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A191705 Dispersion of A016873, (5k+3), by antidiagonals.

%I A191705 #11 Oct 23 2017 20:09:17
%S A191705 1,3,2,13,8,4,63,38,18,5,313,188,88,23,6,1563,938,438,113,28,7,7813,
%T A191705 4688,2188,563,138,33,9,39063,23438,10938,2813,688,163,43,10,195313,
%U A191705 117188,54688,14063,3438,813,213,48,11,976563,585938,273438,70313,17188
%N A191705 Dispersion of A016873, (5k+3), by antidiagonals.
%C A191705 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 A191705 ...
%C A191705 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 A191705 ...
%C A191705 A191702=dispersion of A008587 (5k, k>=1)
%C A191705 A191703=dispersion of A016861 (5k+1, k>=1)
%C A191705 A191704=dispersion of A016873 (5k+2, k>=0)
%C A191705 A191705=dispersion of A016885 (5k+3, k>=0)
%C A191705 A191706=dispersion of A016897 (5k+4, k>=0)
%C A191705 A191707=dispersion of A047201 (1, 2, 3, 4 mod 5 and >1)
%C A191705 A191708=dispersion of A047202 (0, 2, 3, 4 mod 5 and >1)
%C A191705 A191709=dispersion of A047207 (0, 1, 3, 4 mod 5 and >1)
%C A191705 A191710=dispersion of A032763 (0, 1, 2, 4 mod 5 and >1)
%C A191705 A191711=dispersion of A001068 (0, 1, 2, 3 mod 5 and >1)
%C A191705 ...
%C A191705 EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
%C A191705 A191702 has 1st col A047201, all else A008587
%C A191705 A191703 has 1st col A047202, all else A016861
%C A191705 A191704 has 1st col A047207, all else A016873
%C A191705 A191705 has 1st col A032763, all else A016885
%C A191705 A191706 has 1st col A001068, all else A016897
%C A191705 A191707 has 1st col A008587, all else A047201
%C A191705 A191708 has 1st col A042968, all else A047203
%C A191705 A191709 has 1st col A042968, all else A047207
%C A191705 A191710 has 1st col A042968, all else A032763
%C A191705 A191711 has 1st col A042968, all else A001068
%C A191705 ...
%C A191705 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 A191705 ...
%C A191705 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 a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n)+m*floor((n-1)/4)).
%H A191705 Ivan Neretin, <a href="/A191705/b191705.txt">Table of n, a(n) for n = 1..5050</a> (first 100 antidiagonals, flattened)
%e A191705 Northwest corner:
%e A191705 1...3....13....63....313
%e A191705 2...8....38....188...938
%e A191705 4...18...88....438...2188
%e A191705 5...23...113...563...2813
%e A191705 6...28...138...688...3438
%e A191705 7...33...163...813...4063
%t A191705 (* Program generates the dispersion array T of the increasing sequence f[n] *)
%t A191705 r = 40; r1 = 12;  c = 40; c1 = 12;
%t A191705 f[n_] := 5n-2
%t A191705 Table[f[n], {n, 1, 30}] (* A016885 *)
%t A191705 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191705 rows = {NestList[f, 1, c]};
%t A191705 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191705 t[i_, j_] := rows[[i, j]];
%t A191705 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191705 *)
%t A191705 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191705 *)
%Y A191705 Cf. A016885, A032763, A191710, A191426.
%K A191705 nonn,tabl
%O A191705 1,2
%A A191705 _Clark Kimberling_, Jun 12 2011