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.

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

Author

Alex Ratushnyak, May 13 2016

Keywords

• nonn
• base

Crossrefs

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

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

Programs

Mathematica

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 *)

Python

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<
				

Formula

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)

Extensions

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