A273050 Numbers k such that (rol(k) XOR ror(k)) = k, where rol = A006257 and ror = A038572 are rotations of binary digits by one place to the left and right, 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

1,2

Links

Table of n, a(n) for n=1..27.

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)

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 *)

Prog

(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<<BL) + 1
print('0', end=', ')
for n in range(1, 100000):
    if ROR(n) ^ ROL(n) == n: print(n, end=', ')
(PARI) select( {is_A273050(n)=bitxor(A038572(n), A006257(n))==n}, [0..10^5]) \\ M. F. Hasler, Sep 26 2025

Crossrefs

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

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

Sequence in context: A352641 A299168 A219516 * A163481 A298376 A269908

Adjacent sequences: A273047 A273048 A273049 * A273051 A273052 A273053

Keyword

nonn,base

Author

Alex Ratushnyak, May 13 2016

Extensions

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

Status

approved