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%I #20 Nov 29 2023 08:15:00
%S 1,2,3,5,4,6,7,8,11,12,10,9,14,13,15,16,17,20,21,24,25,19,18,23,22,27,
%T 26,28,29,30,33,34,37,38,41,42,32,31,36,35,40,39,44,43,45,46,47,50,51,
%U 54,55,58,59,62,63,49,48,53,52,57,56,61,60,65,64,66,67
%N T(n,k) = ((n+k)^2-4*k+3-2*(-1)^n-(-1)^k-(n+k-4)*(-1)^(n+k))/2-3, if k=1 and (n mod 2)=1; T(n,k) = ((n+k)^2-4*k+3-2*(-1)^n-(-1)^k-(n+k-4)*(-1)^(n+k))/2, else. Table T(n,k) read by antidiagonals; n, k > 0.
%C Permutation of the natural numbers.
%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.
%C Enumeration table T(n,k). Let m be natural number. The order of the list:
%C T(1,1)=1;
%C T(1,2), T(2,1), T(2,2), T(1,3), T(3,1);
%C . . .
%C T(1,2*m), T(2,2*m-1), T(2,2*m), T(1,2*m+1), T(3,2*m-2), ... T(2*m-1,2), T(2*m,1), T(2*m,2), T(2*m-1,3), T(2*m+1,1);
%C . . .
%C Movement along two adjacent antidiagonals - step to the southwest, step east, step to the northeast, 3 steps to the west, 2 steps to the south and so on. The length of each step is 1.
%H Boris Putievskiy, <a href="/A216250/b216250.txt">Rows n = 1..140 of triangle, flattened</a>
%H 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 Eric Weisstein's World of Mathematics, <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 As table
%F T(n,k) = ((n+k)^2-4*k+3-2*(-1)^n-(-1)^k-(n+k-4)*(-1)^(n+k))/2-3, if k=1 and (n mod 2)=1;
%F T(n,k) = ((n+k)^2-4*k+3-2*(-1)^n-(-1)^k-(n+k-4)*(-1)^(n+k))/2, else.
%F As linear sequence
%F a(n) = ((t+2)^2-4*j+3-2*(-1)^i-(-1)^j-(t-2)*(-1)^t)/2-3, if j=1 and (i mod 2)=1;
%F a(n) = ((t+2)^2-4*j+3-2*(-1)^i-(-1)^j-(t-2)*(-1)^t)/2, else; where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).
%e The start of the sequence as table:
%e 1....2...5...7..10..16..19...
%e 3....4...8...9..17..18..30...
%e 6...11..14..20..23..33..36...
%e 12..13..21..22..34..35..51...
%e 15..24..27..37..40..54..57...
%e 25..26..38..39..55..56..76...
%e 28..41..44..58..61..79..82...
%e . . .
%e The start of the sequence as triangular array read by rows:
%e 1;
%e 2,3;
%e 5,4,6;
%e 7,8,11,12;
%e 10,9,14,13,15;
%e 16,17,20,21,24,25;
%e 19,18,23,22,27,26,28;
%e . . .
%e The start of the sequence as array read by rows, with length of row r: 4*r-3:
%e First 2*r-2 numbers are from the row number 2*r-2 of above triangle array.
%e Last 2*r-1 numbers are from the row number 2*r-1 of above triangle array.
%e 1;
%e 2,3,5,4,6;
%e 7,8,11,12,10,9,14,13,15;
%e 16,17,20,21,24,25,19,18,23,22,27,26,28;
%e . . .
%e Row number r contains permutation of the 4*r-3 numbers from 2*r*r-5*r+4 to 2*r*r-r: 2*r*r-5*r+4, 2*r*r-5*r+5, ...2*r*r-r-2, 2*r*r-r.
%o (Python)
%o t=int((math.sqrt(8*n-7) - 1)/ 2)
%o i=n-t*(t+1)/2
%o j=(t*t+3*t+4)/2-n
%o result=((t+2)**2-4*j+3-(-1)**j-2*(-1)**i-(t-2)*(-1)**t)/2
%o if j==1 and (i%2)==1:
%o result=result-3
%Y Cf. A213205, A213171, A213197, A210521; table T(n,k) contains: in rows A130883, A033816, A100037, A100038, A100039; in columns A000384, A014106, A071355, A091823, A130861.
%K nonn,tabl
%O 1,2
%A _Boris Putievskiy_, Mar 14 2013
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