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%I #7 Oct 17 2017 20:50:51
%S 1,2,4,3,6,9,5,8,12,14,7,11,16,18,19,10,15,21,23,25,24,13,20,27,30,32,
%T 31,29,17,26,35,38,41,40,37,34,22,33,45,48,52,51,47,43,39,28,42,57,61,
%U 66,65,60,55,50,44,36,53,72,77,83,82,76,70,63,56,49,46
%N Dispersion of A001068, (numbers >1 and congruent to 0, 1, 2, or 3 mod 5), by antidiagonals.
%C For a background discussion of dispersions and their fractal sequences, see A191426. For dispersions of congruence sequences mod 3 or mod 4, see A191655, A191663, A191667.
%C ...
%C Each of the sequences (5n, n>1), (5n+1, n>1), (5n+2, n>=0), (5n+3, n>=0), (5n+4, n>=0), generates a dispersion. Each complement (beginning with its first term >1) also generates a dispersion. The ten sequences and dispersions are listed here:
%C ...
%C A191702=dispersion of A008587 (5k, k>=1)
%C A191703=dispersion of A016861 (5k+1, k>=1)
%C A191704=dispersion of A016873 (5k+2, k>=0)
%C A191705=dispersion of A016885 (5k+3, k>=0)
%C A191706=dispersion of A016897 (5k+4, k>=0)
%C A191707=dispersion of A047201 (1, 2, 3, 4 mod 5 and >1)
%C A191708=dispersion of A047202 (0, 2, 3, 4 mod 5 and >1)
%C A191709=dispersion of A047207 (0, 1, 3, 4 mod 5 and >1)
%C A191710=dispersion of A032763 (0, 1, 2, 4 mod 5 and >1)
%C A191711=dispersion of A001068 (0, 1, 2, 3 mod 5 and >1)
%C ...
%C EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
%C A191702 has 1st col A047201, all else A008587
%C A191703 has 1st col A047202, all else A016861
%C A191704 has 1st col A047207, all else A016873
%C A191705 has 1st col A032763, all else A016885
%C A191706 has 1st col A001068, all else A016897
%C A191707 has 1st col A008587, all else A047201
%C A191708 has 1st col A042968, all else A047203
%C A191709 has 1st col A042968, all else A047207
%C A191710 has 1st col A042968, all else A032763
%C A191711 has 1st col A042968, all else A001068
%C ...
%C Regarding the dispersions A191670-A191673, there is a formula for sequences of the type "(a or b or c or d mod m)", (as in the relevant Mathematica programs):
%C ...
%C If f(n)=(n mod 3), then (a,b,c,d,a,b,c,d,a,b,c,d,...) is given by a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n); so that for n>=1, "(a, b, c, d mod m)" is given by
%C a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n)+m*floor((n-1)/4)).
%H Ivan Neretin, <a href="/A191711/b191711.txt">Table of n, a(n) for n = 1..5050</a> (first 100 antidiagonals, flattened)
%e Northwest corner:
%e 1....2....3....5....7
%e 4....6....8....11...15
%e 9....12...16...21...27
%e 14...18...23...30...38
%e 19...25...32...41...52
%e 24...31...40...51...65
%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=3; c2=5; d=6; m[n_]:=If[Mod[n,4]==0,1,0];
%t f[n_]:=a*m[n+3]+b*m[n+2]+c2*m[n+1]+d*m[n]+5*Floor[(n-1)/4]
%t Table[f[n], {n, 1, 30}] (* A001068 *)
%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}]] (* A191711 *)
%t Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191711 *)
%Y Cf. A001068, A016897, A191706, A191426.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Jun 12 2011
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