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A191727 Dispersion of A047219, (numbers >1 and congruent to 1 or 3 mod 5), by antidiagonals.

%I A191727 #12 Oct 12 2017 13:06:41
%S A191727 1,3,2,8,6,4,21,16,11,5,53,41,28,13,7,133,103,71,33,18,9,333,258,178,
%T A191727 83,46,23,10,833,646,446,208,116,58,26,12,2083,1616,1116,521,291,146,
%U A191727 66,31,14,5208,4041,2791,1303,728,366,166,78,36,15,13021,10103
%N A191727 Dispersion of A047219, (numbers >1 and congruent to 1 or 3 mod 5), by antidiagonals.
%C A191727 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 A191727 ...
%C A191727 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 A191727 ...
%C A191727 A191722=dispersion of A008851 (0, 1 mod 5 and >1)
%C A191727 A191723=dispersion of A047215 (0, 2 mod 5 and >1)
%C A191727 A191724=dispersion of A047218 (0, 3 mod 5 and >1)
%C A191727 A191725=dispersion of A047208 (0, 4 mod 5 and >1)
%C A191727 A191726=dispersion of A047216 (1, 2 mod 5 and >1)
%C A191727 A191727=dispersion of A047219 (1, 3 mod 5 and >1)
%C A191727 A191728=dispersion of A047209 (1, 4 mod 5 and >1)
%C A191727 A191729=dispersion of A047221 (2, 3 mod 5 and >1)
%C A191727 A191730=dispersion of A047211 (2, 4 mod 5 and >1)
%C A191727 A191731=dispersion of A047204 (3, 4 mod 5 and >1)
%C A191727 ...
%C A191727 A191732=dispersion of A047202 (2,3,4 mod 5 and >1)
%C A191727 A191733=dispersion of A047206 (1,3,4 mod 5 and >1)
%C A191727 A191734=dispersion of A032793 (1,2,4 mod 5 and >1)
%C A191727 A191735=dispersion of A047223 (1,2,3 mod 5 and >1)
%C A191727 A191736=dispersion of A047205 (0,3,4 mod 5 and >1)
%C A191727 A191737=dispersion of A047212 (0,2,4 mod 5 and >1)
%C A191727 A191738=dispersion of A047222 (0,2,3 mod 5 and >1)
%C A191727 A191739=dispersion of A008854 (0,1,4 mod 5 and >1)
%C A191727 A191740=dispersion of A047220 (0,1,3 mod 5 and >1)
%C A191727 A191741=dispersion of A047217 (0,1,2 mod 5 and >1)
%C A191727 ...
%C A191727 For further information about these 20 dispersions, see A191722.
%C A191727 ...
%C A191727 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 A191727 Ivan Neretin, <a href="/A191727/b191727.txt">Table of n, a(n) for n = 1..5050</a> (first 100 antidiagonals, flattened)
%e A191727 Northwest corner:
%e A191727 1....3....8....21....53
%e A191727 2....6....16...41....103
%e A191727 4....11...28...71....178
%e A191727 5....13...33...83....208
%e A191727 7....18...46...116...291
%e A191727 9....23...58...146...366
%t A191727 (* Program generates the dispersion array t of the increasing sequence f[n] *)
%t A191727 r = 40; r1 = 12;  c = 40; c1 = 12;
%t A191727 a=3; b=6; m[n_]:=If[Mod[n,2]==0,1,0];
%t A191727 f[n_]:=a*m[n+1]+b*m[n]+5*Floor[(n-1)/2]
%t A191727 Table[f[n], {n, 1, 30}]  (* A047219 *)
%t A191727 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191727 rows = {NestList[f, 1, c]};
%t A191727 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191727 t[i_, j_] := rows[[i, j]];
%t A191727 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191727 *)
%t A191727 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191727  *)
%Y A191727 Cf. A047212, A047219, A191737, A191722, A191426.
%K A191727 nonn,tabl
%O A191727 1,2
%A A191727 _Clark Kimberling_, Jun 13 2011