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A191728 Dispersion of A047209, (numbers >1 and congruent to 1 or 4 mod 5), by antidiagonals.

%I A191728 #7 Oct 11 2017 16:33:44
%S A191728 1,4,2,11,6,3,29,16,9,5,74,41,24,14,7,186,104,61,36,19,8,466,261,154,
%T A191728 91,49,21,10,1166,654,386,229,124,54,26,12,2916,1636,966,574,311,136,
%U A191728 66,31,13,7291,4091,2416,1436,779,341,166,79,34,15,18229,10229
%N A191728 Dispersion of A047209, (numbers >1 and congruent to 1 or 4 mod 5), by antidiagonals.
%C A191728 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 A191728 ...
%C A191728 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 A191728 ...
%C A191728 A191722=dispersion of A008851 (0, 1 mod 5 and >1)
%C A191728 A191723=dispersion of A047215 (0, 2 mod 5 and >1)
%C A191728 A191724=dispersion of A047218 (0, 3 mod 5 and >1)
%C A191728 A191725=dispersion of A047208 (0, 4 mod 5 and >1)
%C A191728 A191726=dispersion of A047216 (1, 2 mod 5 and >1)
%C A191728 A191727=dispersion of A047219 (1, 3 mod 5 and >1)
%C A191728 A191728=dispersion of A047209 (1, 4 mod 5 and >1)
%C A191728 A191729=dispersion of A047221 (2, 3 mod 5 and >1)
%C A191728 A191730=dispersion of A047211 (2, 4 mod 5 and >1)
%C A191728 A191731=dispersion of A047204 (3, 4 mod 5 and >1)
%C A191728 ...
%C A191728 A191732=dispersion of A047202 (2,3,4 mod 5 and >1)
%C A191728 A191733=dispersion of A047206 (1,3,4 mod 5 and >1)
%C A191728 A191734=dispersion of A032793 (1,2,4 mod 5 and >1)
%C A191728 A191735=dispersion of A047223 (1,2,3 mod 5 and >1)
%C A191728 A191736=dispersion of A047205 (0,3,4 mod 5 and >1)
%C A191728 A191737=dispersion of A047212 (0,2,4 mod 5 and >1)
%C A191728 A191738=dispersion of A047222 (0,2,3 mod 5 and >1)
%C A191728 A191739=dispersion of A008854 (0,1,4 mod 5 and >1)
%C A191728 A191740=dispersion of A047220 (0,1,3 mod 5 and >1)
%C A191728 A191741=dispersion of A047217 (0,1,2 mod 5 and >1)
%C A191728 ...
%C A191728 For further information about these 20 dispersions, see A191722.
%C A191728 ...
%C A191728 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 A191728 Ivan Neretin, <a href="/A191728/b191728.txt">Table of n, a(n) for n = 1..5050</a>
%e A191728 Northwest corner:
%e A191728 1....4....11...29....74
%e A191728 2....6....16...41....104
%e A191728 3....9....24...61....154
%e A191728 5....14...36...91....229
%e A191728 7....19...49...124...311
%e A191728 8....21...54...136...341
%t A191728 (* Program generates the dispersion array t of the increasing sequence f[n] *)
%t A191728 r = 40; r1 = 12;  c = 40; c1 = 12;
%t A191728 a=4; b=6; m[n_]:=If[Mod[n,2]==0,1,0];
%t A191728 f[n_]:=a*m[n+1]+b*m[n]+5*Floor[(n-1)/2]
%t A191728 Table[f[n], {n, 1, 30}]  (* A047209 *)
%t A191728 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191728 rows = {NestList[f, 1, c]};
%t A191728 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191728 t[i_, j_] := rows[[i, j]];
%t A191728 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191728 *)
%t A191728 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191728  *)
%Y A191728 Cf. A047222, A047209, A191738, A191722, A191426.
%K A191728 nonn,tabl
%O A191728 1,2
%A A191728 _Clark Kimberling_, Jun 13 2011