A138148 - OEIS

%I #62 Feb 16 2025 08:33:07

%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 Gennady Eremin, <a href="https://arxiv.org/abs/2405.16143">Partitioning the set of natural numbers into Mersenne trees and into arithmetic progressions; Natural Matrix and Linnik's constant</a>, arXiv:2405.16143 [math.CO], 2024. See pp. 9, 14.

%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="https://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,changed

%O 0,2

%A _Omar E. Pol_, Mar 18 2008

%E More terms from _Omar E. Pol_, Feb 09 2020