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A194054 Natural interspersion of A054347; a rectangular array, by antidiagonals.

%I A194054 #5 Mar 30 2012 18:57:39
%S A194054 1,4,2,8,5,3,14,9,6,7,22,15,10,11,12,31,23,16,17,18,13,42,32,24,25,26,
%T A194054 19,20,54,43,33,34,35,27,28,21,68,55,44,45,46,36,37,29,30,84,69,56,57,
%U A194054 58,47,48,38,39,40,101,85,70,71,72,59,60,49,50,51,41,120,102
%N A194054 Natural interspersion of A054347; a rectangular array, by antidiagonals.
%C A194054 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, A194054 is a permutation of the positive integers; its inverse is A194055.
%e A194054 Northwest corner:
%e A194054 1...4...8...14...22...31
%e A194054 2...5...9...15...23...32
%e A194054 3...6...10..16...24...33
%e A194054 7...11..17..25...34...45
%t A194054 z = 40; g = GoldenRatio
%t A194054 c[k_] := Sum[Floor[j*g], {j, 1, k}];
%t A194054 c = Table[c[k], {k, 1, z}]  (* A054347 *)
%t A194054 f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
%t A194054 f = Table[f[n], {n, 1, 800}]  (* A194053 *)
%t A194054 r[n_] := Flatten[Position[f, n]]
%t A194054 t[n_, k_] := r[n][[k]]
%t A194054 TableForm[Table[t[n, k], {n, 1, 8}, {k, 1, 7}]]
%t A194054 p = Flatten[Table[t[k, n - k + 1], {n, 1, 16}, {k, 1, n}]]  (* A194054 *)
%t A194054 q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 80}]]  (* A194058 *)
%Y A194054 Cf. A194029, A054347, A194053, A194058.
%K A194054 nonn,tabl
%O A194054 1,2
%A A194054 _Clark Kimberling_, Aug 15 2011