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%I #18 Dec 06 2018 10:12:15
%S 1,5,4,2,3,6,10,9,14,13,7,8,11,12,15,19,18,23,22,27,26,16,17,20,21,24,
%T 25,28,32,31,36,35,40,39,44,43,29,30,33,34,37,38,41,42,45,49,48,53,52,
%U 57,56,61,60,65,64,46,47,50,51,54,55,58,59,62,63,66,70
%N T(n,k) = ((k+n)^2-4*k+3+(-1)^k-2*(-1)^n-(k+n)*(-1)^(k+n))/2; n , k > 0, read by antidiagonals.
%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). The order of the list:
%C T(1,1)=1;
%C T(1,3), T(2,2), T(2,1), T(1,2), T(3,1);
%C . . .
%C T(1,2*n+1), T(2,2*n), T(2,2*n-1), T(1,2*n), ...T(2*n-1,3), T(2*n,2), T(2*n,1), T(2*n-1,2), T(2*n+1,1);
%C . . .
%C Movement along two adjacent antidiagonals - step to the southwest, step to the west, step to the northeast, 2 steps to the south, step to the west and so on. The length of each step is 1.
%C Table contains:
%C row 1 accommodates elements A130883 in odd places,
%C row 2 is alternation of elements A100037 and A033816;
%C column 1 is alternation of elements A000384 and A091823,
%C column 2 is alternation of elements A014106 and A071355,
%C column 3 accommodates elements A130861 in even places;
%C main diagonal is alternation of elements A188135 and A033567,
%C diagonal 1, located above the main diagonal accommodates elements A033566 in even places,
%C diagonal 2, located above the main diagonal is alternation of elements A139271 and A024847,
%C diagonal 3, located above the main diagonal accommodates of elements A033585.
%H Boris Putievskiy, <a href="/A213205/b213205.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) = ((k+n)^2-4*k+3+(-1)^k-2*(-1)^n-(k+n)*(-1)^(k+n))/2.
%F As linear sequence
%F a(n) = (A003057(n)^2-4*A004736(n)+3+(-1)^A004736(n)-2*(-1)^A002260(n)-A003057(n)*(-1)^A003056(n))/2;
%F a(n) = ((t+2)^2-4*j+3+(-1)^j-2*(-1)^i-(t+2)*(-1)^t)/2, 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....5...2..10...7..19..16...
%e 4....3...9...8..18..17..31...
%e 6...14..11..23..20..36..33...
%e 13..12..22..21..35..34..52...
%e 15..27..24..40..37..57..54...
%e 26..25..39..38..56..55..77...
%e 28..44..41..61..58..82..79...
%e . . .
%e The start of the sequence as triangle array read by rows:
%e 1;
%e 5,4;
%e 2,3,6;
%e 10,9,14,13;
%e 7,8,11,12,15;
%e 19,18,23,22,27,26;
%e 16,17,20,21,24,25,28;
%e . . .
%e The start of the sequence as array read by rows, the length of row r is 4*r-3.
%e First 2*r-2 numbers are from the row number 2*r-2 of triangle array, located above.
%e Last 2*r-1 numbers are from the row number 2*r-1 of triangle array, located above.
%e 1;
%e 5,4,2,3,6;
%e 10,9,14,13,7,8,11,12,15;
%e 19,18,23,22,27,26,16,17,20,21,24,25,28;
%e . . .
%e Row number r contains permutation 4*r-3 numbers from 2*r*r-5*r+4 to 2*r*r-r:
%e 2*r*r-5*r+7, 2*r*r-5*r+6,...2*r*r-r-4, 2*r*r-r-3, 2*r*r-r.
%p T:=(n,k)->((k+n)^2-4*k+3+(-1)^k-2*(-1)^n-(k+n)*(-1)^(k+n))/2: seq(seq(T(k,n-k),k=1..n-1),n=1..13); # _Muniru A Asiru_, Dec 06 2018
%t T[n_, k_] := ((n+k)^2 - 4k + 3 + (-1)^k - 2(-1)^n - (n+k)(-1)^(n+k))/2;
%t Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Dec 06 2018 *)
%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
%Y Cf. A211377, A130883, A100037, A033816, A000384, A091823, A014106, A071355, A130861, A188135, A033567, A033566, A139271, A024847, A033585, A002260, A004736, A003056, A003057.
%K nonn,tabl
%O 1,2
%A _Boris Putievskiy_, Feb 15 2013
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