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A129868 Binary palindromic numbers with only one 0 bit.

%I A129868 #69 Feb 06 2025 10:21:19
%S A129868 0,5,27,119,495,2015,8127,32639,130815,523775,2096127,8386559,
%T A129868 33550335,134209535,536854527,2147450879,8589869055,34359607295,
%U A129868 137438691327,549755289599,2199022206975,8796090925055,35184367894527,140737479966719,562949936644095
%N A129868 Binary palindromic numbers with only one 0 bit.
%C A129868 Binary expansion is 0, 101, 11011, 1110111, 111101111, ... (see A138148).
%C A129868 9 + 8a(n) = s^2 is a perfect square with s = 2^(n + 2) -1 = 3, 7, 15, 31, 63, ...
%C A129868 Numbers with middle bit 0, that have only one bit 0, and the total number of bits is odd.
%C A129868 The fractional part of the base 2 logarithm of a(n) approaches 1 as n approaches infinity.
%C A129868 Also called binary cyclops numbers.
%C A129868 Last digit of the decimal representation follows the pattern 5, 7, 9, 5, 5, 7, 9, 5, ... . - _Alex Ratushnyak_, Dec 08 2012
%H A129868 Robert Israel, <a href="/A129868/b129868.txt">Table of n, a(n) for n = 0..1630</a>
%H A129868 Gennady Eremin, <a href="https://arxiv.org/abs/2405.16143">Partitioning the set of natural numbers into Mersenne trees and into arithmetic progressions; Natural Matrix and Linnik's constant</a>, arXiv:2405.16143 [math.CO], 2024. See pp. 9, 14.
%H A129868 Brady Haran and Simon Pampena, <a href="https://www.youtube.com/watch?v=HPfAnX5blO0">Glitch Primes and Cyclops Numbers</a>, Numberphile video, video (2015)
%H A129868 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-14,8).
%F A129868 a(n) = 2^(2n + 1) - 2^n - 1 = 2*4^n - 2^n - 1 = (2^n - 1)(2*2^n + 1).
%F A129868 G.f.: x*(8*x-5)/((x-1)*(2*x-1)*(4*x-1)).
%F A129868 Recurrences:
%F A129868 a(n) = (1/2)*(7 + 8*a(n - 1) + sqrt(9 + 8*a(n - 1))), a(0) = 0;
%F A129868 a(n) = 6*a(n - 1) - 8*a(n - 2) - 3, a(0) = 0, a(1) = 5;
%F A129868 a(n) = 7*a(n - 1) - 14*a(n - 2) + 8*a(n - 3), a(0) = 0, a(1) = 5, a(2) = 27.
%F A129868 a(n) = A006516(n+1) - 1.
%p A129868 A129868:=n->2^(2*n + 1) - 2^n - 1: seq(A129868(n), n=0..30); # _Wesley Ivan Hurt_, Dec 08 2015
%t A129868 (* 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_ *)
%t A129868 f[n_] := 2^(2n + 1) - 2^n - 1; Table[f@n, {n, 0, 22}] (* _Robert G. Wilson v_, Aug 24 2007 *)
%t A129868 Table[EulerE[2, 2^n], {n, 1, 60}]/2 - 1 (* _Vladimir Joseph Stephan Orlovsky_, Nov 03 2009 *)
%t A129868 (* After running the program in A134808 *) Select[Range[0, 2^16 - 1], cyclopsQ[#, 2] &] (* _Alonso del Arte_, Dec 17 2010 *)
%t A129868 LinearRecurrence[{7, -14, 8}, {0, 5, 27}, 30] (* _Vincenzo Librandi_, Dec 08 2015 *)
%o A129868 (Magma) [2^(2*n+1)-2^n-1: n in [0..25]]; // _Vincenzo Librandi_, Dec 08 2015
%o A129868 (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
%o A129868 (Python)
%o A129868 def A129868(n): return ((m:=1<<n)-1)*((m<<1)+1) # _Chai Wah Wu_, Mar 19 2024
%Y A129868 Base 10 analog is A134808.
%Y A129868 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, ...
%Y A129868 Cf. A006516, A138148.
%K A129868 nonn,base,easy
%O A129868 0,2
%A A129868 _Zak Seidov_, May 24 2007