A191737 - OEIS

%I #7 Oct 12 2017 19:45:40

%S 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 22,16,57,70,82,67,54,37,27,18,95,117,137,112,90,62,45,30,21,159,195,

%U 229,187,150,104,75,50,35,23,265,325,382,312,250,174,125,84

%N Dispersion of A047212, (numbers >1 and congruent to 0 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="/A191737/b191737.txt">Table of n, a(n) for n = 1..5050</a> (first 100 antidiagonals, flattened)

%e Northwest corner:

%e 1....2....4....7...12

%e 3....5....9...15...25

%e 6....10....17...29...49

%e 8....14...24...40...67

%e 11...19...32...54...90

%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=5; 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}]  (* A047212 *)

%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}]] (* A191737 *)

%t Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191737  *)

%Y Cf. A047219, A047212, A191722, A191727, A191426.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Jun 14 2011