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

Author

Zak Seidov, May 24 2007

Keywords

• nonn
• base
• easy

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
• Gennady Eremin, Partitioning the set of natural numbers into Mersenne trees and into arithmetic progressions; Natural Matrix and Linnik's constant, arXiv:2405.16143 [math.CO], 2024. See pp. 9, 14.
• 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).

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.

Programs

Magma

[2^(2*n+1)-2^n-1: n in [0..25]]; // Vincenzo Librandi, Dec 08 2015

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 *)

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<Chai Wah Wu, Mar 19 2024

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.