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A191736 Dispersion of A047205, (numbers >1 and congruent to 0 or 3 or 4 mod 5), by antidiagonals.

%I A191736 #7 Oct 12 2017 19:45:10
%S A191736 1,3,2,5,4,6,9,8,10,7,15,14,18,13,11,25,24,30,23,19,12,43,40,50,39,33,
%T A191736 20,16,73,68,84,65,55,34,28,17,123,114,140,109,93,58,48,29,21,205,190,
%U A191736 234,183,155,98,80,49,35,22,343,318,390,305,259,164,134,83
%N A191736 Dispersion of A047205, (numbers >1 and congruent to 0 or 3 or 4 mod 5), by antidiagonals.
%C A191736 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 A191736 ...
%C A191736 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 A191736 ...
%C A191736 A191722=dispersion of A008851 (0, 1 mod 5 and >1)
%C A191736 A191723=dispersion of A047215 (0, 2 mod 5 and >1)
%C A191736 A191724=dispersion of A047218 (0, 3 mod 5 and >1)
%C A191736 A191725=dispersion of A047208 (0, 4 mod 5 and >1)
%C A191736 A191726=dispersion of A047216 (1, 2 mod 5 and >1)
%C A191736 A191727=dispersion of A047219 (1, 3 mod 5 and >1)
%C A191736 A191728=dispersion of A047209 (1, 4 mod 5 and >1)
%C A191736 A191729=dispersion of A047221 (2, 3 mod 5 and >1)
%C A191736 A191730=dispersion of A047211 (2, 4 mod 5 and >1)
%C A191736 A191731=dispersion of A047204 (3, 4 mod 5 and >1)
%C A191736 ...
%C A191736 A191732=dispersion of A047202 (2,3,4 mod 5 and >1)
%C A191736 A191733=dispersion of A047206 (1,3,4 mod 5 and >1)
%C A191736 A191734=dispersion of A032793 (1,2,4 mod 5 and >1)
%C A191736 A191735=dispersion of A047223 (1,2,3 mod 5 and >1)
%C A191736 A191736=dispersion of A047205 (0,3,4 mod 5 and >1)
%C A191736 A191737=dispersion of A047212 (0,2,4 mod 5 and >1)
%C A191736 A191738=dispersion of A047222 (0,2,3 mod 5 and >1)
%C A191736 A191739=dispersion of A008854 (0,1,4 mod 5 and >1)
%C A191736 A191740=dispersion of A047220 (0,1,3 mod 5 and >1)
%C A191736 A191741=dispersion of A047217 (0,1,2 mod 5 and >1)
%C A191736 ...
%C A191736 For further information about these 20 dispersions, see A191722.
%C A191736 ...
%C A191736 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 A191736 Ivan Neretin, <a href="/A191736/b191736.txt">Table of n, a(n) for n = 1..5050</a> (first 100 antidiagonals, flattened)
%e A191736 Northwest corner:
%e A191736 1....3....5....9....15
%e A191736 2....4....8....14...24
%e A191736 6....10...18...30...50
%e A191736 7....13...23...39...65
%e A191736 11...19...33...55...93
%e A191736 12...20...34...58...98
%t A191736 (* Program generates the dispersion array t of the increasing sequence f[n] *)
%t A191736 r = 40; r1 = 12;  c = 40; c1 = 12;
%t A191736 a=3; b=4; c2=5; m[n_]:=If[Mod[n,3]==0,1,0];
%t A191736 f[n_]:=a*m[n+2]+b*m[n+1]+c2*m[n]+5*Floor[(n-1)/3]
%t A191736 Table[f[n], {n, 1, 30}]  (* A047205 *)
%t A191736 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191736 rows = {NestList[f, 1, c]};
%t A191736 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191736 t[i_, j_] := rows[[i, j]];
%t A191736 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191736 *)
%t A191736 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191736  *)
%Y A191736 Cf. A047216, A047205, A191722, , A191726, A191426.
%K A191736 nonn,tabl
%O A191736 1,2
%A A191736 _Clark Kimberling_, Jun 14 2011