A138148 Cyclops numbers with binary digits only.

0, 101, 11011, 1110111, 111101111, 11111011111, 1111110111111, 111111101111111, 11111111011111111, 1111111110111111111, 111111111101111111111, 11111111111011111111111, 1111111111110111111111111, 111111111111101111111111111, 11111111111111011111111111111

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

Table of n, a(n) for n=0..14.

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 to Elementary Cellular Automata

Index entries for sequences related to cellular automata

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 Fröhlich, 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