cp's OEIS Frontend

A213928 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.

%I A213928 #20 Mar 23 2025 18:43:20
%S A213928 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 A213928 35,37,64,48,28,22,20,34,38,50,65,63,47,29,21,33,39,51,81,100,66,62,
%U A213928 46,30,32,40,52,80,82,121,99,67,61,45,31,41,53,79,83,101
%N A213928 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 A213928 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 A213928 Boris Putievskiy, <a href="/A213928/b213928.txt">Rows n = 1..140 of triangle, flattened</a>
%H A213928 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 A213928 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PairingFunction.html">Pairing functions</a>
%H A213928 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A213928 For general case.
%F A213928 As table
%F A213928 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 A213928 T(n,k) = ((1+(-1)^b(n))*(n^2-k+1)-(-1+(-1)^b(n))*((n-1)^2 +k))/2, if  n >k.
%F A213928 As linear sequence
%F A213928 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 A213928 a(n) = ((1+(-1)^b(i))*(i^2-j+1)-(-1+(-1)^b(i))*((i-1)^2 +j))/2, if  i >j;
%F A213928 where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).
%F A213928 For this sequence b(z)=z^2 mod 3.
%F A213928 As table
%F A213928 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 A213928 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 A213928 As linear sequence
%F A213928 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 A213928 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 A213928 where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).
%e A213928 The start of the sequence as table.
%e A213928 The direction of the placement denotes by ">" and  "v".
%e A213928   ..........v...........v...........v
%e A213928   >1....4...5..16..25..26..49..64..65...
%e A213928   >2....3...6..15..24..27..48..63..66...
%e A213928   .9....8...7..14..23..28..47..62..67...
%e A213928   >10..11..12..13..22..29..46..61..68...
%e A213928   >17..18..19..20..21..30..45..60..69...
%e A213928   .36..35..34..33..32..31..44..59..70...
%e A213928   >37..38..39..40..41..42..43..58..71...
%e A213928   >50..51..52..53..54..55..56..57..72...
%e A213928   .81..80..79..78..77..76..75..74..73...
%e A213928   . . .
%e A213928 The start of the sequence as triangle array read by rows:
%e A213928   1;
%e A213928   4,2;
%e A213928   5,3,9;
%e A213928   16,6,8,10;
%e A213928   25,15,7,11,17;
%e A213928   26,24,14,12,18,36;
%e A213928   49,27,23,13,19,35,37;
%e A213928   64,48,28,22,20,34,38,50;
%e A213928   65,63,47,29,21,33,39,51,81;
%e A213928   . . .
%o A213928 (Python)
%o A213928 t=int((math.sqrt(8*n-7) - 1)/ 2)
%o A213928 i=n-t*(t+1)/2
%o A213928 j=(t*t+3*t+4)/2-n
%o A213928 if j>=i:
%o A213928    result=((1+(-1)**(j**2%3-1))*(j**2-i+1)-(-1+(-1)**(j**2%3-1))*((j-1)**2 +i))/2
%o A213928 else:
%o A213928    result=((1+(-1)**(i**2%3))*(i**2-j+1)-(-1+(-1)**(i**2%3))*((i-1)**2 +j))/2
%Y A213928 Cf. A219159, A081344, A194280, A042964, A130196, A011655, A220516.
%K A213928 nonn,tabl
%O A213928 1,2
%A A213928 _Boris Putievskiy_, Mar 06 2013