A124166 -id:A124166 - cp's OEIS frontend

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

Alexandre Wajnberg, Apr 25 2005

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

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

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).

Cf. A000918, A002275, A029858, A052379, A052386, A069532, A080674, A124166, A161770.
Row n=10 of A228275.
Partial sums of A178500.

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

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

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

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)

A247817 Sum(4^k, k=2..n).

Original entry on oeis.org

0, 16, 80, 336, 1360, 5456, 21840, 87376, 349520, 1398096, 5592400, 22369616, 89478480, 357913936, 1431655760, 5726623056, 22906492240, 91625968976, 366503875920, 1466015503696, 5864062014800, 23456248059216, 93824992236880, 375299968947536, 1501199875790160

Offset: 1

Vincenzo Librandi, Sep 25 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. Sum(h^k,k=2..n): A028329 (h=2), A168569 (h=3), this sequence (h=4), A168571 (h=5), A247840 (h=6), A168572 (h=7), A247841 (h=8), A247842 (h=9), A124166 (h=10).

[0] cat [&+[4^k: k in [2..n]]: n in [2..30]];

Magma
```
[(4^(n+1)-16)/3: n in [1..30]];
```

RecurrenceTable[{a[1] == 0, a[n] == a[n-1] + 4^n}, a, {n, 30}] (* or *) CoefficientList[ Series[16 x / ((1 - x) (1 - 4 x)),{x, 0, 40}], x]
LinearRecurrence[{5,-4},{0,16},30] (* Harvey P. Dale, Feb 19 2023 *)

a(n) = sum(k=2, n, 4^k); \\ Michel Marcus, Sep 25 2014

G.f.: 16*x^2/((1-x)*(1-4*x)).
a(n) = a(n-1) + 4^n = (4^(n+1) - 16)/3 = 5*a(n-1) - 4*a(n-2).
a(n) = A080674(n) - 4. - Michel Marcus, Sep 25 2014

A161770 n 1's followed by three 0's.

Original entry on oeis.org

1000, 11000, 111000, 1111000, 11111000, 111111000, 1111111000, 11111111000, 111111111000, 1111111111000, 11111111111000, 111111111111000, 1111111111111000, 11111111111111000, 111111111111111000, 1111111111111111000, 11111111111111111000, 111111111111111111000

Offset: 1

Jaroslav Krizek, Jun 18 2009

Sequence A159741 written in base 2.

Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A002275, A105279, A124166, A159741.

Table[FromDigits[PadLeft[{0,0,0},n,1]],{n,4,20}] (* Harvey P. Dale, Aug 07 2023 *)

a(n) = 1000*(10^n - 1)/9 = 1000*A002275(n).
G.f.: 1000*x/((10*x-1)*(x-1)).
From Elmo R. Oliveira, Jun 18 2025: (Start)
E.g.f.: 1000*exp(x)*(exp(9*x) - 1)/9.
a(n) = 100*A105279(n) = 10*A124166(n).
a(n) = 11*a(n-1) - 10*a(n-2) for n > 2. (End)

Edited by Charles R Greathouse IV, Oct 12 2009

cp's OEIS Frontend

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

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A247817 Sum(4^k, k=2..n).

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A161770 n 1's followed by three 0's.

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