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A191441 Dispersion of ([n*x+n+x]), where x=sqrt(2) and [ ]=floor, by antidiagonals.

%I A191441 #12 Oct 21 2024 00:49:36
%S A191441 1,4,2,12,7,3,31,19,9,5,77,48,24,14,6,188,118,60,36,16,8,456,287,147,
%T A191441 89,41,21,10,1103,695,357,217,101,53,26,11,2665,1680,864,526,246,130,
%U A191441 65,28,13,6436,4058,2088,1272,596,316,159,70,33,15,15540,9799,5043
%N A191441 Dispersion of ([n*x+n+x]), where x=sqrt(2) and [ ]=floor, by antidiagonals.
%C A191441 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 A191441 (1) s=A000040 (the primes), D=A114537, u=A114538.
%C A191441 (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
%C A191441 (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
%C A191441 More recent examples of dispersions: A191426-A191455.
%e A191441 Northwest corner:
%e A191441   1....4....12...31...77
%e A191441   2....7....19...48...118
%e A191441   3....9....24...60...147
%e A191441   5....14...36...89...217
%e A191441   6....16...41...101..246
%t A191441 (* Program generates the dispersion array T of increasing sequence f[n] *)
%t A191441 r=40; r1=12; c=40; c1=12;  x = Sqr[2];
%t A191441 f[n_] := Floor[n*x+n+x] (* complement of column 1 *)
%t A191441 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191441 rows = {NestList[f, 1, c]};
%t A191441 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191441 t[i_, j_] := rows[[i, j]];
%t A191441 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
%t A191441 (* A191441 array *)
%t A191441 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191441 sequence *)
%t A191441 (* Program by _Peter J. C. Moses_, Jun 01 2011 *)
%Y A191441 Cf. A114537, A035513, A035506, A191438.
%K A191441 nonn,tabl
%O A191441 1,2
%A A191441 _Clark Kimberling_, Jun 04 2011