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A191536 Dispersion of (3+floor(n*sqrt(2))), by antidiagonals.

%I A191536 #16 Oct 21 2024 00:55:41
%S A191536 1,4,2,8,5,3,14,10,7,6,22,17,12,11,9,34,27,19,18,15,13,51,41,29,28,24,
%T A191536 21,16,75,60,44,42,36,32,25,20,109,87,65,62,53,48,38,31,23,157,126,94,
%U A191536 90,77,70,56,46,35,26,225,181,135,130,111,101,82,68,52,39
%N A191536 Dispersion of (3+floor(n*sqrt(2))), by antidiagonals.
%C A191536 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 A191536 (1) s=A000040 (the primes), D=A114537, u=A114538.
%C A191536 (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
%C A191536 (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
%C A191536 More recent examples of dispersions: A191426-A191455 and A191536-A191545.
%H A191536 G. C. Greubel, <a href="/A191536/b191536.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%e A191536 Northwest corner:
%e A191536   1...4....8....14...22
%e A191536   2...5....10...17...27
%e A191536   3...7....12...19...29
%e A191536   6...11...18...28...42
%e A191536   9...15...24...36...54
%t A191536 (* Program generates the dispersion array T of the increasing sequence f[n] *)
%t A191536 r=40; r1=12; c=40; c1=12; f[n_] :=3+Floor[n*Sqrt[2]]   (* complement of column 1 *)
%t A191536 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191536 rows = {NestList[f, 1, c]};
%t A191536 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191536 t[i_, j_] := rows[[i, j]];
%t A191536 TableForm[Table[t[i, j], {i, 1, r1}, {j, 1, c1}]]
%t A191536 (* A191536 array *)
%t A191536 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191536 sequence *)
%t A191536 (* Program by _Peter J. C. Moses_, Jun 01 2011 *)
%Y A191536 Cf. A114537, A035513, A035506.
%K A191536 nonn,tabl
%O A191536 1,2
%A A191536 _Clark Kimberling_, Jun 06 2011