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A194100 Natural interspersion of A194126; a rectangular array, by antidiagonals.

%I A194100 #6 Mar 30 2012 18:57:39
%S A194100 1,6,2,13,7,3,23,14,8,4,36,24,15,9,5,51,37,25,16,10,11,69,52,38,26,17,
%T A194100 18,12,89,70,53,39,27,28,19,20,112,90,71,54,40,41,29,30,21,138,113,91,
%U A194100 72,55,56,42,43,31,22,166,139,114,92,73,74,57,58,44,32,33,197
%N A194100 Natural interspersion of A194126; a rectangular array, by antidiagonals.
%C A194100 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, A194100 is a permutation of the positive integers; its inverse is A194101.
%e A194100 Northwest corner:
%e A194100 1...6...13...23...36
%e A194100 2...7...14...24...37
%e A194100 3...8...15...25...38
%e A194100 4...9...16...26...39
%e A194100 5...10..17...27...40
%e A194100 11..18..28...41...56
%t A194100 z = 40; g = GoldenRatio;
%t A194100 c[k_] := -1 + Sum[Floor[j + j*g], {j, 1, k}];
%t A194100 c = Table[c[k], {k, 1, z}]  (* 194126 *)
%t A194100 f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
%t A194100 f = Table[f[n], {n, 1, 800}]  (* A193042 *)
%t A194100 r[n_] := Flatten[Position[f, n]]
%t A194100 t[n_, k_] := r[n][[k]]
%t A194100 TableForm[Table[t[n, k], {n, 1, 8}, {k, 1, 7}]]
%t A194100 p = Flatten[Table[t[k, n - k + 1], {n, 1, 16}, {k, 1, n}]]  (* A194100 *)
%t A194100 q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 80}]]  (* A194101 *)
%Y A194100 Cf. A194029, A194126, A193042, A194101.
%K A194100 nonn,tabl
%O A194100 1,2
%A A194100 _Clark Kimberling_, Aug 15 2011