A054073 - OEIS

%I #21 Aug 28 2019 13:41:03

%S 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 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 10,3,8,1,6,11,4,9,2,7,12,5,10,3,8,13,1,6,11

%N 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 generates the interspersion A054077; see A194832 and the Mathematica program.

%H G. C. Greubel, <a href="/A054073/a054073.txt">Table of n, a(n) for n = 1..5000</a>

%e p(1)=(1); p(2)=(1,2); p(3)=(3,1,2); p(4)=(3,1,4,2).

%e When formatted as a triangle, the first 9 rows:

%e 1

%e 1 2

%e 3 1 2

%e 3 1 4 2

%e 5 3 1 4 2

%e 5 3 1 6 4 2

%e 5 3 1 6 4 2 7

%e 5 3 8 1 6 4 2 7

%e 5 3 8 1 6 4 9 2 7

%t r = Sqrt[2];

%t t[n_] := Table[FractionalPart[k*r], {k, 1, n}];

%t f = Flatten[Table[Flatten[(Position[t[n], #1] &) /@ Sort[t[n], Less]],

%t {n, 1, 20}]] (* A054073 *)

%t TableForm[Table[Flatten[(Position[t[n], #1] &) /@ Sort[t[n], Less]], {n, 1, 15}]]

%t row[n_] := Position[f, n];

%t u = TableForm[Table[row[n], {n, 1, 20}]]

%t g[n_, k_] := Part[row[n], k];

%t p = Flatten[Table[g[k, n - k + 1], {n, 1, 13},

%t {k, 1, n}]] (* A054077 *)

%t q[n_] := Position[p, n]; Flatten[

%t Table[q[n], {n, 1, 80}]]  (* A054076 *)

%t (* _Clark Kimberling_, Sep 03 2011 *)

%Y Cf. A054071, A054072, A194832.

%K nonn

%O 1,3

%A _Clark Kimberling_