A105279 a(0)=0; a(n) = 10*a(n-1) + 10.

0, 10, 110, 1110, 11110, 111110, 1111110, 11111110, 111111110, 1111111110, 11111111110, 111111111110, 1111111111110, 11111111111110, 111111111111110, 1111111111111110, 11111111111111110, 111111111111111110, 1111111111111111110, 11111111111111111110, 111111111111111111110

Offset: 0

Author

Alexandre Wajnberg, Apr 25 2005

Keywords

• nonn
• easy

Comments

a(n) is the smallest even number with digits in {0,1} having digit sum n; in other words, the base 10 reading of the binary string of A000918(n). Cf. A069532. - Jason Kimberley, Nov 02 2011

Also, except for a(0), the binary representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 645", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, Jul 19 2017

References

• S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..100
• Robert Price, Diagrams of first 20 stages of the Cellular Automata.
• N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
• Eric Weisstein's World of Mathematics, Elementary Cellular Automaton.
• S. Wolfram, A New Kind of Science.
• Wolfram Research, Wolfram Atlas of Simple Programs.
• Index entries for sequences related to cellular automata.
• Index to 2D 5-Neighbor Cellular Automata.
• Index to Elementary Cellular Automata.
• Index entries for linear recurrences with constant coefficients, signature (11,-10).

Crossrefs

Cf. A000918, A002275, A029858, A052379, A052386, A069532, A080674, A124166, A161770.

Row n=10 of A228275.

Partial sums of A178500.

Programs

Haskell

a105279 n = a105279_list !! n
a105279_list = iterate ((* 10) . (+ 1)) 0
-- Reinhard Zumkeller, Feb 05 2012

Magma

[-10/9+(10/9)*10^n: n in [0..20]]; // Vincenzo Librandi, Jul 04 2011

Mathematica

NestList[10*(# + 1) &, 0, 25] (* Paolo Xausa, Jul 17 2024 *)

Formula

a(n) = (10/9)*(10^n - 1), with n>=0.

a(n) = Sum_{k=1..n} 10^k.

Repunits times 10: a(n) = 10 * A002275(n). - Reinhard Zumkeller, Feb 05 2012

From Stefano Spezia, Sep 15 2023: (Start)

O.g.f.: 10*x/((1 - x)*(1 - 10*x)).

E.g.f.: 10*exp(x)*(exp(9*x) - 1)/9. (End)

From Elmo R. Oliveira, Jun 18 2025: (Start)

a(n) = 11*a(n-1) - 10*a(n-2).

a(n) = A124166(n)/10.

a(n) = A161770(n)/100 for n >= 1. (End)