A109241 Expansion of 1/((1-10*x)*(1-100*x)).

1, 110, 11100, 1111000, 111110000, 11111100000, 1111111000000, 111111110000000, 11111111100000000, 1111111111000000000, 111111111110000000000, 11111111111100000000000, 1111111111111000000000000, 111111111111110000000000000, 11111111111111100000000000000

Offset: 0

Author

Paul Barry, Jun 23 2005

Keywords

• easy
• nonn

Comments

a(n) has n+1 1's and n 0's. Partial sums are A109242.

a(n) = A171476(n) converted from decimal to binary. - Robert Price, Jan 19 2016

Also the binary representation of the n-th iteration of the elementary cellular automaton starting with a single ON (black) cell for Rules 206 and 238. - Robert Price, Feb 21 2016

References

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

Links

• 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 (110,-1000)

Crossrefs

Cf. A006516, A138147.

Programs

Magma

[10^(2*n+1)/9-10^n/9: n in [0..40]]; // Vincenzo Librandi, Feb 22 2016

Maple

A109241 := proc(n)(10^(2*n+1)-10^n)/9 ; end proc:
seq(A109241(n),n=0..20) ; # R. J. Mathar, Mar 21 2011

Mathematica

Table[(10^(2*n+1)-10^n)/9, {n, 0, 100}] (* Robert Price, Feb 21 2016 *)
CoefficientList[Series[1/((1 - 100 x) (1 - 10 x)), {x, 0, 33}], x] (* Vincenzo Librandi, Feb 22 2016 *)

PARI

a(n)=10^(2*n+1)/9-10^n/9 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = (10^(2n+1) - 10^n)/9.

a(n) = A006516(n+1) written in base 2. - Omar E. Pol, Feb 24 2008

a(n) = A138147(n+1)/10. - Omar E. Pol, Nov 08 2008

a(n) = 110*a(n-1) -1000*a(n-2), n>=2. - Vincenzo Librandi, Mar 18 2011

a(n) = A002275(n+1)*10^n. - Wesley Ivan Hurt, Jun 22 2013

E.g.f.: (1/9)*(10*exp(100*x) - exp(10*x)). - G. C. Greubel, Aug 01 2017