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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A117120 a(1)=1. a(n) is smallest positive integer not occurring earlier in the sequence where a(n) is congruent to -1 (mod a(n-1)).

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 6, 11, 10, 9, 8, 15, 14, 13, 12, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 95, 94, 93, 92, 91, 90, 89, 88, 87
Offset: 1

Views

Author

Leroy Quet, Apr 19 2006

Keywords

Comments

Sequence is a permutation of the positive integers.
The permutation is self-inverse. Except for fixed points 1, 2, 3 it consists completely of 2-cycles: (4,5), (6,7), (8,11), (9,10), (12,15), (13,14), (16,23), (17,22), ..., (24,31), ..., (32,47), ... . - Klaus Brockhaus
The permutation transforms enumeration system of positive irreducible fractions A071766/A229742 (HCS) into enumeration system A245325/A245326, and vice versa. - Yosu Yurramendi, Jun 09 2015
A092569(a(n)) = a(A092569(n)), n > 0.
A258746(a(n)) = a(A258746(n)), n > 0.
A258996(a(n)) = a(A258996(n)), n > 0.
A054429(a(n)) = a(A054429(n)), n > 0.
a(n) = A054429(A063946(n)) = A063946(A054429(n)), n > 0. - Yosu Yurramendi, Mar 23 2017

Links

Crossrefs

Cf. A071766, A229742, A245325, A245326.

Programs

  • Maple
    A[1]:= 1: A[2]:= 2: B[1]:= 0: B[2]:= 0:
for n from 3 to 100 do
  for m from A[n-1]-1 by A[n-1] while assigned(B[m]) do od:
  A[n]:= m;
  B[m]:= 0;
od:
seq(A[n],n=1..100); # Robert Israel, Jun 09 2015
  • Mathematica
    f[n_] := Block[{a = {1}, i, k}, Do[k = 1; While[Or[Mod[k, a[[i - 1]]] != a[[i - 1]] - 1, MemberQ[a, k]], k++]; AppendTo[a, k], {i, 2, n}]; a]; f@ 120 (* Michael De Vlieger, Jun 11 2015 *)
A[n_]:= If[n<4, n, If[EvenQ[n], 2A[n/2] + 1, 2A[(n - 1)/2]]]; Table[A[n], {n, 100}] (* Indranil Ghosh, Mar 21 2017 *)
f[lst_List] := Block[{k = 2, m = lst[[-1]]}, While[ MemberQ[lst, k] || 1 + Mod[k, m] != m, k++]; Append[lst, k]]; Nest[f, {1}, 70] (* Robert G. Wilson v, Jan 22 2018 *)
  • PARI
    A(n) = if(n<4, n, if(n%2, 2*A(n\2), 2*A(n/2)+1));
for(n=1, 50, print1(A(n), ", ")) \\ Indranil Ghosh, Mar 21 2017
  • R
    a <- 1:3 # If it were c(1, 3, 2), it would be A054429
maxn <- 50 # by choice
#
for(n in 2:maxn){
  a[2*n  ] <- 2*a[n]+1
  a[2*n+1] <- 2*a[n]
}
#
a
# Yosu Yurramendi, Jun 08 2015

Formula

For n >= 2: If a(n-1) = 2^m, m=positive integer, then a(n)= 2^(m+1)-1. If a(n-1) = 3*2^m, m= nonnegative integer, then a(n) = 3*2^(m+1)-1. Otherwise, a(n) = a(n-1) -1.
For n >= 2: a(2*n) = 2*a(n)+1, a(2*n+1) = 2*a(n). - Yosu Yurramendi, Jun 08 2015

Extensions

More terms from Klaus Brockhaus

A233280 Permutation of nonnegative integers: a(n) = A003188(A054429(n)).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 6, 8, 9, 11, 10, 14, 15, 13, 12, 16, 17, 19, 18, 22, 23, 21, 20, 28, 29, 31, 30, 26, 27, 25, 24, 32, 33, 35, 34, 38, 39, 37, 36, 44, 45, 47, 46, 42, 43, 41, 40, 56, 57, 59, 58, 62, 63, 61, 60, 52, 53, 55, 54, 50, 51, 49, 48, 64, 65, 67, 66
Offset: 0

Views

Author

Antti Karttunen, Dec 18 2013

Keywords

Comments

This permutation transforms the enumeration system of positive irreducible fractions A071766/A229742 (HCS) into the enumeration system A007305/A047679 (Stern-Brocot), and the enumeration system A245325/A245326 into A162909/A162910 (Bird). - Yosu Yurramendi, Jun 09 2015

Links

Crossrefs

Inverse permutation: A233279.
Cf. A000069, A001969, A003188, A054429, A063946, A154436.
Similarly constructed permutation pairs: A003188/A006068, A135141/A227413, A232751/A232752, A233275/A233276, A233277/A233278, A193231 (self-inverse).

Programs

  • Python
    from sympy import floor
def a003188(n): return n^(n>>1)
def a054429(n): return 1 if n==1 else 2*a054429(floor(n/2)) + 1 - n%2
def a(n): return 0 if n==0 else a003188(a054429(n)) # Indranil Ghosh, Jun 11 2017
  • R
    maxrow <- 8 # by choice
a <- 1
for(m in 0:maxrow) for(k in 0:(2^m-1)){
a[2^(m+1)+    k] <- a[2^m+      k] + 2^m
a[2^(m+1)+2^m+k] <- a[2^(m+1)-1-k] + 2^(m+1)
}
a
# Yosu Yurramendi, Apr 05 2017
  • Scheme
    (define (A233280 n) (A003188 (A054429 n)))
;; Alternative version, based on entangling even & odd numbers with odious and evil numbers:
(definec (A233280 n) (cond ((< n 2) n) ((even? n) (A000069 (+ 1 (A233280 (/ n 2))))) (else (A001969 (+ 1 (A233280 (/ (- n 1) 2)))))))

Formula

a(n) = A003188(A054429(n)).
a(n) = A063946(A003188(n)).
a(n) = A054429(A154436(n)).
a(0)=0, a(1)=1, and otherwise, a(2n) = A000069(1+a(n)), a(2n+1) = A001969(1+a(n)). [A recurrence based on entangling even & odd numbers with odious and evil numbers]
a(n) = A258746(A180201(n)) = A180201(A117120(n)), n > 0. - Yosu Yurramendi, Apr 10 2017

A165199 a(n) is obtained by flipping every second bit in the binary representation of n starting at the second-most significant bit and on downwards.

Original entry on oeis.org

0, 1, 3, 2, 6, 7, 4, 5, 13, 12, 15, 14, 9, 8, 11, 10, 26, 27, 24, 25, 30, 31, 28, 29, 18, 19, 16, 17, 22, 23, 20, 21, 53, 52, 55, 54, 49, 48, 51, 50, 61, 60, 63, 62, 57, 56, 59, 58, 37, 36, 39, 38, 33, 32, 35, 34, 45, 44, 47, 46, 41, 40, 43, 42, 106, 107, 104, 105, 110, 111, 108
Offset: 0

Views

Author

Leroy Quet, Sep 07 2009

Keywords

Comments

This is a self-inverse permutation of the positive integers.
Old name was: a(0) = 0, and for n>=1, let b(n,m) be the m-th digit, reading left to right, of binary n. (b(n, 1) is the most significant binary digit, which is 1.) Then a(n) is such that b(a(n),1)=1; and if b(n,m)=b(n,m-1) then b(a(n),m) does not = b(a(n),m-1); and if b(n,m) does not = b(n,m-1) then b(a(n), m) = b(a(n),m-1), for all m where 2 <= m <= number binary digits in n.
From Emeric Deutsch, Oct 06 2020: (Start)
a(n) is the index of the composition that is the conjugate of the composition with index n.
The index of a composition is defined to be the positive integer whose binary form has run-lengths (i.e., runs of 1's, runs of 0's, etc. from left to right) equal to the parts of the composition. Example: the composition 1,1,3,1 has index 46 since the binary form of 46 is 101110.
a(18) = 24. Indeed, since the binary form of 18 is 10010, the composition with index 18 is 1,2,1,1 (the run-lengths of 10010); the conjugate of 1,2,1,1 is 2,3 and so the binary form of a(18) is 11000; consequently, a(18) = 24. (End)

Examples

			a(12) = 9 because 12 = 1100_2 and 1100_2 XOR 0101_2 = 1001_2 = 9.

Links

Crossrefs

Cf. A000035, A000523, A075157, A075158, A241909, A243353, A245443, A245444.
{A001477, A129594, A165199, A056539} form a 4-group.

Programs

  • Maple
    a:= n-> Bits[Xor](n, iquo(2^(1+ilog2(n)), 3)):
seq(a(n), n=0..100);  # Alois P. Heinz, Oct 07 2020
  • PARI
    for(k=0,67,my(b(n)=vector(#digits(n,2),i,!(i%2)));print1(bitxor(k,fromdigits(b(k),2)),", ")) \\ Hugo Pfoertner, Oct 07 2020
  • PARI
    a(n) = if(n, bitxor(n,2<Kevin Ryde, Oct 07 2020
  • R
    maxrow <- 8 # by choice
a <- 1
for(m in 0: maxrow) for(k in 0:(2^m-1)){
a[2^(m+1) +       k] = a[2^(m+1) - 1 - k] + 2^(m+1)
a[2^(m+1) + 2^m + k] = a[2^(m+1) - 1 - k] + 2^m
}
(a <- c(0, a))
# Yosu Yurramendi, Apr 04 2017

Formula

From Antti Karttunen, Jul 22 2014: (Start)
a(0) = 0, and for n >= 1, a(n) = 2*a(floor(n/2)) + A000035(n+A000523(n)).
As a composition of related permutations:
a(n) = A056539(A129594(n)) = A129594(A056539(n)).
a(n) = A245443(A193231(n)) = A193231(A245444(n)).
a(n) = A075158(A243353(n)-1) = A075158((A241909(1+A075157(n))) - 1).
(End)
a(n) = A258746(A054429(n)) = A054429(A258746(n)), n > 0. - Yosu Yurramendi, Mar 29 2017

Extensions

Extended by Ray Chandler, Sep 10 2009
a(0) = 0 prepended by Antti Karttunen, Jul 22 2014
New name from Kevin Ryde, Oct 07 2020

A180201 Inverse permutation to A180200.

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 6, 7, 11, 10, 8, 9, 13, 12, 14, 15, 23, 22, 20, 21, 17, 16, 18, 19, 27, 26, 24, 25, 29, 28, 30, 31, 47, 46, 44, 45, 41, 40, 42, 43, 35, 34, 32, 33, 37, 36, 38, 39, 55, 54, 52, 53, 49, 48, 50, 51, 59, 58, 56, 57, 61, 60, 62, 63, 95, 94, 92, 93, 89, 88, 90, 91, 83
Offset: 0

Views

Author

Reinhard Zumkeller, Aug 15 2010

Keywords

Comments

A180199(n) = a(a(n));
a(A180198(n)) = A180198(a(n)) = A180200(n);
a(A075427(n)) = A075427(n).
This permutation transforms the enumeration system of positive irreducible fractions A245325/A245326 into the enumeration system A007305/A047679 (Stern-Brocot), and enumeration system A071766/A229742 (HCS) into A162909/A162910 (Bird). - Yosu Yurramendi, Jun 09 2015

Links

Programs

  • R
    #
maxn <- 63 # by choice
a <- 1
for(n in 1:maxn){
a[2*n  ] <- 2*a[n] + (n%%2 == 0)
a[2*n+1] <- 2*a[n] + (n%%2 != 0)}
a <- c(0, a)
# Yosu Yurramendi, May 23 2020

Formula

a(n) = A233280(A258746(n)) = A117120(A233280(n)), n > 0. - Yosu Yurramendi, Apr 10 2017 [Corrected by Yosu Yurramendi, Mar 14 2025]
a(0) = 0, a(1) = 1, for n > 0 a(2*n) = 2*a(n) + [n even], a(2*n + 1) = 2*a(n) + [n odd]. - Yosu Yurramendi, May 23 2020
From Alan Michael Gómez Calderón, Mar 04 2025: (Start)
a(n) = A054429(n) XOR floor(n/2) for n > 0.
a(n) = A054429(A003188(n)) for n > 0. (End)
a(n) = A154436(A054429(n)), n > 0. - Yosu Yurramendi, Mar 11 2025
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