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A194108 Natural interspersion of A194106; a rectangular array, by antidiagonals.

%I A194108 #5 Mar 30 2012 18:57:40
%S A194108 1,4,2,9,5,3,15,10,6,7,23,16,11,12,8,33,24,17,18,13,14,45,34,25,26,19,
%T A194108 20,21,58,46,35,36,27,28,29,22,73,59,47,48,37,38,39,30,31,90,74,60,61,
%U A194108 49,50,51,40,41,32,109,91,75,76,62,63,64,52,53,42,43,129,110
%N A194108 Natural interspersion of A194106; a rectangular array, by antidiagonals.
%C A194108 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, A194108 is a permutation of the positive integers; its inverse is A194109.
%e A194108 Northwest corner:
%e A194108 1...4...9...15...23
%e A194108 2...5...10..16...24
%e A194108 3...6...11..17...25
%e A194108 7...12..18..26...36
%e A194108 8...13..19..27...37
%t A194108 z = 40; g = Sqrt[3];
%t A194108 c[k_] := Sum[Floor[j*g], {j, 1, k}];
%t A194108 c = Table[c[k], {k, 1, z}]  (* A194106 *)
%t A194108 f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
%t A194108 f = Table[f[n], {n, 1, 800}]  (* A194107 *)
%t A194108 r[n_] := Flatten[Position[f, n]]
%t A194108 t[n_, k_] := r[n][[k]]
%t A194108 TableForm[Table[t[n, k], {n, 1, 8}, {k, 1, 7}]]
%t A194108 p = Flatten[Table[t[k, n - k + 1], {n, 1, 16}, {k, 1, n}]]  (* A194108 *)
%t A194108 q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 80}]]  (* A194109 *)
%Y A194108 Cf. A194029, A194106, A194107, A194109.
%K A194108 nonn,tabl
%O A194108 1,2
%A A194108 _Clark Kimberling_, Aug 15 2011