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A191672 Dispersion of A042965 (>1 and congruent to 0 or 1 or 3 mod 4), by antidiagonals.

%I A191672 #10 Oct 18 2017 05:07:14
%S A191672 1,3,2,5,4,6,8,7,9,10,12,11,13,15,14,17,16,19,21,20,18,24,23,27,29,28,
%T A191672 25,22,33,32,37,40,39,35,31,26,45,44,51,55,53,48,43,36,30,61,60,69,75,
%U A191672 72,65,59,49,41,34,83,81,93,101,97,88,80,67,56,47,38
%N A191672 Dispersion of A042965 (>1 and congruent to 0 or 1 or 3 mod 4), by antidiagonals.
%C A191672 For a background discussion of dispersions, see A191426.
%C A191672 ...
%C A191672 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 A191672 ...
%C A191672 A191452=dispersion of A008586 (4k, k>=1)
%C A191672 A191667=dispersion of A016813 (4k+1, k>=1)
%C A191672 A191668=dispersion of A016825 (4k+2, k>=0)
%C A191672 A191669=dispersion of A004767 (4k+3, k>=0)
%C A191672 A191670=dispersion of A042968 (1 or 2 or 3 mod 4 and >=2)
%C A191672 A191671=dispersion of A004772 (0 or 1 or 3 mod 4 and >=2)
%C A191672 A191672=dispersion of A004773 (0 or 1 or 2 mod 4 and >=2)
%C A191672 A191673=dispersion of A004773 (0 or 2 or 3 mod 4 and >=2)
%C A191672 ...
%C A191672 EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
%C A191672 A191452 has 1st col A042968, all else A008486
%C A191672 A191667 has 1st col A004772, all else A016813
%C A191672 A191668 has 1st col A042965, all else A016825
%C A191672 A191669 has 1st col A004773, all else A004767
%C A191672 A191670 has 1st col A008486, all else A042968
%C A191672 A191671 has 1st col A016813, all else A004772
%C A191672 A191672 has 1st col A016825, all else A042965
%C A191672 A191673 has 1st col A004767, all else A004773
%C A191672 ...
%C A191672 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 A191672    If f(n)=(n mod 3), then (a,b,c,a,b,c,a,b,c,...) is given by
%C A191672    a*f(n+2)+b*f(n+1)+c*f(n), so that "(a or b or c mod m)" is given by
%C A191672    a*f(n+2)+b*f(n+1)+c*f(n)+m*floor((n-1)/3)), for n>=1.
%H A191672 Ivan Neretin, <a href="/A191672/b191672.txt">Table of n, a(n) for n = 1..5050</a> (first 100 antidiagonals, flattened)
%e A191672 Northwest corner:
%e A191672 1....3...5....8....12
%e A191672 2....4...7....11...16
%e A191672 6....9...13...19...27
%e A191672 10...15..21...29...40
%e A191672 14...20..28...39...53
%t A191672 (* Program generates the dispersion array T of the increasing sequence f[n] *)
%t A191672 r = 40; r1 = 12; c = 40; c1 = 12;
%t A191672 a = 3; b = 4; c2 = 5; m[n_] := If[Mod[n, 3] == 0, 1, 0];
%t A191672 f[n_] := a*m[n + 2] + b*m[n + 1] + c2*m[n] + 4*Floor[(n - 1)/3]
%t A191672 Table[f[n], {n, 1, 30}] (* A042965 *)
%t A191672 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191672 rows = {NestList[f, 1, c]};
%t A191672 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191672 t[i_, j_] := rows[[i, j]];
%t A191672 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191672 *)
%t A191672 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191672 *)
%Y A191672 Cf. A016825, A042965, A191668, A191452.
%K A191672 nonn,tabl
%O A191672 1,2
%A A191672 _Clark Kimberling_, Jun 11 2011