|
%I #65 Mar 19 2024 13:28:51
%S 0,5,27,119,495,2015,8127,32639,130815,523775,2096127,8386559,
%T 33550335,134209535,536854527,2147450879,8589869055,34359607295,
%U 137438691327,549755289599,2199022206975,8796090925055,35184367894527,140737479966719,562949936644095
%N Binary palindromic numbers with only one 0 bit.
%C Binary expansion is 0, 101, 11011, 1110111, 111101111, ... (see A138148).
%C 9 + 8a(n) = s^2 is a perfect square with s = 2^(n + 2) -1 = 3, 7, 15, 31, 63, ...
%C Numbers with middle bit 0, that have only one bit 0, and the total number of bits is odd.
%C The fractional part of the base 2 logarithm of a(n) approaches 1 as n approaches infinity.
%C Also called binary cyclops numbers.
%C Last digit of the decimal representation follows the pattern 5, 7, 9, 5, 5, 7, 9, 5, ... . - _Alex Ratushnyak_, Dec 08 2012
%H Robert Israel, <a href="/A129868/b129868.txt">Table of n, a(n) for n = 0..1630</a>
%H 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 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-14,8).
%F a(n) = 2^(2n + 1) - 2^n - 1 = 2*4^n - 2^n - 1 = (2^n - 1)(2*2^n + 1).
%F G.f.: x*(8*x-5)/((x-1)*(2*x-1)*(4*x-1)).
%F Recurrences:
%F a(n) = (1/2)*(7 + 8*a(n - 1) + sqrt(9 + 8*a(n - 1))), a(0) = 0;
%F a(n) = 6*a(n - 1) - 8*a(n - 2) - 3, a(0) = 0, a(1) = 5;
%F a(n) = 7*a(n - 1) - 14*a(n - 2) + 8*a(n - 3), a(0) = 0, a(1) = 5, a(2) = 27.
%F a(n) = A006516(n+1) - 1.
%p A129868:=n->2^(2*n + 1) - 2^n - 1: seq(A129868(n), n=0..30); # _Wesley Ivan Hurt_, Dec 08 2015
%t (* 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 f[n_] := 2^(2n + 1) - 2^n - 1; Table[f@n, {n, 0, 22}] (* _Robert G. Wilson v_, Aug 24 2007 *)
%t Table[EulerE[2, 2^n], {n, 1, 60}]/2 - 1 (* _Vladimir Joseph Stephan Orlovsky_, Nov 03 2009 *)
%t (* After running the program in A134808 *) Select[Range[0, 2^16 - 1], cyclopsQ[#, 2] &] (* _Alonso del Arte_, Dec 17 2010 *)
%t LinearRecurrence[{7, -14, 8}, {0, 5, 27}, 30] (* _Vincenzo Librandi_, Dec 08 2015 *)
%o (Magma) [2^(2*n+1)-2^n-1: n in [0..25]]; // _Vincenzo Librandi_, Dec 08 2015
%o (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 (Python)
%o def A129868(n): return ((m:=1<<n)-1)*((m<<1)+1) # _Chai Wah Wu_, Mar 19 2024
%Y Base 10 analog is A134808.
%Y 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 Cf. A006516, A138148.
%K nonn,base,easy
%O 0,2
%A _Zak Seidov_, May 24 2007
|
