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A191667 Dispersion of A016813 (4k+1, k>1), by antidiagonals.

%I A191667 #15 Oct 18 2017 05:06:26
%S A191667 1,5,2,21,9,3,85,37,13,4,341,149,53,17,6,1365,597,213,69,25,7,5461,
%T A191667 2389,853,277,101,29,8,21845,9557,3413,1109,405,117,33,10,87381,38229,
%U A191667 13653,4437,1621,469,133,41,11,349525,152917,54613,17749,6485,1877,533
%N A191667 Dispersion of A016813 (4k+1, k>1), by antidiagonals.
%C A191667 For a background discussion of dispersions, see A191426.
%C A191667 ...
%C A191667 Each of the sequences (4n, n>2), (4n+1, n>0), (3n+2, n>=0), generates a dispersion. Each complement (beginning with its first term >1) also generates a dispersion. The six sequences and dispersions are listed here:
%C A191667 ...
%C A191667 A191452=dispersion of A008586 (4k, k>=1)
%C A191667 A191667=dispersion of A016813 (4k+1, k>=1)
%C A191667 A191668=dispersion of A016825 (4k+2, k>=0)
%C A191667 A191669=dispersion of A004767 (4k+3, k>=0)
%C A191667 A191670=dispersion of A042968 (1 or 2 or 3 mod 4 and >=2)
%C A191667 A191671=dispersion of A004772 (0 or 1 or 3 mod 4 and >=2)
%C A191667 A191672=dispersion of A004773 (0 or 1 or 2 mod 4 and >=2)
%C A191667 A191673=dispersion of A004773 (0 or 2 or 3 mod 4 and >=2)
%C A191667 ...
%C A191667 EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
%C A191667 A191452 has 1st col A042968, all else A008486
%C A191667 A191667 has 1st col A004772, all else A016813
%C A191667 A191668 has 1st col A042965, all else A016825
%C A191667 A191669 has 1st col A004773, all else A004767
%C A191667 A191670 has 1st col A008486, all else A042968
%C A191667 A191671 has 1st col A016813, all else A004772
%C A191667 A191672 has 1st col A016825, all else A042965
%C A191667 A191673 has 1st col A004767, all else A004773
%C A191667 ...
%C A191667 Regarding the dispersions A191670-A191673, there is a formula for sequences of the type "(a or b or c mod m)", (as in the Mathematica program below):
%C A191667    If f(n)=(n mod 3), then (a,b,c,a,b,c,a,b,c,...) is given by a*f(n+2)+b*f(n+1)+c*f(n), so that "(a or b or c mod m)" is given by a*f(n+2)+b*f(n+1)+c*f(n)+m*floor((n-1)/3)), for n>=1.
%H A191667 Ivan Neretin, <a href="/A191667/b191667.txt">Table of n, a(n) for n = 1..5050</a> (first 100 antidiagonals, flattened)
%e A191667 Northwest corner:
%e A191667 1....5....21....85....341
%e A191667 2....9....37....149...597
%e A191667 3....13...53....213...853
%e A191667 4....17...69....277...1109
%e A191667 6....25...101...405...1621
%t A191667 (* Program generates the dispersion array T of the increasing sequence f[n] *)
%t A191667 r = 40; r1 = 12;  c = 40; c1 = 12;
%t A191667 f[n_] := 4*n+1
%t A191667 Table[f[n], {n, 1, 30}]  (* A016813 *)
%t A191667 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191667 rows = {NestList[f, 1, c]};
%t A191667 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191667 t[i_, j_] := rows[[i, j]];
%t A191667 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191667 *)
%t A191667 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191667  *)
%Y A191667 Row 1: A002450.
%Y A191667 Cf. A004772, A016813, A191671, A191426.
%K A191667 nonn,tabl
%O A191667 1,2
%A A191667 _Clark Kimberling_, Jun 11 2011