A237126 - OEIS

A237126

a(0)=0, a(1) = 1, a(2n) = nonludic(a(n)), a(2n+1) = ludic(a(n)+1), where ludic = A003309, nonludic = A192607.

0, 1, 4, 2, 9, 7, 6, 3, 16, 25, 14, 17, 12, 13, 8, 5, 26, 61, 36, 115, 22, 47, 27, 67, 20, 41, 21, 43, 15, 23, 10, 11, 38, 119, 81, 359, 51, 179, 146, 791, 33, 91, 64, 247, 39, 121, 88, 407, 31, 83, 57, 221, 32, 89, 59, 227, 24, 53, 34, 97, 18, 29, 19, 37, 54

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OFFSET

0,3

COMMENTS

Shares with permutation A237056 the property that the other bisection consists of only ludic numbers and the other bisection of only nonludic numbers. However, instead of placing terms in those subsets in monotone order this sequence recursively permutes the order of both subsets with the emerging permutation itself, so this is a kind of "deep" variant of A237056.

Alternatively, this can be viewed as yet another "entanglement permutation", where two pairs of complementary subsets of natural numbers are entangled with each other. In this case a complementary pair odd/even numbers (A005408/A005843) is entangled with a complementary pair ludic/nonludic numbers (A003309/A192607).

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..574

Index entries for sequences that are permutations of the natural numbers

FORMULA

a(0)=0, a(1) = 1, a(2n) = nonludic(a(n)), a(2n+1) = ludic(a(n)+1), where ludic = A003309, nonludic = A192607.

EXAMPLE

a(2) = a(2*1) = nonludic(a(1)) = A192607(1) = 4.

a(3) = a(2*1+1) = ludic(a(1)+1) = A003309(1+1) = A003309(2) = 2.

a(4) = a(2*2) = nonludic(a(2)) = A192607(4) = 9.

a(5) = a(2*2+1) = ludic(a(2)+1) = A003309(4+1) = A003309(5) = 7.

MATHEMATICA

nmax = 64;

T = Range[2, 20 nmax];

L = {1};

While[Length[T] > 0, With[{k = First[T]},

AppendTo[L, k]; T = Drop[T, {1, -1, k}]]];

nonL = Complement[Range[Last[L]], L];

a[n_] := a[n] = Which[

n < 2, n,

EvenQ[n] && a[n/2] <= Length[nonL], nonL[[a[n/2]]],

OddQ[n] && a[(n-1)/2]+1 <= Length[L], L[[a[(n-1)/2]+1]],

True, Print[" error: n = ", n, " size of T should be increased"]];

Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Oct 10 2021, after Ray Chandler in A003309 *)

PROG

(Haskell)

import Data.List (transpose)

a237126 n = a237126_list !! n

a237126_list = 0 : es where

es = 1 : concat (transpose [map a192607 es, map (a003309 . (+ 1)) es])

-- Reinhard Zumkeller, Feb 10 2014, Feb 06 2014

(Scheme, with Antti Karttunen's IntSeq-library for memoizing definec-macro)

(definec (A237126 n) (cond ((< n 2) n) ((even? n) (A192607 (A237126 (/ n 2)))) (else (A003309 (+ 1 (A237126 (/ (- n 1) 2))))))) ;; Antti Karttunen, Feb 07 2014

CROSSREFS

Cf. A237427 (inverse), A237056, A235491.

Similarly constructed permutations: A227413/A135141.

Sequence in context: A257730 A246378 A260422 * A246380 A200639 A365255

Adjacent sequences: A237123 A237124 A237125 * A237127 A237128 A237129

KEYWORD

nonn

AUTHOR

Antti Karttunen and Reinhard Zumkeller, Feb 06 2014

STATUS

approved