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A191539 Dispersion of (5*n-floor(n*sqrt(5))), by antidiagonals.

%I A191539 #11 Oct 21 2024 00:57:43
%S A191539 1,3,2,9,6,4,25,17,12,5,70,47,34,14,7,194,130,94,39,20,8,537,360,260,
%T A191539 108,56,23,10,1485,996,719,299,155,64,28,11,4105,2753,1988,827,429,
%U A191539 177,78,31,13,11346,7610,5495,2286,1186,490,216,86,36,15,31360,21034
%N A191539 Dispersion of (5*n-floor(n*sqrt(5))), by antidiagonals.
%C A191539 Background discussion:  Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1.  The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n))), s(s(s(t(n)))), ...).  Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers.  The sequence u given by u(n)=(number of the row of D that contains n) is a fractal sequence.  Examples:
%C A191539 (1) s=A000040 (the primes), D=A114537, u=A114538.
%C A191539 (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
%C A191539 (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
%C A191539 More recent examples of dispersions: A191426-A191455 and A191536-A191545.
%H A191539 G. C. Greubel, <a href="/A191539/b191539.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%e A191539 Northwest corner:
%e A191539   1...3....9....25...70
%e A191539   2...6....17...47...130
%e A191539   4...12...34...94...260
%e A191539   5...14...39...108..299
%e A191539   7...20...56...155..429
%t A191539 (* Program generates the dispersion array T of the increasing sequence f[n] *)
%t A191539 r=40; r1=12; c=40; c1=12; f[n_] :=5n-Floor[n*Sqrt[5]]   (* complement of column 1 *)
%t A191539 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191539 rows = {NestList[f, 1, c]};
%t A191539 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191539 t[i_, j_] := rows[[i, j]];
%t A191539 TableForm[Table[t[i, j], {i, 1, r1}, {j, 1, c1}]]  (* A191539 array *)
%t A191539 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191539 sequence *)
%Y A191539 Cf. A114537, A035513, A035506.
%K A191539 nonn,tabl
%O A191539 1,2
%A A191539 _Clark Kimberling_, Jun 06 2011