

A129868


Binary palindromic numbers with only one 0 bit.


19



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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



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*x5)/((x1)*(2*x1)*(4*x1)).
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.


MAPLE



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 *)
(* After running the program in A134808 *) Select[Range[0, 2^16  1], cyclopsQ[#, 2] &] (* Alonso del Arte, Dec 17 2010 *)


PROG

(PARI) concat(0, Vec(x*(58*x)/(17*x+14*x^28*x^3) + O(x^100))) \\ Altug Alkan, Dec 08 2015
(Python)


CROSSREFS

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, ...


KEYWORD

nonn,base,easy


AUTHOR



STATUS

approved



