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A191740 Dispersion of A047220, (numbers >1 and congruent to 0 or 1 or 3 mod 5), by antidiagonals.

%I A191740 #7 Oct 12 2017 20:00:07
%S A191740 1,3,2,6,5,4,11,10,8,7,20,18,15,13,9,35,31,26,23,16,12,60,53,45,40,28,
%T A191740 21,14,101,90,76,68,48,36,25,17,170,151,128,115,81,61,43,30,19,285,
%U A191740 253,215,193,136,103,73,51,33,22,476,423,360,323,228,173,123
%N A191740 Dispersion of A047220, (numbers >1 and congruent to 0 or 1 or 3 mod 5), by antidiagonals.
%C A191740 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 A191740 ...
%C A191740 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 A191740 ...
%C A191740 A191722=dispersion of A008851 (0, 1 mod 5 and >1)
%C A191740 A191723=dispersion of A047215 (0, 2 mod 5 and >1)
%C A191740 A191724=dispersion of A047218 (0, 3 mod 5 and >1)
%C A191740 A191725=dispersion of A047208 (0, 4 mod 5 and >1)
%C A191740 A191726=dispersion of A047216 (1, 2 mod 5 and >1)
%C A191740 A191727=dispersion of A047219 (1, 3 mod 5 and >1)
%C A191740 A191728=dispersion of A047209 (1, 4 mod 5 and >1)
%C A191740 A191729=dispersion of A047221 (2, 3 mod 5 and >1)
%C A191740 A191730=dispersion of A047211 (2, 4 mod 5 and >1)
%C A191740 A191731=dispersion of A047204 (3, 4 mod 5 and >1)
%C A191740 ...
%C A191740 A191732=dispersion of A047202 (2,3,4 mod 5 and >1)
%C A191740 A191733=dispersion of A047206 (1,3,4 mod 5 and >1)
%C A191740 A191734=dispersion of A032793 (1,2,4 mod 5 and >1)
%C A191740 A191735=dispersion of A047223 (1,2,3 mod 5 and >1)
%C A191740 A191736=dispersion of A047205 (0,3,4 mod 5 and >1)
%C A191740 A191737=dispersion of A047212 (0,2,4 mod 5 and >1)
%C A191740 A191738=dispersion of A047222 (0,2,3 mod 5 and >1)
%C A191740 A191739=dispersion of A008854 (0,1,4 mod 5 and >1)
%C A191740 A191740=dispersion of A047220 (0,1,3 mod 5 and >1)
%C A191740 A191741=dispersion of A047217 (0,1,2 mod 5 and >1)
%C A191740 ...
%C A191740 For further information about these 20 dispersions, see A191722.
%C A191740 ...
%C A191740 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 A191740 Ivan Neretin, <a href="/A191740/b191740.txt">Table of n, a(n) for n = 1..5050</a> (first 100 antidiagonals, flattened)
%e A191740 Northwest corner:
%e A191740 1....3....6....11...20
%e A191740 2....5....10...18...31
%e A191740 4....8....15...26...45
%e A191740 7....13...23...40...68
%e A191740 9....16...28...48...81
%e A191740 12...21...36...61...103
%t A191740 (* Program generates the dispersion array t of the increasing sequence f[n] *)
%t A191740 r = 40; r1 = 12;  c = 40; c1 = 12;
%t A191740 a=3; b=5; c2=6; m[n_]:=If[Mod[n,3]==0,1,0];
%t A191740 f[n_]:=a*m[n+2]+b*m[n+1]+c2*m[n]+5*Floor[(n-1)/3]
%t A191740 Table[f[n], {n, 1, 30}]  (* A047220 *)
%t A191740 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191740 rows = {NestList[f, 1, c]};
%t A191740 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191740 t[i_, j_] := rows[[i, j]];
%t A191740 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
%t A191740   (* A191740 *)
%t A191740 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191740  *)
%Y A191740 Cf. A047211, A047220, A191722, A191730, A191426.
%K A191740 nonn,tabl
%O A191740 1,2
%A A191740 _Clark Kimberling_, Jun 14 2011