

A191702


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


30



1, 5, 2, 25, 10, 3, 125, 50, 15, 4, 625, 250, 75, 20, 6, 3125, 1250, 375, 100, 30, 7, 15625, 6250, 1875, 500, 150, 35, 8, 78125, 31250, 9375, 2500, 750, 175, 40, 9, 390625, 156250, 46875, 12500, 3750, 875, 200, 45, 11, 1953125, 781250, 234375, 62500, 18750
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OFFSET

1,2


COMMENTS

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.
...
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:
...
A191702=dispersion of A008587 (5k, k>=1)
A191703=dispersion of A016861 (5k+1, k>=1)
A191704=dispersion of A016873 (5k+2, k>=0)
A191705=dispersion of A016885 (5k+3, k>=0)
A191706=dispersion of A016897 (5k+4, k>=0)
A191707=dispersion of A047201 (1, 2, 3, 4 mod 5 and >1)
A191708=dispersion of A047202 (0, 2, 3, 4 mod 5 and >1)
A191709=dispersion of A047207 (0, 1, 3, 4 mod 5 and >1)
A191710=dispersion of A032763 (0, 1, 2, 4 mod 5 and >1)
A191711=dispersion of A001068 (0, 1, 2, 3 mod 5 and >1)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191702 has 1st col A047201, all else A008587
A191703 has 1st col A047202, all else A016861
A191704 has 1st col A047207, all else A016873
A191705 has 1st col A032763, all else A016885
A191706 has 1st col A001068, all else A016897
A191707 has 1st col A008587, all else A047201
A191708 has 1st col A042968, all else A047203
A191709 has 1st col A042968, all else A047207
A191710 has 1st col A042968, all else A032763
A191711 has 1st col A042968, all else A001068
...
Regarding the dispersions A191670A191673, there is a formula for sequences of the type
"(a or b or c or d mod m)", used in Mathematica programs for A191707A191711): if f(n)=(n mod 3), then
(a,b,c,d,a,b,c,d,a,b,c,d,...) is given by
a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n), so that for n>=1,
"(a, b, c, d mod m)" is given by
a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n)+m*floor((n1)/4)).


LINKS

Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)


FORMULA

T(i,j) = T(i,1)*T(1,j) = (i1+floor((i+3)/4)*5^(j1), i>=1, j>=1.


EXAMPLE

Northwest corner:
1...5....25....125...625
2...10...50....250...1250
3...15...75....375...1875
4...20...100...500...2500
6...30...150...750...3750


MATHEMATICA

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12; c = 40; c1 = 12;
f[n_] := 5n
Table[f[n], {n, 1, 30}] (* A008587 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191702 *)
Flatten[Table[t[k, n  k + 1], {n, 1, c1}, {k, 1, n}]] (* A191702 *)


CROSSREFS

Cf. A047201, A008587, A191707, A191426.
Sequence in context: A230416 A034079 A090882 * A164309 A104064 A038244
Adjacent sequences: A191699 A191700 A191701 * A191703 A191704 A191705


KEYWORD

nonn,tabl


AUTHOR

Clark Kimberling, Jun 12 2011


STATUS

approved



