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A191722 Dispersion of A008851, (numbers >1 and congruent to 0 or 1 mod 5), by antidiagonals.

%I A191722 #8 Oct 11 2017 16:24:38
%S A191722 1,5,2,15,6,3,40,16,10,4,101,41,26,11,7,255,105,66,30,20,8,640,265,
%T A191722 166,76,51,21,9,1601,665,416,191,130,55,25,12,4005,1665,1041,480,326,
%U A191722 140,65,31,13,10015,4165,2605,1201,816,351,165,80,35,14,25040,10415
%N A191722 Dispersion of A008851, (numbers >1 and congruent to 0 or 1 mod 5), by antidiagonals.
%C A191722 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 A191722 ...
%C A191722 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 A191722 ...
%C A191722 A191722=dispersion of A008851 (0, 1 mod 5 and >1)
%C A191722 A191723=dispersion of A047215 (0, 2 mod 5 and >1)
%C A191722 A191724=dispersion of A047218 (0, 3 mod 5 and >1)
%C A191722 A191725=dispersion of A047208 (0, 4 mod 5 and >1)
%C A191722 A191726=dispersion of A047216 (1, 2 mod 5 and >1)
%C A191722 A191727=dispersion of A047219 (1, 3 mod 5 and >1)
%C A191722 A191728=dispersion of A047209 (1, 4 mod 5 and >1)
%C A191722 A191729=dispersion of A047221 (2, 3 mod 5 and >1)
%C A191722 A191730=dispersion of A047211 (2, 4 mod 5 and >1)
%C A191722 A191731=dispersion of A047204 (3, 4 mod 5 and >1)
%C A191722 ...
%C A191722 A191732=dispersion of A047202 (2,3,4 mod 5 and >1)
%C A191722 A191733=dispersion of A047206 (1,3,4 mod 5 and >1)
%C A191722 A191734=dispersion of A032793 (1,2,4 mod 5 and >1)
%C A191722 A191735=dispersion of A047223 (1,2,3 mod 5 and >1)
%C A191722 A191736=dispersion of A047205 (0,3,4 mod 5 and >1)
%C A191722 A191737=dispersion of A047212 (0,2,4 mod 5 and >1)
%C A191722 A191738=dispersion of A047222 (0,2,3 mod 5 and >1)
%C A191722 A191739=dispersion of A008854 (0,1,4 mod 5 and >1)
%C A191722 A191740=dispersion of A047220 (0,1,3 mod 5 and >1)
%C A191722 A191741=dispersion of A047217 (0,1,2 mod 5 and >1)
%C A191722 ...
%C A191722 EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
%C A191722 A191722 has 1st col A047202, all else A008851
%C A191722 A191723 has 1st col A047206, all else A047215
%C A191722 A191724 has 1st col A032793, all else A047218
%C A191722 A191725 has 1st col A047223, all else A047208
%C A191722 A191726 has 1st col A047205, all else A047216
%C A191722 A191727 has 1st col A047212, all else A047219
%C A191722 A191728 has 1st col A047222, all else A047209
%C A191722 A191729 has 1st col A008854, all else A047221
%C A191722 A191730 has 1st col A047220, all else A047211
%C A191722 A191731 has 1st col A047217, all else A047204
%C A191722 ...
%C A191722 A191732 has 1st col A000851, all else A047202
%C A191722 A191733 has 1st col A047215, all else A047206
%C A191722 A191734 has 1st col A047218, all else A032793
%C A191722 A191735 has 1st col A047208, all else A047223
%C A191722 A191736 has 1st col A047216, all else A047205
%C A191722 A191737 has 1st col A047219, all else A047212
%C A191722 A191738 has 1st col A047209, all else A047222
%C A191722 A191739 has 1st col A047221, all else A008854
%C A191722 A191740 has 1st col A047211, all else A047220
%C A191722 A191741 has 1st col A047204, all else A047217
%C A191722 ...
%C A191722 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 A191722 Ivan Neretin, <a href="/A191722/b191722.txt">Table of n, a(n) for n = 1..5050</a>
%e A191722 Northwest corner:
%e A191722 1....5....15...40...101
%e A191722 2....6....16...41...105
%e A191722 3....10...26...66...166
%e A191722 4....11...30...76...191
%e A191722 7....20...51...130..326
%e A191722 8....21...55...140..351
%t A191722 (* Program generates the dispersion array t of the increasing sequence f[n] *)
%t A191722 r = 40; r1 = 12;  c = 40; c1 = 12;
%t A191722 a=5; b=6; m[n_]:=If[Mod[n,2]==0,1,0];
%t A191722 f[n_]:=a*m[n+1]+b*m[n]+5*Floor[(n-1)/2]
%t A191722 Table[f[n], {n, 1, 30}]  (* A008851 *)
%t A191722 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191722 rows = {NestList[f, 1, c]};
%t A191722 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191722 t[i_, j_] := rows[[i, j]];
%t A191722 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
%t A191722 (* A191722 *)
%t A191722 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191722  *)
%Y A191722 Cf. A047202, A008851, A191732, A191702, A191426.
%K A191722 nonn,tabl
%O A191722 1,2
%A A191722 _Clark Kimberling_, Jun 13 2011