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A191708 Dispersion of A047202, (numbers >1 and congruent to 0, 2, 3, or 4 mod 5), by antidiagonals.

%I A191708 #10 Jun 04 2022 22:04:24
%S A191708 1,2,6,3,8,11,4,10,14,16,5,13,18,20,21,7,17,23,25,27,26,9,22,29,32,34,
%T A191708 33,31,12,28,37,40,43,42,39,36,15,35,47,50,54,53,49,45,41,19,44,59,63,
%U A191708 68,67,62,57,52,46,24,55,74,79,85,84,78,72,65,58,51,30
%N A191708 Dispersion of A047202, (numbers >1 and congruent to 0, 2, 3, or 4 mod 5), by antidiagonals.
%C A191708 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 A191708 ...
%C A191708 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 A191708 ...
%C A191708 A191702=dispersion of A008587 (5k, k>=1)
%C A191708 A191703=dispersion of A016861 (5k+1, k>=1)
%C A191708 A191704=dispersion of A016873 (5k+2, k>=0)
%C A191708 A191705=dispersion of A016885 (5k+3, k>=0)
%C A191708 A191706=dispersion of A016897 (5k+4, k>=0)
%C A191708 A191707=dispersion of A047201 (1, 2, 3, 4 mod 5 and >1)
%C A191708 A191708=dispersion of A047202 (0, 2, 3, 4 mod 5 and >1)
%C A191708 A191709=dispersion of A047207 (0, 1, 3, 4 mod 5 and >1)
%C A191708 A191710=dispersion of A032763 (0, 1, 2, 4 mod 5 and >1)
%C A191708 A191711=dispersion of A001068 (0, 1, 2, 3 mod 5 and >1)
%C A191708 ...
%C A191708 EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
%C A191708 A191702 has 1st col A047201, all else A008587
%C A191708 A191703 has 1st col A047202, all else A016861
%C A191708 A191704 has 1st col A047207, all else A016873
%C A191708 A191705 has 1st col A032763, all else A016885
%C A191708 A191706 has 1st col A001068, all else A016897
%C A191708 A191707 has 1st col A008587, all else A047201
%C A191708 A191708 has 1st col A042968, all else A047203
%C A191708 A191709 has 1st col A042968, all else A047207
%C A191708 A191710 has 1st col A042968, all else A032763
%C A191708 A191711 has 1st col A042968, all else A001068
%C A191708 ...
%C A191708 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 A191708 ...
%C A191708 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 A191708 a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n)+m*floor((n-1)/4)).
%H A191708 Ivan Neretin, <a href="/A191708/b191708.txt">Table of n, a(n) for n = 1..5050</a> (first 100 antidiagonals, flattened)
%e A191708 Northwest corner:
%e A191708 1....2....3....4....5
%e A191708 6....8....10...13...17
%e A191708 11...14...18...23...29
%e A191708 16...20...25...32...40
%e A191708 21...27...34...43...54
%e A191708 26...33...42...53...67
%t A191708 (* Program generates the dispersion array T of the increasing sequence f[n] *)
%t A191708 r = 40; r1 = 12;  c = 40; c1 = 12;
%t A191708 a=2; b=3; c2=4; d=5; m[n_]:=If[Mod[n,4]==0,1,0];
%t A191708 f[n_]:=a*m[n+3]+b*m[n+2]+c2*m[n+1]+d*m[n]+5*Floor[(n-1)/4]
%t A191708 Table[f[n], {n, 1, 30}]  (* A047202 *)
%t A191708 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191708 rows = {NestList[f, 1, c]};
%t A191708 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191708 t[i_, j_] := rows[[i, j]];
%t A191708 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191708 *)
%t A191708 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191708  *)
%Y A191708 Cf. A047202, A016861, A191703, A191426.
%K A191708 nonn,tabl
%O A191708 1,2
%A A191708 _Clark Kimberling_, Jun 12 2011