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%I #10 Feb 14 2014 00:29:48
%S 1,3,2,8,5,4,21,13,11,6,55,34,29,16,7,144,89,76,42,18,9,377,233,199,
%T 110,47,24,10,987,610,521,288,123,63,26,12,2584,1597,1364,754,322,165,
%U 68,32,14,6765,4181,3571,1974,843,432,178,84,37,15,17711,10946
%N Dispersion of ([nx+n+x-2]), where x=(golden ratio) and [ ]=floor, by antidiagonals.
%C Rows 1 and 2: Fibonacci numbers. Rows 3 and 5: Lucas numbers. Row n satisfies the recurrence x(n)=3*x(n-1)-x(n-2).
%C 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 (1) s=A000040 (the primes), D=A114537, u=A114538.
%C (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
%C (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
%C More recent examples of dispersions: A191426-A191455.
%e Northwest corner:
%e 1....3....8....21...55
%e 2....5....13...34...89
%e 4....11...29...76...199
%e 6....16...42...110..288
%e 7....18...47...123..322
%t (* Program generates the dispersion array T of increasing sequence f[n] *)
%t r = 40; r1 = 12; c = 40; c1 = 12; x = GoldenRatio;
%t f[n_] := Floor[n*x+n+x-2] (* complement of column 1 *)
%t mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t rows = {NestList[f, 1, c]};
%t Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t t[i_, j_] := rows[[i, j]];
%t TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
%t (* A191437 array *)
%t Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191437 sequence *)
%t (* Program by _Peter J. C. Moses_, Jun 01 2011 *)
%Y Cf. A114537, A035513, A035506, A000045, A000032, A191426.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Jun 04 2011
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