
%H A213927 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A213927 For the general case.
%F A213927 T(n,k) = (z*(z-1)-(-1+(-1)^b(z))*n+(1+(-1)^b(z))*k)/2, where z=n+k-1 (as a table).
%F A213927 a(n) = (z*(z-1)-(-1+(-1)^b(z))*i+(1+(-1)^b(z))*j)/2, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2), z=i+j-1 (as a linear sequence).
%F A213927 For this sequence b(z)=z^2 mod 3.
%F A213927 T(n,k) = (z*(z-1)-(-1+(-1)^(z^2 mod 3))*n+(1+(-1)^(z^2 mod 3))*k)/2, where z=n+k-1 (as a table).
%F A213927 a(n) = (z*(z-1)-(-1+(-1)^(z^2 mod 3))*i+(1+(-1)^(z^2 mod 3))*j)/2, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2), z=i+j-1 (as linear sequence).
%e A213927 The start of the sequence as table.
%e A213927 The direction of the placement denoted by ">" and  "v".
%e A213927 .v.....v       v...v        v....v
%e A213927 .1.....2...6...7..11...21...22...29...45...
%e A213927 .3.....5...8..12..20...23...30...44...47...
%e A213927 >4.....9..13..19..24...31...43...48...58...
%e A213927 .10...14..18..25..32...42...49...59...75...
%e A213927 .15...17..26..33..41...50...60...74...83...
%e A213927 >16...27..34..40..51...61...73...84...97...
%e A213927 .28...35..39..52..62...72...85...98..114...
%e A213927 .36...38..53..63..71...86...99..113..128...
%e A213927 >37...54..64..70..87..100..112..129..145...
%e A213927 ...
%e A213927 The start of the sequence as triangle array read by rows:
%e A213927    1;
%e A213927    2,  3;
%e A213927    6,  5,  4;
%e A213927    7,  8,  9, 10;
%e A213927   11, 12, 13, 14, 15;
%e A213927   21, 20, 19, 18, 17, 16;
%e A213927   22, 23, 24, 25, 26, 27, 28;
%e A213927   29, 30, 31, 32, 33, 34, 35, 36;
%e A213927   45, 44, 43, 42, 41, 40, 39, 38, 37;
%e A213927   ...
%e A213927 Row r consists of r consecutive numbers from r*r/2-r/2+1 to r*r/2+r.
%e A213927 If r is not divisible by 3, rows are increasing.
%e A213927 If r is     divisible by 3, rows are decreasing.
%t A213927 T[n_, k_] := With[{z = n + k - 1}, (z*(z - 1) - (-1 + (-1)^Mod[z^2, 3])*n + (1 + (-1)^Mod[z^2, 3])*k)/2];
%t A213927 Table[T[n - k + 1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Jul 22 2018 *)
%o A213927 (Python)
%o A213927 t=int((math.sqrt(8*n-7) - 1)/ 2)
%o A213927 i=n-t*(t+1)/2
%o A213927 j=(t*t+3*t+4)/2-n
%o A213927 z=i+j-1
%o A213927 result=(z*(z-1)-(-1+(-1)**(z**2%3))*i+(1+(-1)**(z**2%3))*j)/2
%Y A213927 Cf. A218890, A056011, A056023, A130196, A011655, A001651, A008585.
%K A213927 nonn,tabl,uned
%O A213927 1,2
%A A213927 _Boris Putievskiy_, Mar 06 2013