This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A214929 #23 Feb 16 2025 08:33:18
%S A214929 1,3,4,2,5,9,14,7,6,10,11,20,23,17,12,8,13,19,26,34,43,30,27,16,15,21,
%T A214929 22,35,38,53,58,48,39,31,24,18,25,33,42,52,63,75,88,69,64,47,44,29,28,
%U A214929 36,37,54,57,76,81,102,109,95,82,70,59,49,40,32,41,51,62
%N A214929 A209293 as table read layer by layer - layer clockwise, layer counterclockwise and so on.
%C A214929 Permutation of the natural numbers.
%C A214929 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 A214929 Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1). Table read by boustrophedonic ("ox-plowing") method. Let m be natural number. The order of the list:
%C A214929 T(1,1)=1;
%C A214929 T(2,1), T(2,2), T(1,2);
%C A214929 . . .
%C A214929 T(1,2*m+1), T(2,2*m+1), ... T(2*m,2*m+1), T(2*m+1,2*m+1), T(2*m+1,2*m), ... T(2*m+1,1);
%C A214929 T(2*m,1), T(2*m,2), ... T(2*m,2*m-1), T(2*m,2*m), T(2*m-1,2*m), ... T(1,2*m);
%C A214929 . . .
%C A214929 The first row is layer read clockwise, the second row is layer counterclockwise.
%H A214929 Boris Putievskiy, <a href="/A214929/b214929.txt">Rows n = 1..140 of triangle, flattened</a>
%H A214929 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 A214929 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PairingFunction.html">Pairing functions</a>
%H A214929 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A214929 As table
%F A214929 T(n,k) = n*n/2+4*(floor((k-1)/2)+1)*n+ceiling((k-1)^2/2), n,k > 0.
%F A214929 As linear sequence
%F A214929 a(n)= (m1+m2-1)*(m1+m2-2)/2+m1, where
%F A214929 m1=floor((i+j)/2) + floor(i/2)*(-1)^(2*i+j-1), m2=int((i+j+1)/2)+int(i/2)*(-1)^(2*i+j-2),
%F A214929 where i=(t mod 2)*min(t; n-(t-1)^2) + (t+1 mod 2)*min(t; t^2-n+1), j=(t mod 2)*min(t; t^2-n+1) + (t+1 mod 2)*min(t; n-(t-1)^2), t=floor(sqrt(n-1))+1.
%e A214929 The start of the sequence as table:
%e A214929 1....2...5...8..13..18...
%e A214929 3....4...9..12..19..24...
%e A214929 6....7..14..17..26..31...
%e A214929 10..11..20..23..34..39...
%e A214929 15..16..27..30..43..48...
%e A214929 21..22..35..38..53..58...
%e A214929 . . .
%e A214929 The start of the sequence as triangle array read by rows:
%e A214929 1;
%e A214929 3,4,2;
%e A214929 5,9,14,7,6;
%e A214929 10,11,20,23,17,12,8;
%e A214929 13,19,26,34,43,30,27,16,15;
%e A214929 21,22,35,38,53,58,48,39,31,24,18;
%e A214929 . . .
%e A214929 Row number r contains 2*r-1 numbers.
%o A214929 (Python)
%o A214929 t=int((math.sqrt(n-1)))+1
%o A214929 i=(t % 2)*min(t,n-(t-1)**2) + ((t+1) % 2)*min(t,t**2-n+1)
%o A214929 j=(t % 2)*min(t,t**2-n+1) + ((t+1) % 2)*min(t,n-(t-1)**2)
%o A214929 m1=int((i+j)/2)+int(i/2)*(-1)**(2*i+j-1)
%o A214929 m2=int((i+j+1)/2)+int(i/2)*(-1)**(2*i+j-2)
%o A214929 result=(m1+m2-1)*(m1+m2-2)/2+m1
%Y A214929 Cf. A081344, A209293, A209279, A209278, A185180; table T(n,k) contains: in rows A000982, A097063; in columns A000217, A000124, A000096, A152948, A034856, A152950, A055998, A000982, A097063.
%K A214929 nonn
%O A214929 1,2
%A A214929 _Boris Putievskiy_, Mar 11 2013