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A191729 Dispersion of A047221, (numbers >1 and congruent to 2 or 3 mod 5), by antidiagonals.

%I A191729 #7 Oct 11 2017 16:48:05
%S A191729 1,2,4,3,8,5,7,18,12,6,17,43,28,13,9,42,107,68,32,22,10,103,267,168,
%T A191729 78,53,23,11,257,667,418,193,132,57,27,14,642,1667,1043,482,328,142,
%U A191729 67,33,15,1603,4167,2607,1203,818,353,167,82,37,16,4007,10417,6517
%N A191729 Dispersion of A047221, (numbers >1 and congruent to 2 or 3 mod 5), by antidiagonals.
%C A191729 For a background discussion of dispersions and their fractal sequences, see A191426.  For dispersions of congruence sequences mod 3, mod 4, or mod 5, see A191655, A191663, A191667, A191702.
%C A191729 ...
%C A191729 Suppose that {2,3,4,5,6} is partitioned as {x1, x2} and {x3,x4,x5}.  Let S be the increasing sequence of numbers >1 and congruent to x1 or x2 mod 5, and let T be the increasing sequence of numbers >1 and congruent to x3 or x4 or x5 mod 5.  There are 10 sequences in S, each matched by a (nearly) complementary sequence in T.  Each of the 20 sequences generates a dispersion, as listed here:
%C A191729 ...
%C A191729 A191722=dispersion of A008851 (0, 1 mod 5 and >1)
%C A191729 A191723=dispersion of A047215 (0, 2 mod 5 and >1)
%C A191729 A191724=dispersion of A047218 (0, 3 mod 5 and >1)
%C A191729 A191725=dispersion of A047208 (0, 4 mod 5 and >1)
%C A191729 A191726=dispersion of A047216 (1, 2 mod 5 and >1)
%C A191729 A191727=dispersion of A047219 (1, 3 mod 5 and >1)
%C A191729 A191728=dispersion of A047209 (1, 4 mod 5 and >1)
%C A191729 A191729=dispersion of A047221 (2, 3 mod 5 and >1)
%C A191729 A191730=dispersion of A047211 (2, 4 mod 5 and >1)
%C A191729 A191731=dispersion of A047204 (3, 4 mod 5 and >1)
%C A191729 ...
%C A191729 A191732=dispersion of A047202 (2,3,4 mod 5 and >1)
%C A191729 A191733=dispersion of A047206 (1,3,4 mod 5 and >1)
%C A191729 A191734=dispersion of A032793 (1,2,4 mod 5 and >1)
%C A191729 A191735=dispersion of A047223 (1,2,3 mod 5 and >1)
%C A191729 A191736=dispersion of A047205 (0,3,4 mod 5 and >1)
%C A191729 A191737=dispersion of A047212 (0,2,4 mod 5 and >1)
%C A191729 A191738=dispersion of A047222 (0,2,3 mod 5 and >1)
%C A191729 A191739=dispersion of A008854 (0,1,4 mod 5 and >1)
%C A191729 A191740=dispersion of A047220 (0,1,3 mod 5 and >1)
%C A191729 A191741=dispersion of A047217 (0,1,2 mod 5 and >1)
%C A191729 ...
%C A191729 For further information about these 20 dispersions, see A191722.
%C A191729 ...
%C A191729 Regarding the dispersions A191722-A191741, there are general formulas for sequences of the type "(a or b mod m)" and "(a or b or c mod m)" used in the relevant Mathematica programs.
%H A191729 Ivan Neretin, <a href="/A191729/b191729.txt">Table of n, a(n) for n = 1..5050</a>
%e A191729 Northwest corner:
%e A191729 1....2....3....7.....17
%e A191729 4....8....18...43....107
%e A191729 5....12...28...68....168
%e A191729 6....13...32...78....193
%e A191729 9....22...53...132...328
%e A191729 10...23...57...142...353
%t A191729 (* Program generates the dispersion array t of the increasing sequence f[n] *)
%t A191729 r = 40; r1 = 12;  c = 40; c1 = 12;
%t A191729 a=2; b=3; m[n_]:=If[Mod[n,2]==0,1,0];
%t A191729 f[n_]:=a*m[n+1]+b*m[n]+5*Floor[(n-1)/2]
%t A191729 Table[f[n], {n, 1, 30}]  (* A047221 *)
%t A191729 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191729 rows = {NestList[f, 1, c]};
%t A191729 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191729 t[i_, j_] := rows[[i, j]];
%t A191729 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191729 *)
%t A191729 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191729  *)
%Y A191729 Cf. A008854, A047221, A191739, A191722, A191426.
%K A191729 nonn,tabl
%O A191729 1,2
%A A191729 _Clark Kimberling_, Jun 13 2011