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A129868
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Binary palindromic numbers with only one 0 bit.
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5
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0, 5, 27, 119, 495, 2015, 8127, 32639, 130815, 523775, 2096127, 8386559, 33550335, 134209535, 536854527, 2147450879, 8589869055, 34359607295, 137438691327, 549755289599, 2199022206975, 8796090925055, 35184367894527
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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.
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FORMULA
| a(n) = 2^(2n + 1) - 2^n - 1 = 2*4^n - 2^n - 1 = (2^n - 1)(2*2^n + 1), n = 0, 1, ...;
G.f. = x(5 - 8*x)/(1 - 7*x + 14*x^2 - 8*x^3)
Recurrences: a(n) = (1/2)(7 + 8a(n - 1) + sqrt(9 + 8a(n - 1))), a(0) = 0; a(n) = 6a(n - 1) - 8a(n - 2) - 3, a(0) = 0, a(1) = 5; a(n) = 7a(n - 1) - 14a(n - 2) + 8a(n - 3), a(0) = 0, a(1) = 5, a(2) = 27.
a(n) = A006516(n+1) - 1.
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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 from Zak Seidov *)
f[n_] := 2^(2n + 1) - 2^n - 1; Table[f@n, {n, 0, 22}] (* Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 24 2007 *)
Table[EulerE[2, 2^n], {n, 1, 60}]/2 - 1 (* From Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 03 2009 *)
(* After running the program in A134808 *) Select[Range[0, 2^16 - 1], cyclopsQ[#, 2] &] (* From Alonso del Arte, Dec 17 2010 *)
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CROSSREFS
| Base 10 analogue is A134808.
Binary palindromic numbers, including repunits (or Mersenne numbers A000225) are in A006995. The sequence of binary pandigital (having both 0s and 1s) palindromic numbers begins 5, 9, 17, 21, 27, 33, 45, 51, 65, 73, ...
Sequence in context: A085740 A201436 A202508 * A069993 A009027 A037498
Adjacent sequences: A129865 A129866 A129867 * A129869 A129870 A129871
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KEYWORD
| nonn,base
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AUTHOR
| Zak Seidov (zakseidov(AT)yahoo.com), May 24 2007
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