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A191673 Dispersion of A004773 (>1 and congruent to 0 or 1 or 2 mod 4), by antidiagonals.

%I A191673 #10 Oct 18 2017 05:07:33
%S A191673 1,2,3,4,5,7,6,8,10,11,9,12,14,16,15,13,17,20,22,21,19,18,24,28,30,29,
%T A191673 26,23,25,33,38,41,40,36,32,27,34,45,52,56,54,49,44,37,31,46,61,70,76,
%U A191673 73,66,60,50,42,35,62,82,94,102,98,89,81,68,57,48,39,84
%N A191673 Dispersion of A004773 (>1 and congruent to 0 or 1 or 2 mod 4), by antidiagonals.
%C A191673 For a background discussion of dispersions, see A191426.
%C A191673 ...
%C A191673 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 A191673 ...
%C A191673 A191452=dispersion of A008586 (4k, k>=1)
%C A191673 A191667=dispersion of A016813 (4k+1, k>=1)
%C A191673 A191668=dispersion of A016825 (4k+2, k>=0)
%C A191673 A191669=dispersion of A004767 (4k+3, k>=0)
%C A191673 A191670=dispersion of A042968 (1 or 2 or 3 mod 4 and >=2)
%C A191673 A191671=dispersion of A004772 (0 or 1 or 3 mod 4 and >=2)
%C A191673 A191672=dispersion of A004773 (0 or 1 or 2 mod 4 and >=2)
%C A191673 A191673=dispersion of A004773 (0 or 2 or 3 mod 4 and >=2)
%C A191673 ...
%C A191673 EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
%C A191673 A191452 has 1st col A042968, all else A008486
%C A191673 A191667 has 1st col A004772, all else A016813
%C A191673 A191668 has 1st col A042965, all else A016825
%C A191673 A191669 has 1st col A004773, all else A004767
%C A191673 A191670 has 1st col A008486, all else A042968
%C A191673 A191671 has 1st col A016813, all else A004772
%C A191673 A191672 has 1st col A016825, all else A042965
%C A191673 A191673 has 1st col A004767, all else A004773
%C A191673 ...
%C A191673 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 A191673    If f(n)=(n mod 3), then (a,b,c,a,b,c,a,b,c,...) is given by
%C A191673    a*f(n+2)+b*f(n+1)+c*f(n), so that "(a or b or c mod m)" is given by
%C A191673    a*f(n+2)+b*f(n+1)+c*f(n)+m*floor((n-1)/3)), for n>=1.
%H A191673 Ivan Neretin, <a href="/A191673/b191673.txt">Table of n, a(n) for n = 1..5050</a> (first 100 antidiagonals, flattened)
%e A191673 Northwest corner:
%e A191673 1....2....4....6....9
%e A191673 3....5....8....12...17
%e A191673 7....10...14...20...28
%e A191673 11...16...22...30...41
%e A191673 15...21...29...40...54
%t A191673 (* Program generates the dispersion array T of the increasing sequence f[n] *)
%t A191673 r = 40; r1 = 12; c = 40; c1 = 12;
%t A191673 a = 2; b = 4; c2 = 5; m[n_] := If[Mod[n, 3] == 0, 1, 0];
%t A191673 f[n_] := a*m[n + 2] + b*m[n + 1] + c2*m[n] + 4*Floor[(n - 1)/3]
%t A191673 Table[f[n], {n, 1, 30}] (* A004773 *)
%t A191673 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191673 rows = {NestList[f, 1, c]};
%t A191673 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191673 t[i_, j_] := rows[[i, j]];
%t A191673 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191673 *)
%t A191673 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191673 *)
%Y A191673 Cf. A004767, A004773, A191669, A191452.
%K A191673 nonn,tabl
%O A191673 1,2
%A A191673 _Clark Kimberling_, Jun 11 2011