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A194032 Natural interspersion of the squares (1,4,9,16,25,...), a rectangular array, by antidiagonals.

%I A194032 #21 Jul 08 2025 10:48:56
%S A194032 1,4,2,9,5,3,16,10,6,7,25,17,11,12,8,36,26,18,19,13,14,49,37,27,28,20,
%T A194032 21,15,64,50,38,39,29,30,22,23,81,65,51,52,40,41,31,32,24,100,82,66,
%U A194032 67,53,54,42,43,33,34,121,101,83,84,68,69,55,56,44,45
%N A194032 Natural interspersion of the squares (1,4,9,16,25,...), a rectangular array, by antidiagonals.
%C A194032 See A194029 for definitions of natural fractal sequence and natural interspersion.  Every positive integer occurs exactly once (and every pair of rows intersperse), so that as a sequence, A194032 is a permutation of the positive integers; its inverse is A194033.
%H A194032 Zhuorui He, <a href="/A194032/b194032.txt">Table of n, a(n) for n = 1..11325</a>
%F A194032 T(n, k) = (k + max(floor(n/2)-1,0))^2 + n - 1. - _Zhuorui He_, Jul 08 2025
%e A194032 Northwest corner:
%e A194032   1...4...9...16...25
%e A194032   2...5...10..17...26
%e A194032   3...6...11..18...27
%e A194032   7...12..19..28...39
%e A194032   8...13..20..29...40
%t A194032 z = 30;
%t A194032 c[k_] := k^2;
%t A194032 c = Table[c[k], {k, 1, z}]  (* A000290 *)
%t A194032 f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]] (* A071797 *)
%t A194032 f = Table[f[n], {n, 1, 255}]
%t A194032 r[n_] := Flatten[Position[f, n]]
%t A194032 t[n_, k_] := r[n][[k]]
%t A194032 TableForm[Table[t[n, k], {n, 1, 7}, {k, 1, 7}]]
%t A194032 p = Flatten[Table[t[k, n - k + 1], {n, 1, 14}, {k, 1, n}]] (* A194032 *)
%t A194032 q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 70}]] (* A194033 *)
%Y A194032 Cf. A000290, A071797, A194033, A192872.
%K A194032 nonn,easy,tabl
%O A194032 1,2
%A A194032 _Clark Kimberling_, Aug 12 2011