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A191737 Dispersion of A047212, (numbers >1 and congruent to 0 or 2 or 4 mod 5), by antidiagonals.

%I A191737 #7 Oct 12 2017 19:45:40
%S A191737 1,2,3,4,5,6,7,9,10,8,12,15,17,14,11,20,25,29,24,19,13,34,42,49,40,32,
%T A191737 22,16,57,70,82,67,54,37,27,18,95,117,137,112,90,62,45,30,21,159,195,
%U A191737 229,187,150,104,75,50,35,23,265,325,382,312,250,174,125,84
%N A191737 Dispersion of A047212, (numbers >1 and congruent to 0 or 2 or 4 mod 5), by antidiagonals.
%C A191737 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 A191737 ...
%C A191737 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 A191737 ...
%C A191737 A191722=dispersion of A008851 (0, 1 mod 5 and >1)
%C A191737 A191723=dispersion of A047215 (0, 2 mod 5 and >1)
%C A191737 A191724=dispersion of A047218 (0, 3 mod 5 and >1)
%C A191737 A191725=dispersion of A047208 (0, 4 mod 5 and >1)
%C A191737 A191726=dispersion of A047216 (1, 2 mod 5 and >1)
%C A191737 A191727=dispersion of A047219 (1, 3 mod 5 and >1)
%C A191737 A191728=dispersion of A047209 (1, 4 mod 5 and >1)
%C A191737 A191729=dispersion of A047221 (2, 3 mod 5 and >1)
%C A191737 A191730=dispersion of A047211 (2, 4 mod 5 and >1)
%C A191737 A191731=dispersion of A047204 (3, 4 mod 5 and >1)
%C A191737 ...
%C A191737 A191732=dispersion of A047202 (2,3,4 mod 5 and >1)
%C A191737 A191733=dispersion of A047206 (1,3,4 mod 5 and >1)
%C A191737 A191734=dispersion of A032793 (1,2,4 mod 5 and >1)
%C A191737 A191735=dispersion of A047223 (1,2,3 mod 5 and >1)
%C A191737 A191736=dispersion of A047205 (0,3,4 mod 5 and >1)
%C A191737 A191737=dispersion of A047212 (0,2,4 mod 5 and >1)
%C A191737 A191738=dispersion of A047222 (0,2,3 mod 5 and >1)
%C A191737 A191739=dispersion of A008854 (0,1,4 mod 5 and >1)
%C A191737 A191740=dispersion of A047220 (0,1,3 mod 5 and >1)
%C A191737 A191741=dispersion of A047217 (0,1,2 mod 5 and >1)
%C A191737 ...
%C A191737 For further information about these 20 dispersions, see A191722.
%C A191737 ...
%C A191737 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 A191737 Ivan Neretin, <a href="/A191737/b191737.txt">Table of n, a(n) for n = 1..5050</a> (first 100 antidiagonals, flattened)
%e A191737 Northwest corner:
%e A191737 1....2....4....7...12
%e A191737 3....5....9...15...25
%e A191737 6....10....17...29...49
%e A191737 8....14...24...40...67
%e A191737 11...19...32...54...90
%e A191737 13...22...37...62...104
%t A191737 (* Program generates the dispersion array t of the increasing sequence f[n] *)
%t A191737 r = 40; r1 = 12;  c = 40; c1 = 12;
%t A191737 a=2; b=4; c2=5; m[n_]:=If[Mod[n,3]==0,1,0];
%t A191737 f[n_]:=a*m[n+2]+b*m[n+1]+c2*m[n]+5*Floor[(n-1)/3]
%t A191737 Table[f[n], {n, 1, 30}]  (* A047212 *)
%t A191737 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191737 rows = {NestList[f, 1, c]};
%t A191737 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191737 t[i_, j_] := rows[[i, j]];
%t A191737 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191737 *)
%t A191737 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191737  *)
%Y A191737 Cf. A047219, A047212, A191722, A191727, A191426.
%K A191737 nonn,tabl
%O A191737 1,2
%A A191737 _Clark Kimberling_, Jun 14 2011