A191702 - OEIS

%I #16 Oct 21 2024 01:19:57

%S 1,5,2,25,10,3,125,50,15,4,625,250,75,20,6,3125,1250,375,100,30,7,

%T 15625,6250,1875,500,150,35,8,78125,31250,9375,2500,750,175,40,9,

%U 390625,156250,46875,12500,3750,875,200,45,11,1953125,781250,234375,62500,18750

%N Dispersion of A008587 (5,10,15,20,25,30,...), by antidiagonals.

%C For a background discussion of dispersions and their fractal sequences, see A191426. For dispersions of congruence sequences mod 3 or mod 4, see A191655, A191663, A191667.

%C ...

%C Each of the sequences (5n, n>1), (5n+1, n>1), (5n+2, n>=0), (5n+3, n>=0), (5n+4, n>=0), generates a dispersion. Each complement (beginning with its first term >1) also generates a dispersion. The ten sequences and dispersions are listed here:

%C ...

%C A191702=dispersion of A008587 (5k, k>=1)

%C A191703=dispersion of A016861 (5k+1, k>=1)

%C A191704=dispersion of A016873 (5k+2, k>=0)

%C A191705=dispersion of A016885 (5k+3, k>=0)

%C A191706=dispersion of A016897 (5k+4, k>=0)

%C A191707=dispersion of A047201 (1, 2, 3, 4 mod 5 and >1)

%C A191708=dispersion of A047202 (0, 2, 3, 4 mod 5 and >1)

%C A191709=dispersion of A047207 (0, 1, 3, 4 mod 5 and >1)

%C A191710=dispersion of A032763 (0, 1, 2, 4 mod 5 and >1)

%C A191711=dispersion of A001068 (0, 1, 2, 3 mod 5 and >1)

%C ...

%C EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):

%C A191702 has 1st col A047201, all else A008587

%C A191703 has 1st col A047202, all else A016861

%C A191704 has 1st col A047207, all else A016873

%C A191705 has 1st col A032763, all else A016885

%C A191706 has 1st col A001068, all else A016897

%C A191707 has 1st col A008587, all else A047201

%C A191708 has 1st col A042968, all else A047203

%C A191709 has 1st col A042968, all else A047207

%C A191710 has 1st col A042968, all else A032763

%C A191711 has 1st col A042968, all else A001068

%C ...

%C Regarding the dispersions A191670-A191673, there is a formula for sequences of the type

%C "(a or b or c or d mod m)", used in Mathematica programs for A191707-A191711):  if f(n)=(n mod 3), then

%C   (a,b,c,d,a,b,c,d,a,b,c,d,...) is given by

%C    a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n), so that for n>=1,

%C "(a, b, c, d mod m)" is given by

%C    a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n)+m*floor((n-1)/4)).

%H Ivan Neretin, <a href="/A191702/b191702.txt">Table of n, a(n) for n = 1..5050</a> (first 100 antidiagonals, flattened)

%F T(i,j) = T(i,1)*T(1,j) = (i-1+floor((i+3)/4))*5^(j-1), i>=1, j>=1.

%e Northwest corner:

%e   1...5....25....125...625

%e   2...10...50....250...1250

%e   3...15...75....375...1875

%e   4...20...100...500...2500

%e   6...30...150...750...3750

%t (* Program generates the dispersion array T of the increasing sequence f[n] *)

%t r = 40; r1 = 12;  c = 40; c1 = 12;

%t f[n_] := 5n

%t Table[f[n], {n, 1, 30}]  (* A008587 *)

%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}]]

%t (* A191702 *)

%t Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191702  *)

%Y Cf. A047201, A008587, A191707, A191426.

%K nonn,tabl,easy

%O 1,2

%A _Clark Kimberling_, Jun 12 2011