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A191707 Dispersion of A016873, (numbers >1 and congruent to 1, 2, 3, or 4 mod 5), by antidiagonals.

%I A191707 #7 Oct 17 2017 19:32:04
%S A191707 1,2,5,3,7,10,4,9,13,15,6,12,17,19,20,8,16,22,24,26,25,11,21,28,31,33,
%T A191707 32,30,14,27,36,39,42,41,38,35,18,34,46,49,53,52,48,44,40,23,43,58,62,
%U A191707 67,66,61,56,51,45,29,54,73,78,84,83,77,71,64,57,50,37
%N A191707 Dispersion of A016873, (numbers >1 and congruent to 1, 2, 3, or 4 mod 5), by antidiagonals.
%C A191707 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 A191707 ...
%C A191707 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 A191707 ...
%C A191707 A191702=dispersion of A008587 (5k, k>=1)
%C A191707 A191703=dispersion of A016861 (5k+1, k>=1)
%C A191707 A191704=dispersion of A016873 (5k+2, k>=0)
%C A191707 A191705=dispersion of A016885 (5k+3, k>=0)
%C A191707 A191706=dispersion of A016897 (5k+4, k>=0)
%C A191707 A191707=dispersion of A047201 (1, 2, 3, 4 mod 5 and >1)
%C A191707 A191708=dispersion of A047202 (0, 2, 3, 4 mod 5 and >1)
%C A191707 A191709=dispersion of A047207 (0, 1, 3, 4 mod 5 and >1)
%C A191707 A191710=dispersion of A032763 (0, 1, 2, 4 mod 5 and >1)
%C A191707 A191711=dispersion of A001068 (0, 1, 2, 3 mod 5 and >1)
%C A191707 ...
%C A191707 EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
%C A191707 A191702 has 1st col A047201, all else A008587
%C A191707 A191703 has 1st col A047202, all else A016861
%C A191707 A191704 has 1st col A047207, all else A016873
%C A191707 A191705 has 1st col A032763, all else A016885
%C A191707 A191706 has 1st col A001068, all else A016897
%C A191707 A191707 has 1st col A008587, all else A047201
%C A191707 A191708 has 1st col A042968, all else A047203
%C A191707 A191709 has 1st col A042968, all else A047207
%C A191707 A191710 has 1st col A042968, all else A032763
%C A191707 A191711 has 1st col A042968, all else A001068
%C A191707 ...
%C A191707 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 A191707 ...
%C A191707 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 A191707 a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n)+m*floor((n-1)/4)).
%H A191707 Ivan Neretin, <a href="/A191707/b191707.txt">Table of n, a(n) for n = 1..5050</a> (first 100 antidiagonals, flattened)
%e A191707 Northwest corner:
%e A191707 1....2....3....4....6
%e A191707 5....7....9....12...16
%e A191707 10...13...17...22...28
%e A191707 15...19...24...31...39
%e A191707 20...26...33...42...53
%e A191707 25...32...41...52...66
%t A191707 (* Program generates the dispersion array T of the increasing sequence f[n] *)
%t A191707 r = 40; r1 = 12;  c = 40; c1 = 12;
%t A191707 a=2; b=3; c2=4; d=6; m[n_]:=If[Mod[n,4]==0,1,0];
%t A191707 f[n_]:=a*m[n+3]+b*m[n+2]+c2*m[n+1]+d*m[n]+5*Floor[(n-1)/4]
%t A191707 Table[f[n], {n, 1, 30}]  (* A047201 *)
%t A191707 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191707 rows = {NestList[f, 1, c]};
%t A191707 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191707 t[i_, j_] := rows[[i, j]];
%t A191707 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191707 *)
%t A191707 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191707  *)
%Y A191707 Cf. A047201, A008587, A191702, A191426.
%K A191707 nonn,tabl
%O A191707 1,2
%A A191707 _Clark Kimberling_, Jun 12 2011