%I #7 Oct 11 2017 16:33:44

%S 1,4,2,11,6,3,29,16,9,5,74,41,24,14,7,186,104,61,36,19,8,466,261,154,

%T 91,49,21,10,1166,654,386,229,124,54,26,12,2916,1636,966,574,311,136,

%U 66,31,13,7291,4091,2416,1436,779,341,166,79,34,15,18229,10229

%N Dispersion of A047209, (numbers >1 and congruent to 1 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="/A191728/b191728.txt">Table of n, a(n) for n = 1..5050</a>

%e Northwest corner:

%e 1....4....11...29....74

%e 2....6....16...41....104

%e 3....9....24...61....154

%e 5....14...36...91....229

%e 7....19...49...124...311

%e 8....21...54...136...341

%t (* Program generates the dispersion array t of the increasing sequence f[n] *)

%t r = 40; r1 = 12; c = 40; c1 = 12;

%t a=4; b=6; m[n_]:=If[Mod[n,2]==0,1,0];

%t f[n_]:=a*m[n+1]+b*m[n]+5*Floor[(n-1)/2]

%t Table[f[n], {n, 1, 30}] (* A047209 *)

%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}]] (* A191728 *)

%t Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191728 *)

%Y Cf. A047222, A047209, A191738, A191722, A191426.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Jun 13 2011