A116625 -id:A116625 - cp's OEIS frontend

A116626 a(1)=1; a(odd n) = a(n-1) XOR a(n-2), for a(even n) we find the first i > 1 such that neither i nor (i XOR A116626(n-1)) is present in A116626(1..n-1), in which case a(n) = (i XOR A116626(n-1)).

1, 3, 2, 6, 4, 12, 8, 13, 5, 15, 10, 26, 16, 23, 7, 14, 9, 24, 17, 49, 32, 43, 11, 25, 18, 51, 33, 50, 19, 48, 35, 55, 20, 54, 34, 52, 22, 62, 40, 61, 21, 60, 41, 53, 28, 56, 36, 63, 27, 91, 64, 93, 29, 59, 38, 57, 31, 58, 37, 100, 65, 95, 30, 92, 66, 101, 39, 99, 68, 110, 42

Offset: 1

Paul D. Hanna and Antti Karttunen, Feb 21 2006

nonn

This is a permutation of the natural numbers provided that A116625 is the complement of A116624. XOR is A003987.

Index entries for sequences that are permutations of the natural numbers

Cf. a(2n) = a(2n-1) XOR a(2n+1), a(2n+1) = A116624(n+1). Inverse: A116627. Bisections: A116624, A116625. Cf. A116648.

A116624 a(1)=1; for n>1, a(n) = least positive integer not appearing earlier such that {a(k) | 1 <= k <= n} and {a(k) XOR a(k-1) | 1 <= k <= n} are disjoint sets of distinct numbers.

Original entry on oeis.org

1, 2, 4, 8, 5, 10, 16, 7, 9, 17, 32, 11, 18, 33, 19, 35, 20, 34, 22, 40, 21, 41, 28, 36, 27, 64, 29, 38, 31, 37, 65, 30, 66, 39, 68, 42, 67, 44, 70, 45, 69, 128, 46, 72, 47, 77, 129, 71, 131, 73, 130, 74, 132, 75, 134, 79, 136, 80, 133, 81, 135, 84, 137, 82, 139, 85

Offset: 1

Paul D. Hanna and Antti Karttunen, Feb 21 2006

nonn

Another way to define this: A116624(1) = 1; A116624(n) = the least positive integer i distinct from any of A116624(1..n-1) and A116625(1..n-2), such that also (i XOR A116624(n-1)) is not present in A116625(1..n-2) nor in A116624(1..n-1).

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. Bisection of A116626. Complement of A116625?
Cf. A003987, A116624, A116628.
Cf. A235262

a = {1}; used = {}; Do[k = 1; While[MemberQ[Join[a, used], k] || MemberQ[Join[a, used], r = BitXor[a[[-1]], k]], k++]; AppendTo[a, k]; AppendTo[used, r], {n, 2, 66}]; a (* Ivan Neretin, Mar 13 2017 *)

A116628 Positions where A116624 is a power of 2.

Original entry on oeis.org

1, 2, 3, 4, 7, 11, 26, 42, 109, 166, 373, 772, 1532, 2930, 6154, 10933, 24184, 44069, 105575, 155528, 374727, 681122, 1630332, 3077586, 6523332

Offset: 1

Paul D. Hanna and Antti Karttunen, Feb 21 2006

Conjecture: all the powers of 2 occur in A116624 (implying that none occur in A116625) and they occur in ascending order, in which case A116624(a(n)) = A000079(n-1). Checked up to n=16.

Rémy Sigrist, C program

Cf. A000079, A116624, A116625.

C
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a(17)-a(25) from Rémy Sigrist, Feb 14 2023

cp's OEIS Frontend

A116626 a(1)=1; a(odd n) = a(n-1) XOR a(n-2), for a(even n) we find the first i > 1 such that neither i nor (i XOR A116626(n-1)) is present in A116626(1..n-1), in which case a(n) = (i XOR A116626(n-1)).

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A116624 a(1)=1; for n>1, a(n) = least positive integer not appearing earlier such that {a(k) | 1 <= k <= n} and {a(k) XOR a(k-1) | 1 <= k <= n} are disjoint sets of distinct numbers.

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A116628 Positions where A116624 is a power of 2.

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