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A054073 Fractal sequence induced by sqrt(2): for k >= 1 let p(k) be the permutation of 1,2,...,k obtained by ordering the fractional parts {h*sqrt(2)} for h=1,2,...,k; then juxtapose p(1),p(2),p(3),...

%I A054073 #21 Aug 28 2019 13:41:03
%S A054073 1,1,2,3,1,2,3,1,4,2,5,3,1,4,2,5,3,1,6,4,2,5,3,1,6,4,2,7,5,3,8,1,6,4,
%T A054073 2,7,5,3,8,1,6,4,9,2,7,5,10,3,8,1,6,4,9,2,7,5,10,3,8,1,6,11,4,9,2,7,5,
%U A054073 10,3,8,1,6,11,4,9,2,7,12,5,10,3,8,13,1,6,11
%N A054073 Fractal sequence induced by sqrt(2): for k >= 1 let p(k) be the permutation of 1,2,...,k obtained by ordering the fractional parts {h*sqrt(2)} for h=1,2,...,k; then juxtapose p(1),p(2),p(3),...
%C A054073 A054073 generates the interspersion A054077; see A194832 and the Mathematica program.
%H A054073 G. C. Greubel, <a href="/A054073/a054073.txt">Table of n, a(n) for n = 1..5000</a>
%e A054073 p(1)=(1); p(2)=(1,2); p(3)=(3,1,2); p(4)=(3,1,4,2).
%e A054073 When formatted as a triangle, the first 9 rows:
%e A054073 1
%e A054073 1 2
%e A054073 3 1 2
%e A054073 3 1 4 2
%e A054073 5 3 1 4 2
%e A054073 5 3 1 6 4 2
%e A054073 5 3 1 6 4 2 7
%e A054073 5 3 8 1 6 4 2 7
%e A054073 5 3 8 1 6 4 9 2 7
%t A054073 r = Sqrt[2];
%t A054073 t[n_] := Table[FractionalPart[k*r], {k, 1, n}];
%t A054073 f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@ Sort[t[n], Less]],
%t A054073 {n, 1, 20}]] (* A054073 *)
%t A054073 TableForm[Table[Flatten[(Position[t[n], #1] &) /@ Sort[t[n], Less]], {n, 1, 15}]]
%t A054073 row[n_] := Position[f, n];
%t A054073 u = TableForm[Table[row[n], {n, 1, 20}]]
%t A054073 g[n_, k_] := Part[row[n], k];
%t A054073 p = Flatten[Table[g[k, n - k + 1], {n, 1, 13},
%t A054073 {k, 1, n}]] (* A054077 *)
%t A054073 q[n_] := Position[p, n]; Flatten[
%t A054073 Table[q[n], {n, 1, 80}]]  (* A054076 *)
%t A054073 (* _Clark Kimberling_, Sep 03 2011 *)
%Y A054073 Cf. A054071, A054072, A194832.
%K A054073 nonn
%O A054073 1,3
%A A054073 _Clark Kimberling_