A088163 -id:A088163 - cp's OEIS frontend

A088161 n rotated one binary place to the right less n rotated one binary place to the left.

0, 0, 0, 0, 1, 3, -2, 0, 3, 9, 0, 6, -3, 3, -6, 0, 7, 21, 4, 18, 1, 15, -2, 12, -5, 9, -8, 6, -11, 3, -14, 0, 15, 45, 12, 42, 9, 39, 6, 36, 3, 33, 0, 30, -3, 27, -6, 24, -9, 21, -12, 18, -15, 15, -18, 12, -21, 9, -24, 6, -27, 3, -30, 0, 31, 93, 28, 90, 25, 87, 22, 84, 19, 81, 16, 78, 13, 75, 10, 72, 7, 69, 4

Offset: 0

Robert G. Wilson v, Sep 13 2003

f(n) is negative about 2/7 of the time to 10^7. f(n) is zero, see A088163.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A006257, A038572, A088163.

f:= proc(n) local a,b;
  if n::even then a:= n/2 else a:= 2^ilog2(n) + (n-1)/2 fi;
  if n = 0 then b:= 0 else b:= 2*n + 1 - 2^(ilog2(n)+1) fi;
  a-b
end proc:
map(f, [$0..200]); # Robert Israel, Mar 03 2025

f[n_] := FromDigits[ RotateRight[ IntegerDigits[n, 2]], 2] - FromDigits[ RotateLeft[ IntegerDigits[n, 2]], 2]; Table[ f[n], {n, 0, 82}]

a(n) = A038572(n) - A006257(n).

A273105 a(n) = A038572(n) + A006257(n), sum of the two numbers obtained by rotating the binary representation of n by one place to the right and to the left.

Original entry on oeis.org

0, 2, 2, 6, 3, 9, 8, 14, 5, 15, 10, 20, 15, 25, 20, 30, 9, 27, 14, 32, 19, 37, 24, 42, 29, 47, 34, 52, 39, 57, 44, 62, 17, 51, 22, 56, 27, 61, 32, 66, 37, 71, 42, 76, 47, 81, 52, 86, 57, 91, 62, 96, 67, 101, 72, 106, 77, 111, 82, 116, 87, 121, 92, 126, 33, 99

Offset: 0

Alex Ratushnyak, May 15 2016

Cf. A006257, A038572, A088161, A088163.

Table[FromDigits[RotateRight@ #, 2] + FromDigits[RotateLeft@ #, 2] &@ IntegerDigits[n, 2], {n, 0, 65}] (* Michael De Vlieger, May 17 2016 *)

print('0', end=',')
for n in range(1,1000):
    BL = len(bin(n))-2
    x = (n>>1) + ((n&1) << (BL-1))   # A038572(n)
    x+= (n*2) - (1<A006257(n)  for n>0
    print(str(x), end=',')

A368427 A permutation related to the Christmas tree pattern map (A367508): a(1) = 1, and for any n > 1, a(n) = A053644(n) + A367562(n-1).

Original entry on oeis.org

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

Offset: 1

Rémy Sigrist, Dec 24 2023

This sequence is a permutation of the positive integers (with inverse A368428):
- the Christmas tree pattern map runs through all finite nonempty binary words,
- by prefixing these words with a 1, we obtain the binary expansions of all integers >= 2,
- hence, with the leading term a(1) = 1, we have a permutation of the positive integers.
Apparently, A088163 \ {0} corresponds to the fixed points.
We can also obtain this sequence by applying the Christmas tree pattern map starting from the chain "1" (instead of "0 1") and converting the resulting binary words to decimal.

			The first terms, alongside their binary expansion and the corresponding word in the Christmas tree pattern map, are:
  n   a(n)  bin(a(n))  Xmas word
  --  ----  ---------  ---------
   1     1          1        N/A
   2     2         10          0
   3     3         11          1
   4     6        110         10
   5     4        100         00
   6     5        101         01
   7     7        111         11
   8    12       1100        100
   9    13       1101        101
  10    10       1010        010
  11    14       1110        110
  12     8       1000        000
  13     9       1001        001
  14    11       1011        011
  15    15       1111        111

Rémy Sigrist, Table of n, a(n) for n = 1..8191
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A053644, A088163, A367508, A367562, A368428 (inverse).

With[{imax=7},Map[FromDigits[#,2]&,Flatten[NestList[Map[Delete[{If[Length[#]>1,Map[#<>"0"&,Rest[#]],Nothing],Join[{#[[1]]<>"0"},Map[#<>"1"&,#]]},0]&],{{"1"}},imax-1]]]] (* Generates terms up to order 7 *) (* Paolo Xausa, Dec 28 2023 *)

PARI
```
See Links section.
```

from itertools import islice
from functools import reduce
def uniq(r): return reduce(lambda u, e: u if e in u else u+[e], r, [])
def agen():  # generator of terms
    R = [["1"]]
    while R:
        r = R.pop(0)
        yield from map(lambda b: int(b, 2), r)
        if len(r) > 1: R.append(uniq([r[k]+"0" for k in range(1, len(r))]))
        R.append(uniq([r[0]+"0", r[0]+"1"] + [r[k]+"1" for k in range(1, len(r))]))
print(list(islice(agen(), 66))) # Michael S. Branicky, Dec 24 2023

A273106 Numbers representable as ror(k)+rol(k), where ror(k)=A038572(k) is k rotated one binary place to the right, rol(k)=A006257(k) is k rotated one binary place to the left.

Original entry on oeis.org

0, 2, 3, 5, 6, 8, 9, 10, 14, 15, 17, 19, 20, 22, 24, 25, 27, 29, 30, 32, 33, 34, 37, 38, 39, 42, 43, 44, 47, 48, 51, 52, 53, 56, 57, 58, 61, 62, 63, 65, 66, 67, 68, 70, 71, 72, 73, 75, 76, 77, 78, 80, 81, 82, 83, 85, 86, 87, 88, 90, 91, 92, 93, 95, 96, 98

Offset: 0

Alex Ratushnyak, May 15 2016

Cf. A006257, A038572, A088161, A088163, A273105.

Take[#, 66] &@ Union@ Table[FromDigits[RotateRight@ #, 2] + FromDigits[RotateLeft@ #, 2] &@ IntegerDigits[n, 2], {n, 0, 10^3}] (* Michael De Vlieger, May 17 2016 *)

def ROR(n):                # returns A038572(n)
    BL = len(bin(n))-2
    return (n>>1) + ((n&1) << (BL-1))
def ROL(n):                # returns A006257(n) for n>0
    BL = len(bin(n))-2
    return (n*2) - (1<

A273180 Numbers n such that ror(n) + rol(n) is a power of 2, where ror(n)=A038572(n) is n rotated one binary place to the right, rol(n)=A006257(n) is n rotated one binary place to the left.

Original entry on oeis.org

1, 2, 6, 19, 38, 102, 307, 614, 1638, 4915, 9830, 26214, 78643, 157286, 419430, 1258291, 2516582, 6710886, 20132659, 40265318, 107374182, 322122547, 644245094, 1717986918, 5153960755, 10307921510, 27487790694, 82463372083, 164926744166, 439804651110

Offset: 1

Alex Ratushnyak, May 17 2016

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,17,0,0,-16).

Cf. A006257, A038572, A088161, A088163, A112627, A273105, A273106.

#include 
int main(int argc, char** argv)
{
  unsigned long long x, n, BL=0;
  for (n=1; n>0; ++n) {
    if ((n & (n-1))==0)  ++BL;
    x = (n>>1) + ((n&1) << (BL-1));   // A038572(n)
    x+= (n*2) - (1ull<A006257(n)  for n>0
    if ((x & (x-1))==0)  printf("%lld, ", n);
  }
}

Select[Range[10^6], IntegerQ@ Log2[FromDigits[RotateRight@ #, 2] + FromDigits[RotateLeft@ #, 2]] &@ IntegerDigits[#, 2] &] (* or *)
Rest@ CoefficientList[Series[x (1 + 2 x + 6 x^2 + 2 x^3 + 4 x^4)/((1 - x) (1 + x + x^2) (1 - 16 x^3)), {x, 0, 30}], x] (* Michael De Vlieger, May 19 2016 *)

Vec(x*(1+2*x+6*x^2+2*x^3+4*x^4)/((1-x)*(1+x+x^2)*(1-16*x^3)) + O(x^50)) \\ Colin Barker, May 19 2016

From Colin Barker, May 19 2016: (Start)
a(n) = 17*a(n-3) - 16*a(n-6) for n>6.
G.f.: x*(1+2*x+6*x^2+2*x^3+4*x^4) / ((1-x)*(1+x+x^2)*(1-16*x^3)).
(End)

A273050 Numbers k such that ror(k) XOR rol(k) = k, where ror(x)=A038572(x) is x rotated one binary place to the right, rol(x)=A006257(x) is x rotated one binary place to the left, and XOR is the binary exclusive-or operator.

Original entry on oeis.org

0, 5, 6, 45, 54, 365, 438, 2925, 3510, 23405, 28086, 187245, 224694, 1497965, 1797558, 11983725, 14380470, 95869805, 115043766, 766958445, 920350134, 6135667565, 7362801078, 49085340525, 58902408630, 392682724205, 471219269046

Offset: 1

Alex Ratushnyak, May 13 2016

Cf. A006257, A038572, A088163, A125836 (bisection?), A125837 (bisection?).

Cf. A020988 (numbers k such that ror(k) + rol(k) = k).

ok[n_] := Block[{x = IntegerDigits[n, 2]}, x == BitXor @@@ Transpose@ {RotateLeft@ x, RotateRight@ x}]; Select[ Range[0, 10^5], ok] (* Giovanni Resta, May 14 2016 *)
ok[n_] := Block[{x = IntegerDigits[n, 2]}, x == BitXor @@@ Transpose[ {RotateLeft[x], RotateRight[x]}]]; Select[LinearRecurrence[{0, 9, 0, -8}, {0, 5, 6, 45}, 100], ok] (* Jean-François Alcover, May 22 2016, after Giovanni Resta *)

def ROR(n):                # returns A038572(n)
    BL = len(bin(n))-2
    return (n>>1) + ((n&1) << (BL-1))
def ROL(n):                # returns A006257(n) for n>0
    BL = len(bin(n))-2
    return (n*2) - (1<

Conjectures from Colin Barker, May 22 2016: (Start)
a(n) = (-11+(-1)^n+2^(-1/2+(3*n)/2)*(3-3*(-1)^n+5*sqrt(2)+5*(-1)^n*sqrt(2)))/14.
a(n) = 5*(2^(3*n/2)-1)/7 for n even.
a(n) = 3*(2^((3*n)/2-1/2)-2)/7 for n odd.
a(n) = 9*a(n-2)-8*a(n-4) for n>4.
G.f.: x^2*(5+6*x) / ((1-x)*(1+x)*(1-8*x^2)).
(End)

a(19)-a(27) from Giovanni Resta, May 14 2016

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A088161 n rotated one binary place to the right less n rotated one binary place to the left.

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A273105 a(n) = A038572(n) + A006257(n), sum of the two numbers obtained by rotating the binary representation of n by one place to the right and to the left.

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A368427 A permutation related to the Christmas tree pattern map (A367508): a(1) = 1, and for any n > 1, a(n) = A053644(n) + A367562(n-1).

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A273106 Numbers representable as ror(k)+rol(k), where ror(k)=A038572(k) is k rotated one binary place to the right, rol(k)=A006257(k) is k rotated one binary place to the left.

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A273180 Numbers n such that ror(n) + rol(n) is a power of 2, where ror(n)=A038572(n) is n rotated one binary place to the right, rol(n)=A006257(n) is n rotated one binary place to the left.

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A273050 Numbers k such that ror(k) XOR rol(k) = k, where ror(x)=A038572(x) is x rotated one binary place to the right, rol(x)=A006257(x) is x rotated one binary place to the left, and XOR is the binary exclusive-or operator.

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