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A194061 Natural interspersion of A002620; a rectangular array, by antidiagonals.

%I A194061 #5 Mar 30 2012 18:57:39
%S A194061 1,2,3,4,5,8,6,7,11,15,9,10,14,19,24,12,13,18,23,29,35,16,17,22,28,34,
%T A194061 41,48,20,21,27,33,40,47,55,63,25,26,32,39,46,54,62,71,80,30,31,38,45,
%U A194061 53,61,70,79,89,99,36,37,44,52,60,69,78,88,98,109,120
%N A194061 Natural interspersion of A002620; a rectangular array, by antidiagonals.
%C A194061 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, A194061 is a permutation of the positive integers; its inverse is A194062.
%e A194061 Northwest corner:
%e A194061 1...2...4...6...9...12
%e A194061 3...5...7...10..13..17
%e A194061 8...11..14..18..22..27
%e A194061 15..19..23..28..33..39
%e A194061 24..29..34..40..46..53
%t A194061 z = 50;
%t A194061 c[k_] := Floor[((k + 1)^2)/4];
%t A194061 c = Table[c[k], {k, 1, z}]  (* A002620 *)
%t A194061 f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
%t A194061 f = Table[f[n], {n, 1, 400}]   (* [A122197] *)
%t A194061 r[n_] := Flatten[Position[f, n]]
%t A194061 t[n_, k_] := r[n][[k]]
%t A194061 TableForm[Table[t[n, k], {n, 1, 7}, {k, 1, 7}]]
%t A194061 p = Flatten[
%t A194061   Table[t[k, n - k + 1], {n, 1, 11}, {k, 1, n}]] (* A194061 *)
%t A194061 q[n_] := Position[p, n]; Flatten[
%t A194061  Table[q[n], {n, 1, 90}]]  (* A194062 *)
%Y A194061 Cf. A194029, A002620, A122197, A194062.
%K A194061 nonn,tabl
%O A194061 1,2
%A A194061 _Clark Kimberling_, Aug 14 2011