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 A138148 Cyclops numbers with binary digits only. 101
 0, 101, 11011, 1110111, 111101111, 11111011111, 1111110111111, 111111101111111, 11111111011111111, 1111111110111111111, 111111111101111111111, 11111111111011111111111, 1111111111110111111111111, 111111111111101111111111111, 11111111111111011111111111111 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS All members are palindromes A002113. The first five members are mentioned in A129868. Also, binary representation of A129868. a(A090748(n)) is equal to A138831(n), the n-th perfect number minus 1, written in base 2. Except for the first term (replace 0 with 1) the binary representation of the n-th iteration of the elementary cellular automaton, Rule 219 starting with a single ON (black) cell. - Robert Price, Feb 21 2016 a(1) = 101 is only prime number in this sequence since a(n) = (10^(n+1)+1)*(10^n-1)/9. - Altug Alkan, May 11 2016 REFERENCES S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55. LINKS Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video, video (2015). Eric Weisstein's World of Mathematics, Elementary Cellular Automaton S. Wolfram, A New Kind of Science Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000). FORMULA From Colin Barker, Feb 21 2013: (Start) a(n) = (-1-9*10^n+10^(1+2*n))/9. G.f.: x*(200*x-101) / ((x-1)*(10*x-1)*(100*x-1)). (End) a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n>2. - Wesley Ivan Hurt, Dec 08 2015 a(n) = A000533(n+1)*A002275(n). - Altug Alkan, May 12 2016 E.g.f.: (-1 - 9*exp(9*x) + 10*exp(99*x))*exp(x)/9. - Ilya Gutkovskiy, May 12 2016 a(n) = A002275(2n+1) - A011557(n). - M. F. Hasler, Feb 08 2020 EXAMPLE n ........ a(n) .... A129868(n): value of a(n) read in base 2. 0 ......... 0 ......... 0 1 ........ 101 ........ 5 2 ....... 11011 ....... 27 3 ...... 1110111 ...... 119 4 ..... 111101111 ..... 495 5 .... 11111011111 .... 2015 6 ... 1111110111111 ... 8127 MAPLE A138148:=n->(-1-9*10^n+10^(1+2*n))/9: seq(A138148(n), n=0..15); # Wesley Ivan Hurt, Dec 08 2015 MATHEMATICA Table[(-1 - 9*10^n + 10^(1 + 2*n))/9, {n, 0, 15}] (* Wesley Ivan Hurt, Dec 08 2015 *) PROG (PARI) for(n=1, 20, if(n%2==1, c=((10^n-1)/9)-10^((n-1)/2); print1(c, ", "))) \\ Felix FrÃ¶hlich, Jul 07 2014 (PARI) apply( {A138148(n)=10^(n*2+1)\9-10^n}, [0..15]) \\ M. F. Hasler, Feb 08 2020 (MAGMA) [(-1 - 9*10^n + 10^(1 + 2*n))/9 : n in [0..15]]; // Wesley Ivan Hurt, Dec 08 2015 CROSSREFS Cyclops numbers: A134808. Cf. A002113, A129868. Cf. A000396, A090748, A135627, A138831. Cf. A002275 (repunits R_n = (10^n-1)/9), A011557 (10^n). Sequence in context: A097727 A083981 A267779 * A138831 A267920 A300570 Adjacent sequences:  A138145 A138146 A138147 * A138149 A138150 A138151 KEYWORD easy,nonn,base AUTHOR Omar E. Pol, Mar 18 2008 EXTENSIONS More terms from Omar E. Pol, Feb 09 2020 STATUS approved

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Last modified November 24 10:26 EST 2020. Contains 338612 sequences. (Running on oeis4.)