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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. 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
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=', ')
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
KEYWORD
nonn,base
AUTHOR
Alex Ratushnyak, May 13 2016
EXTENSIONS
a(19)-a(27) from Giovanni Resta, May 14 2016
STATUS
approved

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Last modified May 22 20:45 EDT 2024. Contains 372758 sequences. (Running on oeis4.)