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%I #57 Sep 08 2022 08:45:33
%S 0,101,11011,1110111,111101111,11111011111,1111110111111,
%T 111111101111111,11111111011111111,1111111110111111111,
%U 111111111101111111111,11111111111011111111111,1111111111110111111111111,111111111111101111111111111,11111111111111011111111111111
%N Cyclops numbers with binary digits only.
%C All members are palindromes A002113. The first five members are mentioned in A129868.
%C Also, binary representation of A129868.
%C a(A090748(n)) is equal to A138831(n), the n-th perfect number minus 1, written in base 2.
%C 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
%C 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
%D S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
%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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (111,-1110,1000).
%F From _Colin Barker_, Feb 21 2013: (Start)
%F a(n) = (-1-9*10^n+10^(1+2*n))/9.
%F G.f.: x*(200*x-101) / ((x-1)*(10*x-1)*(100*x-1)). (End)
%F a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n>2. - _Wesley Ivan Hurt_, Dec 08 2015
%F a(n) = A000533(n+1)*A002275(n). - _Altug Alkan_, May 12 2016
%F E.g.f.: (-1 - 9*exp(9*x) + 10*exp(99*x))*exp(x)/9. - _Ilya Gutkovskiy_, May 12 2016
%F a(n) = A002275(2n+1) - A011557(n). - _M. F. Hasler_, Feb 08 2020
%e n ........ a(n) .... A129868(n): value of a(n) read in base 2.
%e 0 ......... 0 ......... 0
%e 1 ........ 101 ........ 5
%e 2 ....... 11011 ....... 27
%e 3 ...... 1110111 ...... 119
%e 4 ..... 111101111 ..... 495
%e 5 .... 11111011111 .... 2015
%e 6 ... 1111110111111 ... 8127
%p A138148:=n->(-1-9*10^n+10^(1+2*n))/9: seq(A138148(n), n=0..15); # _Wesley Ivan Hurt_, Dec 08 2015
%t Table[(-1 - 9*10^n + 10^(1 + 2*n))/9, {n, 0, 15}] (* _Wesley Ivan Hurt_, Dec 08 2015 *)
%o (PARI) for(n=1, 20, if(n%2==1, c=((10^n-1)/9)-10^((n-1)/2); print1(c, ", "))) \\ _Felix Fröhlich_, Jul 07 2014
%o (PARI) apply( {A138148(n)=10^(n*2+1)\9-10^n}, [0..15]) \\ _M. F. Hasler_, Feb 08 2020
%o (Magma) [(-1 - 9*10^n + 10^(1 + 2*n))/9 : n in [0..15]]; // _Wesley Ivan Hurt_, Dec 08 2015
%Y Cyclops numbers: A134808. Cf. A002113, A129868.
%Y Cf. A000396, A090748, A135627, A138831.
%Y Cf. A002275 (repunits R_n = (10^n-1)/9), A011557 (10^n).
%K easy,nonn,base
%O 0,2
%A _Omar E. Pol_, Mar 18 2008
%E More terms from _Omar E. Pol_, Feb 09 2020
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