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%I #30 Feb 15 2022 12:58:20
%S 1,2,5,3,6,9,13,8,4,7,12,18,25,19,14,10,15,20,26,33,41,32,24,17,11,16,
%T 23,31,40,50,61,51,42,34,27,21,28,35,43,52,62,73,85,72,60,49,39,30,22,
%U 29,38,48,59,71,84,98,113
%N Inverse permutation to A081344.
%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 Call a "layer" a pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1). This sequence is A188568 as table read layer by layer clockwise.
%C The same table A188568 read by boustrophedon ("ox-plowing") method - layer clockwise, layer counterclockwise and so on - is A064790. - _Boris Putievskiy_, Mar 14 2013
%H Boris Putievskiy, <a href="/A194280/b194280.txt">Rows n = 1..140 of triangle, flattened</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 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F a(n) = (i+j-1)*(i+j-2)/2+j, where
%F i = mod(t;2)*min{t; n - (t - 1)^2} + mod(t + 1; 2)*min{t; t^2 - n + 1}
%F j = mod(t;2)*min{t; t^2 - n + 1} + mod(t + 1; 2)*min{t; n - (t - 1)^2},
%F t = int(math.sqrt(n-1))+1.
%e From _Boris Putievskiy_, Mar 14 2013: (Start)
%e The start of the sequence as table:
%e 1....2...6...7..15..16..28...
%e 3....5...9..12..20..23..35...
%e 4....8..13..18..26..31..43...
%e 10..14..19..25..33..40..52...
%e 11..17..24..32..41..50..62...
%e 21..27..34..42..51..61..73...
%e 22..30..39..49..60..72..85...
%e ...
%e The start of the sequence as triangular array read by rows:
%e 1;
%e 2,5,3;
%e 6,9,13,8,4;
%e 7,12,18,25,19,14,10;
%e 15,20,26,33,41,32,24,17,11;
%e 16,23,31,40,50,61,51,42,34,27,21;
%e 28,35,43,52,62,73,85,72,60,49,39,30,22;
%e ...
%e Row number r contains 2*r-1 numbers. (End)
%o (Python)
%o t=int(math.sqrt(n-1))+1
%o i=(t % 2)*min(t,n-(t-1)**2) + ((t+1) % 2)*min(t,t**2-n+1)
%o j=(t % 2)*min(t,t**2-n+1) + ((t+1) % 2)*min(t,n-(t-1)**2)
%o m=(i+j-1)*(i+j-2)/2+j
%Y Cf. A081344, A064790, A188568.
%K nonn
%O 1,2
%A _Boris Putievskiy_, Dec 23 2012
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