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A191431 Dispersion of ([n*x+x]), where x=sqrt(2) and [ ]=floor, by antidiagonals.

%I A191431 #14 Oct 20 2024 21:02:51
%S A191431 1,2,3,4,5,6,7,8,9,10,11,12,14,15,13,16,18,21,22,19,17,24,26,31,32,28,
%T A191431 25,20,35,38,45,46,41,36,29,23,50,55,65,66,59,52,42,33,27,72,79,93,94,
%U A191431 84,74,60,48,39,30,103,113,132,134,120,106,86,69,56,43,34,147,161,188,190,171,151,123,98,80,62,49,37
%N A191431 Dispersion of ([n*x+x]), where x=sqrt(2) and [ ]=floor, by antidiagonals.
%C A191431 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 A191431 (1) s=A000040 (the primes), D=A114537, u=A114538.
%C A191431 (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
%C A191431 (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
%C A191431 More recent examples of dispersions: A191426-A191455.
%e A191431 Northwest corner:
%e A191431   1.....2....4....7...11...16
%e A191431   3.....5....8...12...18...26
%e A191431   6.....9...14...21...31...45
%e A191431   10...15...22...32...46...66
%e A191431   13...19...28...41...59...84
%t A191431 (* Program generates the dispersion array T of increasing sequence f[n] *)
%t A191431 r = 40; r1 = 12; (* r=# rows of T, r1=# rows to show *)
%t A191431 c = 40; c1 = 12; (* c=# cols of T, c1=# cols to show *)
%t A191431 x = Sqrt[2];
%t A191431 f[n_] := Floor[n*x + x] (* f(n) is complement of column 1 *)
%t A191431 mex[list_] :=
%t A191431 NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1,
%t A191431   Length[Union[list]]]
%t A191431 rows = {NestList[f, 1, c]};
%t A191431 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191431 t[i_, j_] := rows[[i, j]];
%t A191431 TableForm[
%t A191431 Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
%t A191431 (* A191431 array *)
%t A191431 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]]
%t A191431 (* A191431 sequence *)
%t A191431 (* Program by _Peter J. C. Moses_, Jun 01 2011 *)
%Y A191431 Cf. A114537, A035513, A035506.
%K A191431 nonn,tabl
%O A191431 1,2
%A A191431 _Clark Kimberling_, Jun 03 2011