A273105 -id:A273105 - cp's OEIS frontend

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.

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)

A274133 Numbers representable as ror(x)+rol(x) in two or more ways, 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.

Original entry on oeis.org

2, 9, 14, 15, 20, 27, 32, 37, 42, 47, 52, 57, 62, 129, 134, 139, 144, 149, 154, 159, 164, 169, 174, 179, 184, 189, 194, 195, 199, 200, 204, 205, 209, 210, 214, 215, 219, 220, 224, 225, 229, 230, 234, 235, 239, 240, 244, 245, 249, 250, 254, 255, 260, 265, 270, 275

Offset: 1

Alex Ratushnyak, Jun 10 2016

These are the duplicates in A273105.

			ror(5) + rol(5) = 6 + 3 = 9, and also ror(16) + rol(16) = 8 + 1 = 9, therefore 9 is in the sequence.

Cf. A006257, A038572, A273105.

A274341 Numbers that cannot be represented as ror(x)+rol(x), 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.

Original entry on oeis.org

1, 4, 7, 11, 12, 13, 16, 18, 21, 23, 26, 28, 31, 35, 36, 40, 41, 45, 46, 49, 50, 54, 55, 59, 60, 64, 69, 74, 79, 84, 89, 94, 97, 102, 107, 112, 117, 122, 127, 131, 132, 136, 137, 141, 142, 146, 147, 151, 152, 156, 157, 161, 162, 166, 167, 171, 172, 176, 177, 181

Offset: 1

Alex Ratushnyak, Jun 26 2016

Numbers that are not in A273105.

Cf. A006257, A038572, A273105.

cp's OEIS Frontend

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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A274133 Numbers representable as ror(x)+rol(x) in two or more ways, 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.

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A274341 Numbers that cannot be represented as ror(x)+rol(x), 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.

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