|
%I #11 Oct 18 2017 05:07:00
%S 1,2,5,3,7,9,4,10,12,13,6,14,16,18,17,8,19,22,24,23,21,11,26,30,32,31,
%T 28,25,15,35,40,43,42,38,34,29,20,47,54,58,56,51,46,39,33,27,63,72,78,
%U 75,68,62,52,44,37,36,84,96,104,100,91,83,70,59,50,41
%N Dispersion of A004772 (>1 and congruent to 0 or 2 or 3 mod 4), by antidiagonals.
%C For a background discussion of dispersions, see A191426.
%C ...
%C 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 ...
%C A191452=dispersion of A008586 (4k, k>=1)
%C A191667=dispersion of A016813 (4k+1, k>=1)
%C A191668=dispersion of A016825 (4k+2, k>=0)
%C A191669=dispersion of A004767 (4k+3, k>=0)
%C A191670=dispersion of A042968 (1 or 2 or 3 mod 4 and >=2)
%C A191671=dispersion of A004772 (0 or 1 or 3 mod 4 and >=2)
%C A191672=dispersion of A004773 (0 or 1 or 2 mod 4 and >=2)
%C A191673=dispersion of A004773 (0 or 2 or 3 mod 4 and >=2)
%C ...
%C EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
%C A191452 has 1st col A042968, all else A008486
%C A191667 has 1st col A004772, all else A016813
%C A191668 has 1st col A042965, all else A016825
%C A191669 has 1st col A004773, all else A004767
%C A191670 has 1st col A008486, all else A042968
%C A191671 has 1st col A016813, all else A004772
%C A191672 has 1st col A016825, all else A042965
%C A191673 has 1st col A004767, all else A004773
%C ...
%C 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 If f(n)=(n mod 3), then (a,b,c,a,b,c,a,b,c,...) is given by
%C a*f(n+2)+b*f(n+1)+c*f(n), so that "(a or b or c mod m)" is given by
%C a*f(n+2)+b*f(n+1)+c*f(n)+m*floor((n-1)/3)), for n>=1.
%H Ivan Neretin, <a href="/A191671/b191671.txt">Table of n, a(n) for n = 1..5050</a> (first 100 antidiagonals, flattened)
%e Northwest corner:
%e 1....2....3....4....6
%e 5....7....10...14...19
%e 9....12...16...22...30
%e 13...18...24...32...43
%e 17...23...31...42...56
%t (* Program generates the dispersion array T of the increasing sequence f[n] *)
%t r = 40; r1 = 12; c = 40; c1 = 12;
%t a = 2; b = 3; c2 = 4; m[n_] := If[Mod[n, 3] == 0, 1, 0];
%t f[n_] := a*m[n + 2] + b*m[n + 1] + c2*m[n] + 4*Floor[(n - 1)/3]
%t Table[f[n], {n, 1, 30}] (* A004772 *)
%t mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t rows = {NestList[f, 1, c]};
%t Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t t[i_, j_] := rows[[i, j]];
%t TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191671 *)
%t Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191671 *)
%Y Cf. A016813, A004772, A191667, A191452.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Jun 11 2011
|
