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A191669 Dispersion of A004767 (4k+3, k>=0), by antidiagonals.

%I A191669 #12 Oct 18 2017 05:06:43
%S A191669 1,3,2,11,7,4,43,27,15,5,171,107,59,19,6,683,427,235,75,23,8,2731,
%T A191669 1707,939,299,91,31,9,10923,6827,3755,1195,363,123,35,10,43691,27307,
%U A191669 15019,4779,1451,491,139,39,12,174763,109227,60075,19115,5803,1963,555,155
%N A191669 Dispersion of A004767 (4k+3, k>=0), by antidiagonals.
%C A191669 For a background discussion of dispersions, see A191426.
%C A191669 ...
%C A191669 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 A191669 ...
%C A191669 A191452=dispersion of A008586 (4k, k>=1)
%C A191669 A191667=dispersion of A016813 (4k+1, k>=1)
%C A191669 A191668=dispersion of A016825 (4k+2, k>=0)
%C A191669 A191669=dispersion of A004767 (4k+3, k>=0)
%C A191669 A191670=dispersion of A042968 (1 or 2 or 3 mod 4 and >=2)
%C A191669 A191671=dispersion of A004772 (0 or 1 or 3 mod 4 and >=2)
%C A191669 A191672=dispersion of A004773 (0 or 1 or 2 mod 4 and >=2)
%C A191669 A191673=dispersion of A004773 (0 or 2 or 3 mod 4 and >=2)
%C A191669 ...
%C A191669 EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
%C A191669 A191452 has 1st col A042968, all else A008486
%C A191669 A191667 has 1st col A004772, all else A016813
%C A191669 A191668 has 1st col A042965, all else A016825
%C A191669 A191669 has 1st col A004773, all else A004767
%C A191669 A191670 has 1st col A008486, all else A042968
%C A191669 A191671 has 1st col A016813, all else A004772
%C A191669 A191672 has 1st col A016825, all else A042965
%C A191669 A191673 has 1st col A004767, all else A004773
%C A191669 ...
%C A191669 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 A191669    If f(n)=(n mod 3), then (a,b,c,a,b,c,a,b,c,...) is given by
%C A191669    a*f(n+2)+b*f(n+1)+c*f(n), so that "(a or b or c mod m)" is given by
%C A191669    a*f(n+2)+b*f(n+1)+c*f(n)+m*floor((n-1)/3)), for n>=1.
%H A191669 Ivan Neretin, <a href="/A191669/b191669.txt">Table of n, a(n) for n = 1..5050</a> (first 100 antidiagonals, flattened)
%e A191669 Northwest corner:
%e A191669 1...3....11....43....171
%e A191669 2...7....27....107...427
%e A191669 4...15...59....235...939
%e A191669 5...19...75....299...1195
%e A191669 6...23...91....363...1451
%t A191669 (* Program generates the dispersion array T of the increasing sequence f[n] *)
%t A191669 r = 40; r1 = 12; c = 40; c1 = 12;
%t A191669 f[n_] := 4*n-1
%t A191669 Table[f[n], {n, 1, 30}] (* A004767 *)
%t A191669 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191669 rows = {NestList[f, 1, c]};
%t A191669 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191669 t[i_, j_] := rows[[i, j]];
%t A191669 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191669 *)
%t A191669 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191669 *)
%Y A191669 Row 1: A007583, Row 2: A136412.
%Y A191669 Cf. A004773, A004767, A191673, A191426.
%K A191669 nonn,tabl
%O A191669 1,2
%A A191669 _Clark Kimberling_, Jun 11 2011