A109242 -id:A109242 - cp's OEIS frontend

A180153 a(n) = 10*a(n-1) + A109242(n).

1, 121, 12421, 1246421, 124696421, 12470296421, 1247037296421, 124703817296421, 12470382717296421, 1247038282717296421, 124703828392717296421, 12470382840592717296421, 1247038284073592717296421, 124703828407513592717296421, 12470382840753013592717296421

Offset: 1

Mark Dols, Aug 13 2010

Colin Barker, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (121,-2220,12100,-10000).

Cf. A109242, A000292, A002620, A059259, A094028.

Vec(x / ((x-1)*(10*x-1)^2*(100*x-1)) + O(x^30)) \\ Colin Barker, Oct 03 2015

From Colin Barker, Oct 03 2015: (Start)
a(n) = 121*a(n-1)-2220*a(n-2)+12100*a(n-3)-10000*a(n-4) for n>4.
G.f.: x / ((x-1)*(10*x-1)^2*(100*x-1)).
(End)

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

Original entry on oeis.org

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

Offset: 0

Paul Barry, Jun 23 2005

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

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

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)

Cf. A006516, A138147.

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

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

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 *)

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

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

A349805 a(n) = A261138(n)/11.

Original entry on oeis.org

1, 111, 11211, 1122211, 112232211, 11223332211, 1122334332211, 112233444332211, 11223344544332211, 112233444636544332211, 1122334446373736544332211, 11223344463737473736544332211, 112233444637374666473736544332211, 1122334446373746649466473736544332211, 11223344463737466492319466473736544332211

Offset: 1

N. J. A. Sloane, Dec 01 2021

Cf. A261138.

A109242 has the same initial terms.

a:= n-> (s-> parse(cat(s, seq(s[-i], i=1..length(s))))/11)(cat("", $1..n)):
seq(a(n), n=1..14);  # Alois P. Heinz, Dec 01 2021

def A349805(n): return int((lambda x: x+x[::-1])(''.join(str(d) for d in range(1,n+1))))//11 # Chai Wah Wu, Dec 01 2021

More than the usual number of terms are shown in order to distinguish this from several similar sequences.

A244842 a(n) = (10^n - 1)*(10^n - 10)/90.

Original entry on oeis.org

0, 99, 10989, 1109889, 111098889, 11110988889, 1111109888889, 111111098888889, 11111110988888889, 1111111109888888889, 111111111098888888889, 11111111110988888888889, 1111111111109888888888889, 111111111111098888888888889, 11111111111110988888888888889

Offset: 1

Wesley Ivan Hurt, Jul 08 2014

For n > 1, the digit sums of a(n) are 18, 27, 36, ..., 9n (see example).

			-------------------------------------------------------
n    a(n)                    digitsum(a(n))    9*n
-------------------------------------------------------
2:   99                            18          9*2
3:   10989                         27          9*3
4:   1109889                       36          9*4
5:   111098889                     45          9*5
6:   11110988889                   54          9*6
7:   1111109888889                 63          9*7
8:   111111098888889               72          9*8
9:   11111110988888889             81          9*9
10:  1111111109888888889           90         9*10
11:  111111111098888888889         99         9*11
12:  11111111110988888888889      108         9*12, etc.

Jens Kruse Andersen, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A053422, A109242.

Magma
```
[(10^n-1)*(10^n-10)/90: n in [1..15]];
```

A244842:=n->(10^n - 1)*(10^n - 10)/90: seq(A244842(n), n=1..15);

Table[(10^n - 1) (10^n - 10)/90, {n, 15}]

G.f.: -99*x / ( (x-1)*(100*x-1)*(10*x-1) ). - R. J. Mathar, Jul 11 2014

a(n) = 99*A109242(n-2). - R. J. Mathar, Jul 11 2014

A210257 Construct a triangle with centered rows in which row n contains n copies of n; form the componentwise sum of the first n rows and concatenate these decimal numbers to get a(n).

Original entry on oeis.org

1, 212, 32423, 4364634, 548696845, 651081291281056, 7612101512161215101267, 87141218152016201518121478, 981614211824202520241821141689, 109181624212824302530242821241618910

Offset: 1

Dave Durgin, Mar 19 2012

A109242 may be constructed using a similar pyramid algorithm:
....1..........a(1) = 1
...1 1.........a(2) = 111
..1 1 1........a(3) = 11211
.1 1 1 1.......a(4) = 1122211

			...1................a(1) = 1
..2 2...............a(2) = 212
.3 3 3..............a(3) = 32423
4 4 4 4.............a(4) = 4364634

Definition edited by N. J. A. Sloane, Jan 11 2020

A275944 Gaussian binomial coefficient [n,3] for q = 10.

Original entry on oeis.org

1, 1111, 1122211, 1123333211, 1123445443211, 1123456666543211, 1123457788877543211, 1123457901110987543211, 1123457912334332087543211, 1123457913456666543087543211, 1123457913568899988653087543211, 1123457913580123333209753087543211, 1123457913581245667665420753087543211

Offset: 3

Ilya Gutkovskiy, Aug 13 2016

More generally, the ordinary generation function for the Gaussian binomial coefficients [n,k]q is x^k/Product{m=0..k} (1 - q^m*x).
Convolution of A002275 and A147816 (considering offset: 0, 0, 1, 1100, 1110000, ...).
The first seven members are palindromes.

Index entries related to Gaussian binomial coefficients
Index entries for linear recurrences with constant coefficients, signature (1111,-112110,1111000,-1000000)

Cf. A002275, A022174, A109242, A147816.

Table[((10^n - 100) (10^n - 10) (10^n - 1))/890109000, {n, 0, 15}]
Table[QBinomial[n, 3, 10], {n, 3, 15}]

O.g.f.: x^3/((1 - x)*(1 - 10*x)*(1 - 100*x)*(1 - 1000*x)).
E.g.f.: (-1000 + 1110*exp(9*x) - 111*exp(99*x) + exp(999*x))*exp(x)/890109000.
a(n) = 1111*a(n-1) - 112110*a(n-2) + 1111000*a(n-3) - 1000000*a(n-4).
a(n) = ((10^n - 100)*(10^n - 10)*(10^n - 1))/890109000.
a(n) = Product_{i=0..2} (1 - 10^(n-i))/(1 - 10^(i+1)).

cp's OEIS Frontend

A180153 a(n) = 10*a(n-1) + A109242(n).

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

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A349805 a(n) = A261138(n)/11.

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A244842 a(n) = (10^n - 1)*(10^n - 10)/90.

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A210257 Construct a triangle with centered rows in which row n contains n copies of n; form the componentwise sum of the first n rows and concatenate these decimal numbers to get a(n).

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Examples

Extensions

A275944 Gaussian binomial coefficient [n,3] for q = 10.

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Mathematica

Formula

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