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A191670 Dispersion of A042968 (>1 and congruent to 1 or 2 or 3 mod 4), by antidiagonals.

%I A191670 #13 Oct 18 2017 05:06:51
%S A191670 1,2,4,3,6,8,5,9,11,12,7,13,15,17,16,10,18,21,23,22,20,14,25,29,31,30,
%T A191670 27,24,19,34,39,42,41,37,33,28,26,46,53,57,55,50,45,38,32,35,62,71,77,
%U A191670 74,67,61,51,43,36,47,83,95,103,99,90,82,69,58,49,40,63
%N A191670 Dispersion of A042968 (>1 and congruent to 1 or 2 or 3 mod 4), by antidiagonals.
%C A191670 For a background discussion of dispersions, see A191426.
%C A191670 ...
%C A191670 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 A191670 ...
%C A191670 A191452=dispersion of A008586 (4k, k>=1)
%C A191670 A191667=dispersion of A016813 (4k+1, k>=1)
%C A191670 A191668=dispersion of A016825 (4k+2, k>=0)
%C A191670 A191669=dispersion of A004767 (4k+3, k>=0)
%C A191670 A191670=dispersion of A042968 (1 or 2 or 3 mod 4 and >=2)
%C A191670 A191671=dispersion of A004772 (0 or 1 or 3 mod 4 and >=2)
%C A191670 A191672=dispersion of A004773 (0 or 1 or 2 mod 4 and >=2)
%C A191670 A191673=dispersion of A004773 (0 or 2 or 3 mod 4 and >=2)
%C A191670 ...
%C A191670 EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
%C A191670 A191452 has 1st col A042968, all else A008486
%C A191670 A191667 has 1st col A004772, all else A016813
%C A191670 A191668 has 1st col A042965, all else A016825
%C A191670 A191669 has 1st col A004773, all else A004767
%C A191670 A191670 has 1st col A008486, all else A042968
%C A191670 A191671 has 1st col A016813, all else A004772
%C A191670 A191672 has 1st col A016825, all else A042965
%C A191670 A191673 has 1st col A004767, all else A004773
%C A191670 ...
%C A191670 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 A191670    If f(n)=(n mod 3), then (a,b,c,a,b,c,a,b,c,...) is given by
%C A191670    a*f(n+2)+b*f(n+1)+c*f(n), so that "(a or b or c mod m)" is given by
%C A191670    a*f(n+2)+b*f(n+1)+c*f(n)+m*floor((n-1)/3)), for n>=1.
%H A191670 Ivan Neretin, <a href="/A191670/b191670.txt">Table of n, a(n) for n = 1..5050</a> (first 100 antidiagonals, flattened)
%e A191670 Northwest corner:
%e A191670 1....2....3....5....7
%e A191670 4....6....9....13...18
%e A191670 8....11...15...21...29
%e A191670 12...17...23...31...42
%e A191670 16...22...30...41...55
%t A191670 (* Program generates the dispersion array T of the increasing sequence f[n] *)
%t A191670 r = 40; r1 = 12; c = 40; c1 = 12;
%t A191670 a = 2; b = 3; c2 = 5; m[n_] := If[Mod[n, 3] == 0, 1, 0];
%t A191670 f[n_] := a*m[n + 2] + b*m[n + 1] + c2*m[n] + 4*Floor[(n - 1)/3]
%t A191670 Table[f[n], {n, 1, 30}] (* A042968 *)
%t A191670 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191670 rows = {NestList[f, 1, c]};
%t A191670 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191670 t[i_, j_] := rows[[i, j]];
%t A191670 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191670 *)
%t A191670 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191670 *)
%Y A191670 Row 1: A155167, Row 2: A171861.
%Y A191670 Cf. A008486, A042968, A191673, A191452.
%K A191670 nonn,tabl
%O A191670 1,2
%A A191670 _Clark Kimberling_, Jun 11 2011