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A191731 Dispersion of A047204, (numbers >1 and congruent to 3 or 4 mod 5), by antidiagonals.

%I A191731 #7 Oct 11 2017 16:43:25
%S A191731 1,3,2,8,4,5,19,9,13,6,48,23,33,14,7,119,58,83,34,18,10,298,144,208,
%T A191731 84,44,24,11,744,359,519,209,109,59,28,12,1859,898,1298,523,273,148,
%U A191731 69,29,15,4648,2244,3244,1308,683,369,173,73,38,16,11619,5609,8109
%N A191731 Dispersion of A047204, (numbers >1 and congruent to 3 or 4 mod 5), by antidiagonals.
%C A191731 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 A191731 ...
%C A191731 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 A191731 ...
%C A191731 A191722=dispersion of A008851 (0, 1 mod 5 and >1)
%C A191731 A191723=dispersion of A047215 (0, 2 mod 5 and >1)
%C A191731 A191724=dispersion of A047218 (0, 3 mod 5 and >1)
%C A191731 A191725=dispersion of A047208 (0, 4 mod 5 and >1)
%C A191731 A191726=dispersion of A047216 (1, 2 mod 5 and >1)
%C A191731 A191727=dispersion of A047219 (1, 3 mod 5 and >1)
%C A191731 A191728=dispersion of A047209 (1, 4 mod 5 and >1)
%C A191731 A191729=dispersion of A047221 (2, 3 mod 5 and >1)
%C A191731 A191730=dispersion of A047211 (2, 4 mod 5 and >1)
%C A191731 A191731=dispersion of A047204 (3, 4 mod 5 and >1)
%C A191731 ...
%C A191731 A191732=dispersion of A047202 (2,3,4 mod 5 and >1)
%C A191731 A191733=dispersion of A047206 (1,3,4 mod 5 and >1)
%C A191731 A191734=dispersion of A032793 (1,2,4 mod 5 and >1)
%C A191731 A191735=dispersion of A047223 (1,2,3 mod 5 and >1)
%C A191731 A191736=dispersion of A047205 (0,3,4 mod 5 and >1)
%C A191731 A191737=dispersion of A047212 (0,2,4 mod 5 and >1)
%C A191731 A191738=dispersion of A047222 (0,2,3 mod 5 and >1)
%C A191731 A191739=dispersion of A008854 (0,1,4 mod 5 and >1)
%C A191731 A191740=dispersion of A047220 (0,1,3 mod 5 and >1)
%C A191731 A191741=dispersion of A047217 (0,1,2 mod 5 and >1)
%C A191731 ...
%C A191731 For further information about these 20 dispersions, see A191722.
%C A191731 ...
%C A191731 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 A191731 Ivan Neretin, <a href="/A191731/b191731.txt">Table of n, a(n) for n = 1..5050</a>
%e A191731 Northwest corner:
%e A191731 1....3....8.....19.....48
%e A191731 2....4....9....23....58
%e A191731 5....13...33...83....208
%e A191731 6....14...34...84....209
%e A191731 7....18...44...109...273
%e A191731 10...24...59...148...369
%t A191731 (* Program generates the dispersion array t of the increasing sequence f[n] *)
%t A191731 r = 40; r1 = 12;  c = 40; c1 = 12;
%t A191731 a=3; b=4; m[n_]:=If[Mod[n,2]==0,1,0];
%t A191731 f[n_]:=a*m[n+1]+b*m[n]+5*Floor[(n-1)/2]
%t A191731 Table[f[n], {n, 1, 30}]  (* A047204 *)
%t A191731 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191731 rows = {NestList[f, 1, c]};
%t A191731 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191731 t[i_, j_] := rows[[i, j]];
%t A191731 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191731 *)
%t A191731 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191731  *)
%Y A191731 Cf. A047217, A047204, A191741, A191722, A191426.
%K A191731 nonn,tabl
%O A191731 1,2
%A A191731 _Clark Kimberling_, Jun 13 2011