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%I #7 Oct 11 2017 17:05:46
%S 1,2,3,4,6,5,7,11,9,8,12,19,16,14,10,21,32,27,24,17,13,36,54,46,41,29,
%T 22,15,61,91,77,69,49,37,26,18,102,152,129,116,82,62,44,31,20,171,254,
%U 216,194,137,104,74,52,34,23,286,424,361,324,229,174,124,87
%N Dispersion of A032793, (numbers >1 and congruent to 1 or 2 or 4 mod 5), by antidiagonals.
%C 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 ...
%C 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 ...
%C A191722=dispersion of A008851 (0, 1 mod 5 and >1)
%C A191723=dispersion of A047215 (0, 2 mod 5 and >1)
%C A191724=dispersion of A047218 (0, 3 mod 5 and >1)
%C A191725=dispersion of A047208 (0, 4 mod 5 and >1)
%C A191726=dispersion of A047216 (1, 2 mod 5 and >1)
%C A191727=dispersion of A047219 (1, 3 mod 5 and >1)
%C A191728=dispersion of A047209 (1, 4 mod 5 and >1)
%C A191729=dispersion of A047221 (2, 3 mod 5 and >1)
%C A191730=dispersion of A047211 (2, 4 mod 5 and >1)
%C A191731=dispersion of A047204 (3, 4 mod 5 and >1)
%C ...
%C A191732=dispersion of A047202 (2,3,4 mod 5 and >1)
%C A191733=dispersion of A047206 (1,3,4 mod 5 and >1)
%C A191734=dispersion of A032793 (1,2,4 mod 5 and >1)
%C A191735=dispersion of A047223 (1,2,3 mod 5 and >1)
%C A191736=dispersion of A047205 (0,3,4 mod 5 and >1)
%C A191737=dispersion of A047212 (0,2,4 mod 5 and >1)
%C A191738=dispersion of A047222 (0,2,3 mod 5 and >1)
%C A191739=dispersion of A008854 (0,1,4 mod 5 and >1)
%C A191740=dispersion of A047220 (0,1,3 mod 5 and >1)
%C A191741=dispersion of A047217 (0,1,2 mod 5 and >1)
%C ...
%C For further information about these 20 dispersions, see A191722.
%C ...
%C 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 Ivan Neretin, <a href="/A191734/b191734.txt">Table of n, a(n) for n = 1..5050</a>
%e Northwest corner:
%e 1....2....4....7...12
%e 3....6....11...19...32
%e 5....9....16...27...46
%e 8....14...24...41...69
%e 10...17...29...49...82
%e 13...22...37...62...104
%t (* Program generates the dispersion array t of the increasing sequence f[n] *)
%t r = 40; r1 = 12; c = 40; c1 = 12;
%t a=2; b=4; c2=6; m[n_]:=If[Mod[n,3]==0,1,0];
%t f[n_]:=a*m[n+2]+b*m[n+1]+c2*m[n]+5*Floor[(n-1)/3]
%t Table[f[n], {n, 1, 30}] (* A032793 *)
%t mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t rows = {NestList[f, 1, c]};
%t Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t t[i_, j_] := rows[[i, j]];
%t TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191734 *)
%t Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191734 *)
%Y Cf. A047218, A082793, A191722, , A191724, A191426.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Jun 14 2011
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