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%I #10 Feb 14 2014 05:46:01
%S 1,2,3,4,6,5,7,10,9,8,12,17,15,14,11,20,28,25,23,19,13,33,46,41,38,31,
%T 22,16,54,75,67,62,51,36,27,18,88,122,109,101,83,59,44,30,21,143,198,
%U 177,164,135,96,72,49,35,24,232,321,287,266,219,156,117,80,57,40,26,376,520,465,431,355,253,190,130,93,65,43,29
%N Interspersion related to the Wythoff Array.
%C The rows satisfy the recurrence r(n)=r(n-1)+r(n-2)+1.
%C Every positive integer occurs exactly once, so that as a sequence, A175004 is a permutation of the natural numbers. As an array, it is an interspersion, hence also a dispersion. Specifically, it is the dispersion of the sequence floor(n*x+2/x), where x=(golden ratio). For a discussion of dispersions, see A191426.
%F Let W'=W-1, where W is the Wythoff array, given by A035513.
%F Row 1 of W' is (0,1,2,4,7,12,...); replace this by (1,2,4,7,12,...).
%F The resulting array is A175004.
%e Corner of the array:
%e 1....2....4....7....12...20... (cf. A000071)
%e 3....6....10...17...28...46... (cf. A001610)
%e 5....9....15...25...41...67... (cf. A001595)
%e 8....14...23...38...62...101..
%t (* program generates the dispersion array T of the complement of increasing sequence f[n] *)
%t r = 40; r1 = 12; (* r=# rows of T, r1=# rows to show *)
%t c = 40; c1 = 12; (* c= # cols of T, c1=# cols to show *)
%t x = GoldenRatio; f[n_] := Floor[n*x + 2/x]
%t (* f(n) is 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 (* A175004 *)
%t Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* array as a sequence *)
%t (* Program by _Peter J. C. Moses_, Jun 01 2011, added here Jun 03 2011 by _Clark Kimberling_ *)
%Y Cf. A035513.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Apr 03 2010
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