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A194064 Natural interspersion of A006578; a rectangular array, by antidiagonals.

%I A194064 #5 Mar 30 2012 18:57:39
%S A194064 1,4,2,8,5,3,14,9,6,7,21,15,10,11,12,30,22,16,17,18,13,40,31,23,24,25,
%T A194064 19,20,52,41,32,33,34,26,27,28,65,53,42,43,44,35,36,37,29,80,66,54,55,
%U A194064 56,45,46,47,38,39,96,81,67,68,69,57,58,59,48,49,50
%N A194064 Natural interspersion of A006578; a rectangular array, by antidiagonals.
%C A194064 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, A194064 is a permutation of the positive integers; its inverse is A194065.
%e A194064 Northwest corner:
%e A194064 1...4...8...14...21...30
%e A194064 2...5...9...15...22...31
%e A194064 3...6...10..16...23...32
%e A194064 7...11..17..24...33...43
%e A194064 12..18..25..34...44...56
%t A194064 z = 50;
%t A194064 c[k_] := k (k + 1)/2 + Floor[(k^2)/4];
%t A194064 c = Table[c[k], {k, 1, z}]  (* A006578 *)
%t A194064 f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
%t A194064 f = Table[f[n], {n, 1, 400}]   (* A194063 *)
%t A194064 r[n_] := Flatten[Position[f, n]]
%t A194064 t[n_, k_] := r[n][[k]]
%t A194064 TableForm[Table[t[n, k], {n, 1, 7}, {k, 1, 7}]]
%t A194064 p = Flatten[Table[t[k, n - k + 1], {n, 1, 11}, {k, 1, n}]] (* A194064 *)
%t A194064 q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 90}]]  (* A194065 *)
%Y A194064 Cf. A194029, A194063, A194065.
%K A194064 nonn,tabl
%O A194064 1,2
%A A194064 _Clark Kimberling_, Aug 14 2011