A057932 -id:A057932 - cp's OEIS frontend

A099669 Partial sums of repdigits of A002276.

2, 24, 246, 2468, 24690, 246912, 2469134, 24691356, 246913578, 2469135800, 24691358022, 246913580244, 2469135802466, 24691358024688, 246913580246910, 2469135802469132, 24691358024691354, 246913580246913576, 2469135802469135798, 24691358024691358020, 246913580246913580242

Offset: 1

Labos Elemer, Nov 17 2004

			2 + 22 + 222 + 2222 = a(4) = 2468.

Matthew House, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (12,-21,10).

Cf. A002275-A002283, A014824, A057932.

A099669:=n->(2/81)*(10^(n+1) - 9*n - 10): seq(A099669(n), n=1..30); # Wesley Ivan Hurt, Apr 18 2017

<Robert G. Wilson v, Nov 20 2004 *)
LinearRecurrence[{12,-21,10},{2,24,246},30] (* Harvey P. Dale, Jun 01 2025 *)

a(n) = (2/81)*(10^(n+1) - 9*n - 10). - R. Piyo (nagoya314(AT)yahoo.com), Dec 10 2004
a(n) = 12*a(n-1) - 21*a(n-2) + 10*a(n-3). - Matthew House, Jun 30 2016
G.f.: 2*x/((1 - x)^2*(1 - 10*x)). - Ilya Gutkovskiy, Jun 30 2016
From Elmo R. Oliveira, Apr 02 2025: (Start)
E.g.f.: 2*exp(x)*(10*exp(9*x) - 9*x - 10)/81.
a(n) = 2*A014824(n). (End)

A099675 Partial sums of repdigits of A002282.

Original entry on oeis.org

8, 96, 984, 9872, 98760, 987648, 9876536, 98765424, 987654312, 9876543200, 98765432088, 987654320976, 9876543209864, 98765432098752, 987654320987640, 9876543209876528, 98765432098765416, 987654320987654304, 9876543209876543192, 98765432098765432080, 987654320987654320968

Offset: 1

Labos Elemer, Nov 17 2004

			8 + 88 + 888 + 8888 + 88888 = a(5) = 98760.

Index entries for linear recurrences with constant coefficients, signature (12,-21,10).

Cf. A002275-A002283, A014824, A057932, A099669-A099675.

Mathematica
```
<Robert G. Wilson v, Nov 20 2004 *)
```

a(n) = (8/81)*(10^(n+1) - 9*n - 10). - R. Piyo (nagoya314(AT)yahoo.com), Dec 10 2004
a(n) = 12*a(n-1) - 21*a(n-2) + 10*a(n-3). - Wesley Ivan Hurt, Jan 20 2024
From Elmo R. Oliveira, Apr 02 2025: (Start)
G.f.: 8*x/((1 - x)^2*(1 - 10*x)).
E.g.f.: 8*exp(x)*(10*exp(9*x) - 9*x - 10)/81.
a(n) = 8*A014824(n). (End)

More terms from Elmo R. Oliveira, Apr 02 2025

A070189 a(n) = 12345679*n.

Original entry on oeis.org

0, 12345679, 24691358, 37037037, 49382716, 61728395, 74074074, 86419753, 98765432, 111111111, 123456790, 135802469, 148148148, 160493827, 172839506, 185185185, 197530864, 209876543, 222222222, 234567901, 246913580, 259259259, 271604938, 283950617, 296296296, 308641975

Offset: 0

Henry Bottomley, Apr 24 2002

a(82)=1012345678 is the first term which has a digit appearing more than once without an obvious pattern, although a(-82)=-1012345678 might be seen as the concatenation of ten consecutive numbers.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See entry 12345679 at p. 188.

Tanya Khovanova, Recursive Sequences.
"TyYann", MegaFavNumbers: Lewis Carroll's Number, video (2020).
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A027889, A057932, A060073, A085959.

Table[12345679*n,{n,0,30}] (* or *) LinearRecurrence[{2,-1},{0,12345679},30] (* Harvey P. Dale, Oct 16 2015 *)

a(n)=12345679*n \\ Charles R Greathouse IV, Jan 09 2012

concat(0, Vec(12345679*x/(1-x)^2 + O(x^26))) \\ Elmo R. Oliveira, Jun 26 2025

a(n) = n*(10^(10-1)-1)/(10-1)^2.
From Elmo R. Oliveira, Jun 26 2025: (Start)
G.f.: 12345679*x/(1-x)^2.
E.g.f.: 12345679*x*exp(x).
a(n) = 333667*A085959(n).
a(n) = 2*a(n-1) - a(n-2). (End)

More terms from Elmo R. Oliveira, Jun 26 2025

A099674 Partial sums of repdigits of A002281.

Original entry on oeis.org

0, 7, 84, 861, 8638, 86415, 864192, 8641969, 86419746, 864197523, 8641975300, 86419753077, 864197530854, 8641975308631, 86419753086408, 864197530864185, 8641975308641962, 86419753086419739, 864197530864197516, 8641975308641975293, 86419753086419753070, 864197530864197530847

Offset: 0

Labos Elemer, Nov 17 2004

			7 + 77 + 777 + 7777 + 77777 = a(5) = 86415.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-21,10).

Cf. A002275-A002283, A014824, A057932, A099669-A099675.

<Robert G. Wilson v, Nov 20 2004 *)
Accumulate[LinearRecurrence[{11,-10},{0,7},25]] (* Harvey P. Dale, Jul 22 2025 *)

a(n) = (7/81)*(10^(n+1) - 9*n - 10). - R. Piyo (nagoya314(AT)yahoo.com), Dec 10 2004
From Elmo R. Oliveira, Apr 02 2025: (Start)
G.f.: 7*x/((1 - x)^2*(1 - 10*x)).
E.g.f.: 7*exp(x)*(10*exp(9*x) - 9*x - 10)/81.
a(n) = 7*A014824(n).
a(n) = 12*a(n-1) - 21*a(n-2) + 10*a(n-3) for n > 3. (End)

More terms from Elmo R. Oliveira, Apr 02 2025

a(0)=0 prepended by Harvey P. Dale, Jul 22 2025

A099676 Partial sums of repdigits of A002283.

Original entry on oeis.org

9, 108, 1107, 11106, 111105, 1111104, 11111103, 111111102, 1111111101, 11111111100, 111111111099, 1111111111098, 11111111111097, 111111111111096, 1111111111111095, 11111111111111094, 111111111111111093, 1111111111111111092, 11111111111111111091

Offset: 1

Labos Elemer, Nov 17 2004

a(n) is the maximal positive integer k such that the sequence 1, 2, 3, 4, ..., k-1, k has a total of n*k digits. - Bui Quang Tuan, Mar 12 2015

			9 + 99 + 999 + 9999 + 99999 = a(5) = 111105.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (12,-21,10).

Cf. A057932, A002275-A002283, A099669-A099674.

[(10/9)*(10^n-1)-n: n in [1..20]]; // Vincenzo Librandi, Mar 14 2014

a:=n->sum((10^(n-j)-1^(n-j)), j=0..n): seq(a(n), n=1..17); # Zerinvary Lajos, Jan 15 2007

<Vincenzo Librandi, Mar 14 2014 *)
LinearRecurrence[{12,-21,10},{9,108,1107},20] (* Harvey P. Dale, Apr 18 2015 *)

Vec(-9*x/((x-1)^2*(10*x-1)) + O(x^100)) \\ Colin Barker, Mar 12 2014

[gaussian_binomial(n,1,10)-n for n in range(2,19)] # Zerinvary Lajos, May 29 2009

a(n) = (10/9)*(10^n-1) - n. - R. Piyo (nagoya314(AT)yahoo.com), Dec 10 2004
From Colin Barker, Mar 12 2014: (Start)
a(n) = 12*a(n-1)-21*a(n-2)+10*a(n-3).
G.f.: -9*x / ((x-1)^2*(10*x-1)). (End)
E.g.f.: exp(x)*(10*(exp(9*x) - 1) - 9*x)/9. - Stefano Spezia, Sep 13 2023

A057933 Floor[(80/81)*10^n].

Original entry on oeis.org

9, 98, 987, 9876, 98765, 987654, 9876543, 98765432, 987654320, 9876543209, 98765432098, 987654320987, 9876543209876, 98765432098765, 987654320987654, 9876543209876543, 98765432098765432, 987654320987654320

Offset: 1

Henry Bottomley, Oct 04 2000

nonn

Index entries for linear recurrences with constant coefficients, signature (10,0,0,0,0,0,0,0,1,-10).

Cf. A057932.

Floor[80/81*10^Range[20]] (* Harvey P. Dale, May 28 2017 *)

PARI
```
a(n)=10^n*80\81
```

G.f.:(9x+8x^2+7x^3+6x^4+5x^5+4x^6+3x^7+2x^8)/((1-10x)(1-x^9)).

a(n)=10*a(n-1)-(n%9)+10*(n%9!=0).

A099672 Partial sums of repdigits of A002279.

Original entry on oeis.org

5, 60, 615, 6170, 61725, 617280, 6172835, 61728390, 617283945, 6172839500, 61728395055, 617283950610, 6172839506165, 61728395061720, 617283950617275, 6172839506172830, 61728395061728385, 617283950617283940, 6172839506172839495, 61728395061728395050, 617283950617283950605

Offset: 1

Labos Elemer, Nov 17 2004

			5 + 55 + 555 + 5555 + 55555 = a(5) = 61725.

Colin Barker, Table of n, a(n) for n = 1..999
Index entries for linear recurrences with constant coefficients, signature (12,-21,10).

Cf. A002275-A002283, A014824, A057932, A099669-A099675.

<Robert G. Wilson v, Nov 20 2004 *)
Accumulate[Table[FromDigits[PadRight[{},n,5]],{n,0,20}]] (* Harvey P. Dale, Oct 05 2013 *)

Vec(5*x/((1 - x)^2*(1 - 10*x)) + O(x^40)) \\ Colin Barker, Nov 30 2017

a(n) = (5/81)*(10^(n+1) - 9*n - 10). - R. Piyo (nagoya314(AT)yahoo.com), Dec 10 2004.
From Colin Barker, Nov 30 2017: (Start)
G.f.: 5*x/((1 - x)^2*(1 - 10*x)).
a(n) = 12*a(n-1) - 21*a(n-2) + 10*a(n-3) for n > 3. (End)
From Elmo R. Oliveira, Apr 03 2025: (Start)
E.g.f.: 5*exp(x)*(10*exp(9*x) - 9*x - 10)/81.
a(n) = 5*A014824(n). (End)

A099670 Partial sums of repdigits of A002277.

Original entry on oeis.org

3, 36, 369, 3702, 37035, 370368, 3703701, 37037034, 370370367, 3703703700, 37037037033, 370370370366, 3703703703699, 37037037037032, 370370370370365, 3703703703703698, 37037037037037031, 370370370370370364, 3703703703703703697, 37037037037037037030, 370370370370370370363

Offset: 1

Labos Elemer, Nov 17 2004

			3 + 33 + 333 + 3333 = a(4) = 3702.

Index entries for linear recurrences with constant coefficients, signature (12,-21,10).

Cf. A002275-A002283, A014824, A057932.

a:=n->sum((10^(n-j)-1^(n-j))/3,j=0..n): seq(a(n), n=1..18); # Zerinvary Lajos, Jan 15 2007

Mathematica
```
<Robert G. Wilson v, Nov 20 2004 *)
```

a(n) = (3/81)*(10^(n+1) - 9*n - 10). - R. Piyo (nagoya314(AT)yahoo.com), Dec 10 2004
From Elmo R. Oliveira, Apr 02 2025: (Start)
G.f.: 3*x/((1 - x)^2*(1 - 10*x)).
E.g.f.: 3*exp(x)*(10*exp(9*x) - 9*x - 10)/81.
a(n) = 3*A014824(n).
a(n) = 12*a(n-1) - 21*a(n-2) + 10*a(n-3) for n > 3. (End)

More terms from Elmo R. Oliveira, Apr 02 2025

A099671 Partial sums of repdigits of A002278.

Original entry on oeis.org

4, 48, 492, 4936, 49380, 493824, 4938268, 49382712, 493827156, 4938271600, 49382716044, 493827160488, 4938271604932, 49382716049376, 493827160493820, 4938271604938264, 49382716049382708, 493827160493827152, 4938271604938271596, 49382716049382716040, 493827160493827160484

Offset: 1

Labos Elemer, Nov 17 2004

			4 + 44 + 444 + 4444 + 44444 = a(5) = 49380.

Index entries for linear recurrences with constant coefficients, signature (12,-21,10).

Cf. A002275-A002283, A014824, A057932.

Mathematica
```
<Robert G. Wilson v, Nov 20 2004 *)
```

a(n) = (4/81)*(10^(n+1) - 9*n - 10). - R. Piyo (nagoya314(AT)yahoo.com), Dec 10 2004
From Chai Wah Wu, Feb 28 2018: (Start)
a(n) = 12*a(n-1) - 21*a(n-2) + 10*a(n-3) for n > 3.
G.f.: 4*x/((1 - x)^2*(1 - 10*x)). (End)
From Elmo R. Oliveira, Apr 03 2025: (Start)
E.g.f.: 4*exp(x)*(10*exp(9*x) - 9*x - 10)/81.
a(n) = 4*A014824(n). (End)

More terms from Elmo R. Oliveira, Apr 03 2025

A099673 Partial sums of repdigits of A002280.

Original entry on oeis.org

6, 72, 738, 7404, 74070, 740736, 7407402, 74074068, 740740734, 7407407400, 74074074066, 740740740732, 7407407407398, 74074074074064, 740740740740730, 7407407407407396, 74074074074074062, 740740740740740728, 7407407407407407394, 74074074074074074060, 740740740740740740726

Offset: 1

Labos Elemer, Nov 17 2004

			6 + 66 + 666 + 6666 + 66666 = a(5) = 74070.

Index entries for linear recurrences with constant coefficients, signature (12,-21,10).

Cf. A002275-A002283, A014824, A057932, A099669-A099675.

Mathematica
```
<Robert G. Wilson v, Nov 20 2004 *)
```

a(n) = (2/27)*(10^(n+1) - 9*n - 10). - R. Piyo (nagoya314(AT)yahoo.com), Dec 10 2004
From Elmo R. Oliveira, Apr 02 2025: (Start)
G.f.: 6*x/((1 - x)^2*(1 - 10*x)).
a(n) = 6*A014824(n).
E.g.f.: 2*exp(x)*(10*exp(9*x) - 9*x - 10)/27.
a(n) = 12*a(n-1) - 21*a(n-2) + 10*a(n-3) for n > 3. (End)

More terms from Elmo R. Oliveira, Apr 02 2025

cp's OEIS Frontend

A099669 Partial sums of repdigits of A002276.

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A099675 Partial sums of repdigits of A002282.

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A070189 a(n) = 12345679*n.

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A099674 Partial sums of repdigits of A002281.

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A099676 Partial sums of repdigits of A002283.

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A057933 Floor[(80/81)*10^n].

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A099672 Partial sums of repdigits of A002279.

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