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 A213922 #21 Feb 16 2025 08:33:17
%S A213922 1,3,4,8,2,9,15,6,7,16,24,13,5,14,25,35,22,11,12,23,36,48,33,20,10,21,
%T A213922 34,49,63,46,31,18,19,32,47,64,80,61,44,29,17,30,45,62,81,99,78,59,42,
%U A213922 27,28,43,60,79,100,120,97,76,57,40,26,41,58,77,98,121
%N A213922 Natural numbers placed in table T(n,k) layer by layer. The order of placement: T(n,n), T(n-1,n), T(n,n-1), ... T(1,n), T(n,1). Table T(n,k) read by antidiagonals.
%C A213922 Permutation of the natural numbers.
%C A213922 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 A213922 Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1). Enumeration table T(n,k) is layer by layer. The order of the list:
%C A213922 T(1,1)=1;
%C A213922 T(2,2), T(1,2), T(2,1);
%C A213922 ...
%C A213922 T(n,n), T(n-1,n), T(n,n-1), ... T(1,n), T(n,1);
%C A213922 ...
%H A213922 Boris Putievskiy, <a href="/A213922/b213922.txt">Rows n = 1..140 of triangle, flattened</a>
%H A213922 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 A213922 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/PairingFunction.html">MathWorld: Pairing functions</a>
%H A213922 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A213922 As a table,
%F A213922 T(n,k) = n*n - 2*k + 2, if n >= k;
%F A213922 T(n,k) = k*k - 2*n + 1, if n < k.
%F A213922 As a linear sequence,
%F A213922 a(n) = i*i - 2*j + 2, if i >= j;
%F A213922 a(n) = j*j - 2*i + 1, if i < j
%F A213922 where
%F A213922 i = n - t*(t+1)/2,
%F A213922 j = (t*t + 3*t + 4)/2 - n,
%F A213922 t = floor((-1 + sqrt(8*n-7))/2).
%e A213922 The start of the sequence as a table:
%e A213922 1, 3, 8, 15, 24, 35, ...
%e A213922 4, 2, 6, 13, 22, 33, ...
%e A213922 9, 7, 5, 11, 20, 31, ...
%e A213922 16, 14, 12, 10, 18, 29, ...
%e A213922 25, 23, 21, 19, 17, 27, ...
%e A213922 36, 34, 32, 30, 28, 26, ...
%e A213922 ...
%e A213922 The start of the sequence as triangular array read by rows:
%e A213922 1;
%e A213922 3, 4;
%e A213922 8, 2, 9;
%e A213922 15, 6, 7, 16;
%e A213922 24, 13, 5, 14, 25;
%e A213922 35, 22, 11, 12, 23, 36;
%e A213922 ...
%t A213922 f[n_, k_] := n^2 - 2*k + 2 /; n >= k; f[n_, k_] := k^2 - 2*n + 1 /; n < k; TableForm[Table[f[n, k], {n, 1, 5}, {k, 1, 10}]]; Table[f[n - k + 1, k], {n, 5}, {k, n, 1, -1}] // Flatten (* _G. C. Greubel_, Aug 19 2017 *)
%o A213922 (Python)
%o A213922 t=int((math.sqrt(8*n-7) - 1)/ 2)
%o A213922 i=n-t*(t+1)/2
%o A213922 j=(t*t+3*t+4)/2-n
%o A213922 if i >= j:
%o A213922 result=i*i-2*j+2
%o A213922 else:
%o A213922 result=j*j-2*i+1
%Y A213922 Cf. A060734, A060736; table T(n,k) contains: in rows A005563, A028872, A028875, A028881, A028560, A116711; in columns A000290, A008865, A028347, A028878, A028884.
%K A213922 nonn,tabl
%O A213922 1,2
%A A213922 _Boris Putievskiy_, Mar 05 2013