A138148 Cyclops numbers with binary digits only.

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

Offset: 0

Author

Omar E. Pol, Mar 18 2008

Keywords

• easy
• nonn
• base

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

Examples

			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
		

References

• S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Links

• Gennady Eremin, Partitioning the set of natural numbers into Mersenne trees and into arithmetic progressions; Natural Matrix and Linnik's constant, arXiv:2405.16143 [math.CO], 2024. See pp. 9, 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).

Crossrefs

Cyclops numbers: A134808. Cf. A002113, A129868.

Cf. A000396, A090748, A135627, A138831.

Cf. A002275 (repunits R_n = (10^n-1)/9), A011557 (10^n).

Programs

Magma

[(-1 - 9*10^n + 10^(1 + 2*n))/9 : n in [0..15]]; // Wesley Ivan Hurt, Dec 08 2015

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 *)

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

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

Extensions

More terms from Omar E. Pol, Feb 09 2020