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 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 Robert Israel, Table of n, a(n) for n = 0..1630 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). 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. 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 *) PROG (Magma) [2^(2*n+1)-2^n-1: n in [0..25]]; // 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<

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Last modified June 13 09:03 EDT 2024. Contains 373383 sequences. (Running on oeis4.)