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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

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

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