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A194048 Natural interspersion of A000330, a rectangular array, by antidiagonals.

%I A194048 #5 Mar 30 2012 18:57:39
%S A194048 1,5,2,14,6,3,30,15,7,4,55,31,16,8,9,91,56,32,17,18,10,140,92,57,33,
%T A194048 34,19,11,204,141,93,58,59,35,20,12,285,205,142,94,95,60,36,21,13,385,
%U A194048 286,206,143,144,96,61,37,22,23
%N A194048 Natural interspersion of A000330, a rectangular array, by antidiagonals.
%C A194048 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, A194048 is a permutation of the positive integers; its inverse is A194049.
%e A194048 Northwest corner:
%e A194048 1...5...14...30...55
%e A194048 2...6...15...31...56
%e A194048 3...7...16...32...57
%e A194048 4...8...17...33...58
%e A194048 9...18..34...59...95
%t A194048 Remove["Global`*"];
%t A194048 z = 30;
%t A194048 c[k_] := k (k + 1) (2 k + 1)/6;
%t A194048 c = Table[c[k], {k, 1, z}]  (* A000330 *)
%t A194048 f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
%t A194048 f = Table[f[n], {n, 1, 500}]  (* fractal sequence [A064866] *)
%t A194048 r[n_] := Flatten[Position[f, n]]
%t A194048 t[n_, k_] := r[n][[k]]
%t A194048 TableForm[Table[t[n, k], {n, 1, 7}, {k, 1, 7}]]
%t A194048 p = Flatten[Table[t[k, n - k + 1], {n, 1, 10}, {k, 1, n}]]  (* A194048 *)
%t A194048 q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 70}]] (* A194049 *)
%Y A194048 Cf. A194029, A194049.
%K A194048 nonn,tabl
%O A194048 1,2
%A A194048 _Clark Kimberling_, Aug 13 2011