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A191702 Dispersion of A008587 (5,10,15,20,25,30,...), by antidiagonals.

%I A191702 #16 Oct 21 2024 01:19:57
%S A191702 1,5,2,25,10,3,125,50,15,4,625,250,75,20,6,3125,1250,375,100,30,7,
%T A191702 15625,6250,1875,500,150,35,8,78125,31250,9375,2500,750,175,40,9,
%U A191702 390625,156250,46875,12500,3750,875,200,45,11,1953125,781250,234375,62500,18750
%N A191702 Dispersion of A008587 (5,10,15,20,25,30,...), by antidiagonals.
%C A191702 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 A191702 ...
%C A191702 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 A191702 ...
%C A191702 A191702=dispersion of A008587 (5k, k>=1)
%C A191702 A191703=dispersion of A016861 (5k+1, k>=1)
%C A191702 A191704=dispersion of A016873 (5k+2, k>=0)
%C A191702 A191705=dispersion of A016885 (5k+3, k>=0)
%C A191702 A191706=dispersion of A016897 (5k+4, k>=0)
%C A191702 A191707=dispersion of A047201 (1, 2, 3, 4 mod 5 and >1)
%C A191702 A191708=dispersion of A047202 (0, 2, 3, 4 mod 5 and >1)
%C A191702 A191709=dispersion of A047207 (0, 1, 3, 4 mod 5 and >1)
%C A191702 A191710=dispersion of A032763 (0, 1, 2, 4 mod 5 and >1)
%C A191702 A191711=dispersion of A001068 (0, 1, 2, 3 mod 5 and >1)
%C A191702 ...
%C A191702 EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
%C A191702 A191702 has 1st col A047201, all else A008587
%C A191702 A191703 has 1st col A047202, all else A016861
%C A191702 A191704 has 1st col A047207, all else A016873
%C A191702 A191705 has 1st col A032763, all else A016885
%C A191702 A191706 has 1st col A001068, all else A016897
%C A191702 A191707 has 1st col A008587, all else A047201
%C A191702 A191708 has 1st col A042968, all else A047203
%C A191702 A191709 has 1st col A042968, all else A047207
%C A191702 A191710 has 1st col A042968, all else A032763
%C A191702 A191711 has 1st col A042968, all else A001068
%C A191702 ...
%C A191702 Regarding the dispersions A191670-A191673, there is a formula for sequences of the type
%C A191702 "(a or b or c or d mod m)", used in Mathematica programs for A191707-A191711):  if f(n)=(n mod 3), then
%C A191702   (a,b,c,d,a,b,c,d,a,b,c,d,...) is given by
%C A191702    a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n), so that for n>=1,
%C A191702 "(a, b, c, d mod m)" is given by
%C A191702    a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n)+m*floor((n-1)/4)).
%H A191702 Ivan Neretin, <a href="/A191702/b191702.txt">Table of n, a(n) for n = 1..5050</a> (first 100 antidiagonals, flattened)
%F A191702 T(i,j) = T(i,1)*T(1,j) = (i-1+floor((i+3)/4))*5^(j-1), i>=1, j>=1.
%e A191702 Northwest corner:
%e A191702   1...5....25....125...625
%e A191702   2...10...50....250...1250
%e A191702   3...15...75....375...1875
%e A191702   4...20...100...500...2500
%e A191702   6...30...150...750...3750
%t A191702 (* Program generates the dispersion array T of the increasing sequence f[n] *)
%t A191702 r = 40; r1 = 12;  c = 40; c1 = 12;
%t A191702 f[n_] := 5n
%t A191702 Table[f[n], {n, 1, 30}]  (* A008587 *)
%t A191702 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191702 rows = {NestList[f, 1, c]};
%t A191702 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191702 t[i_, j_] := rows[[i, j]];
%t A191702 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
%t A191702 (* A191702 *)
%t A191702 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191702  *)
%Y A191702 Cf. A047201, A008587, A191707, A191426.
%K A191702 nonn,tabl,easy
%O A191702 1,2
%A A191702 _Clark Kimberling_, Jun 12 2011