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%I #14 Nov 29 2023 06:57:32
%S 1,4,2,5,3,9,16,6,8,10,25,15,7,11,17,26,24,14,12,18,36,49,27,23,13,19,
%T 35,37,64,48,28,22,20,34,38,50,65,63,47,29,21,33,39,51,81,100,66,62,
%U 46,30,32,40,52,80,82,121,99,67,61,45,31,41,53,79,83,101
%N Natural numbers placed in table T(n,k) layer by layer. The order of placement - at the beginning 2 layers counterclockwise, next 1 layer clockwise and so on. T(n,k) read by antidiagonals.
%C Permutation of the natural numbers. 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.In general, let b(z) be a sequence of integer numbers. Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1). Natural numbers placed in table T(n,k) layer by layer. The order of placement - layer is counterclockwise, if b(z) is odd; layer is clockwise if b(z) is even. T(n,k) read by antidiagonals.For A219159 - the order of the placement - at the beginning m layers counterclockwise, next m layers clockwise and so on - b(z)=floor((z-1)/m)+1. For this sequence b(z)=z^2 mod 3.
%H Boris Putievskiy, <a href="/A213928/b213928.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 For general case.
%F As table
%F T(n,k) = ((1+(-1)^(b(k)-1))*(k^2-n+1)-(-1+(-1)^(b(k)-1))*((k-1)^2 +n))/2, if k >= n;
%F T(n,k) = ((1+(-1)^b(n))*(n^2-k+1)-(-1+(-1)^b(n))*((n-1)^2 +k))/2, if n >k.
%F As linear sequence
%F a(n) = ((1+(-1)^(b(j)-1))*(j^2-i+1)-(-1+(-1)^(b(j)-1))*((j-1)^2 +i))/2, if j >= i;
%F a(n) = ((1+(-1)^b(i))*(i^2-j+1)-(-1+(-1)^b(i))*((i-1)^2 +j))/2, if i >j;
%F where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).
%F For this sequence b(z)=z^2 mod 3.
%F As table
%F T(n,k) = ((1+(-1)^(k^2 mod 3-1))*(k^2-n+1)-(-1+(-1)^(k^2 mod 3-1))*((k-1)^2 +n))/2, if k >= n;
%F T(n,k) = ((1+(-1)^(n^2 mod 3))*(n^2-k+1)-(-1+(-1)^(n^2 mod 3))*((n-1)^2 +k))/2, if n >k.
%F As linear sequence
%F a(n) = ((1+(-1)^(j^2 mod 3-1))*(j^2-i+1)-(-1+(-1)^(j^2 mod 3-1))*((j-1)^2 +i))/2, if j >= i;
%F a(n) = ((1+(-1)^(i^2 mod 3))*(i^2-j+1)-(-1+(-1)^(i^2 mod 3))*((i-1)^2 +j))/2, if i >j;
%F 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 The direction of the placement denotes by ">" and "v".
%e ..........v...........v...........v
%e >1....4...5..16..25..26..49..64..65...
%e >2....3...6..15..24..27..48..63..66...
%e .9....8...7..14..23..28..47..62..67...
%e >10..11..12..13..22..29..46..61..68...
%e >17..18..19..20..21..30..45..60..69...
%e .36..35..34..33..32..31..44..59..70...
%e >37..38..39..40..41..42..43..58..71...
%e >50..51..52..53..54..55..56..57..72...
%e .81..80..79..78..77..76..75..74..73...
%e . . .
%e The start of the sequence as triangle array read by rows:
%e 1;
%e 4,2;
%e 5,3,9;
%e 16,6,8,10;
%e 25,15,7,11,17;
%e 26,24,14,12,18,36;
%e 49,27,23,13,19,35,37;
%e 64,48,28,22,20,34,38,50;
%e 65,63,47,29,21,33,39,51,81;
%e . . .
%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 if j>=i:
%o result=((1+(-1)**(j**2%3-1))*(j**2-i+1)-(-1+(-1)**(j**2%3-1))*((j-1)**2 +i))/2
%o else:
%o result=((1+(-1)**(i**2%3))*(i**2-j+1)-(-1+(-1)**(i**2%3))*((i-1)**2 +j))/2
%Y Cf. A219159, A081344, A194280, A042964, A130196, A011655, A220516.
%K nonn,tabl
%O 1,2
%A _Boris Putievskiy_, Mar 06 2013
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