A129868 Binary palindromic numbers with only one 0 bit.

0, 5, 27, 119, 495, 2015, 8127, 32639, 130815, 523775, 2096127, 8386559, 33550335, 134209535, 536854527, 2147450879, 8589869055, 34359607295, 137438691327, 549755289599, 2199022206975, 8796090925055, 35184367894527, 140737479966719, 562949936644095

Offset

0,2

Comments

Binary expansion is 0, 101, 11011, 1110111, 111101111, ... (see A138148).

9 + 8a(n) = s^2 is a perfect square with s = 2^(n + 2) -1 = 3, 7, 15, 31, 63, ...

Numbers with middle bit 0, that have only one bit 0, and the total number of bits is odd.

The fractional part of the base 2 logarithm of a(n) approaches 1 as n approaches infinity.

Also called binary cyclops numbers.

Last digit of the decimal representation follows the pattern 5, 7, 9, 5, 5, 7, 9, 5, ... . - Alex Ratushnyak, Dec 08 2012

Links

Robert Israel, Table of n, a(n) for n = 0..1630

Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video, video (2015)

Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Formula

a(n) = 2^(2n + 1) - 2^n - 1 = 2*4^n - 2^n - 1 = (2^n - 1)(2*2^n + 1).

G.f.: x*(8*x-5)/((x-1)*(2*x-1)*(4*x-1)).

Recurrences:

a(n) = (1/2)*(7 + 8*a(n - 1) + sqrt(9 + 8*a(n - 1))), a(0) = 0;

a(n) = 6*a(n - 1) - 8*a(n - 2) - 3, a(0) = 0, a(1) = 5;

a(n) = 7*a(n - 1) - 14*a(n - 2) + 8*a(n - 3), a(0) = 0, a(1) = 5, a(2) = 27.

a(n) = A006516(n+1) - 1.

Maple

A129868:=n->2^(2*n + 1) - 2^n - 1: seq(A129868(n), n=0..30); # Wesley Ivan Hurt, Dec 08 2015

Mathematica

(* 1st *) FromDigits[ #, 2]&/@NestList[Append[Prepend[ #, 1], 1]&, {0}, 25] (* 2nd *) NestList[(1/2)(7 + 8# + Sqrt[9 + 8# ])&, 0, 22] (* both of these are from Zak Seidov *)
f[n_] := 2^(2n + 1) - 2^n - 1; Table[f@n, {n, 0, 22}] (* Robert G. Wilson v, Aug 24 2007 *)
Table[EulerE[2, 2^n], {n, 1, 60}]/2 - 1 (* Vladimir Joseph Stephan Orlovsky, Nov 03 2009 *)
(* After running the program in A134808 *) Select[Range[0, 2^16 - 1], cyclopsQ[#, 2] &] (* Alonso del Arte, Dec 17 2010 *)
LinearRecurrence[{7, -14, 8}, {0, 5, 27}, 30] (* Vincenzo Librandi, Dec 08 2015 *)

Prog

(Magma) [2^(2*n+1)-2^n-1: n in [0..25]]; // Vincenzo Librandi, Dec 08 2015
(PARI) concat(0, Vec(x*(5-8*x)/(1-7*x+14*x^2-8*x^3) + O(x^100))) \\ Altug Alkan, Dec 08 2015
(Python)
def A129868(n): return ((m:=1<<n)-1)*((m<<1)+1) # Chai Wah Wu, Mar 19 2024

Crossrefs

Base 10 analog is A134808.

Binary palindromic numbers, including repunits (or Mersenne numbers A000225) are in A006995. The sequence of binary pandigital (having both 0's and 1's) palindromic numbers begins 5, 9, 17, 21, 27, 33, 45, 51, 65, 73, ...

Cf. A006516, A138148.

Sequence in context: A226315 A201436 A202508 * A069993 A249995 A009027

Adjacent sequences: A129865 A129866 A129867 * A129869 A129870 A129871

Keyword

nonn,base,easy

Author

Zak Seidov, May 24 2007

Status

approved