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A191741 Dispersion of A047217, (numbers >1 and congruent to 0 or 1 or 2 mod 5), by antidiagonals.

%I A191741 #7 Oct 12 2017 20:01:46
%S A191741 1,2,3,5,6,4,10,11,7,8,17,20,12,15,9,30,35,21,26,16,13,51,60,36,45,27,
%T A191741 22,14,86,101,61,76,46,37,25,18,145,170,102,127,77,62,42,31,19,242,
%U A191741 285,171,212,130,105,71,52,32,23,405,476,286,355,217,176,120
%N A191741 Dispersion of A047217, (numbers >1 and congruent to 0 or 1 or 2 mod 5), by antidiagonals.
%C A191741 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 A191741 ...
%C A191741 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 A191741 ...
%C A191741 A191722=dispersion of A008851 (0, 1 mod 5 and >1)
%C A191741 A191723=dispersion of A047215 (0, 2 mod 5 and >1)
%C A191741 A191724=dispersion of A047218 (0, 3 mod 5 and >1)
%C A191741 A191725=dispersion of A047208 (0, 4 mod 5 and >1)
%C A191741 A191726=dispersion of A047216 (1, 2 mod 5 and >1)
%C A191741 A191727=dispersion of A047219 (1, 3 mod 5 and >1)
%C A191741 A191728=dispersion of A047209 (1, 4 mod 5 and >1)
%C A191741 A191729=dispersion of A047221 (2, 3 mod 5 and >1)
%C A191741 A191730=dispersion of A047211 (2, 4 mod 5 and >1)
%C A191741 A191731=dispersion of A047204 (3, 4 mod 5 and >1)
%C A191741 ...
%C A191741 A191732=dispersion of A047202 (2,3,4 mod 5 and >1)
%C A191741 A191733=dispersion of A047206 (1,3,4 mod 5 and >1)
%C A191741 A191734=dispersion of A032793 (1,2,4 mod 5 and >1)
%C A191741 A191735=dispersion of A047223 (1,2,3 mod 5 and >1)
%C A191741 A191736=dispersion of A047205 (0,3,4 mod 5 and >1)
%C A191741 A191737=dispersion of A047212 (0,2,4 mod 5 and >1)
%C A191741 A191738=dispersion of A047222 (0,2,3 mod 5 and >1)
%C A191741 A191739=dispersion of A008854 (0,1,4 mod 5 and >1)
%C A191741 A191740=dispersion of A047220 (0,1,3 mod 5 and >1)
%C A191741 A191741=dispersion of A047217 (0,1,2 mod 5 and >1)
%C A191741 ...
%C A191741 For further information about these 20 dispersions, see A191722.
%C A191741 ...
%C A191741 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 A191741 Ivan Neretin, <a href="/A191741/b191741.txt">Table of n, a(n) for n = 1..5050</a> (first 100 antidiagonals, flattened)
%e A191741 Northwest corner:
%e A191741 1....2....5....10...17
%e A191741 3....6....11...20...35
%e A191741 4....7....12...21...36
%e A191741 8....15...26...45...76
%e A191741 9....16...27...46...77
%e A191741 13...22...37...62...105
%t A191741 (* Program generates the dispersion array t of the increasing sequence f[n] *)
%t A191741 r = 40; r1 = 12;  c = 40; c1 = 12;
%t A191741 a=2; b=5; c2=6; m[n_]:=If[Mod[n,3]==0,1,0];
%t A191741 f[n_]:=a*m[n+2]+b*m[n+1]+c2*m[n]+5*Floor[(n-1)/3]
%t A191741 Table[f[n], {n, 1, 30}]  (* A047217 *)
%t A191741 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191741 rows = {NestList[f, 1, c]};
%t A191741 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191741 t[i_, j_] := rows[[i, j]];
%t A191741 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191741 *)
%t A191741 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191741  *)
%Y A191741 Cf. A047204, A047217, A191722, A191731, A191426.
%K A191741 nonn,tabl
%O A191741 1,2
%A A191741 _Clark Kimberling_, Jun 14 2011