A059482 -id:A059482 - cp's OEIS frontend

A002282 a(n) = 8*(10^n - 1)/9.

0, 8, 88, 888, 8888, 88888, 888888, 8888888, 88888888, 888888888, 8888888888, 88888888888, 888888888888, 8888888888888, 88888888888888, 888888888888888, 8888888888888888, 88888888888888888, 888888888888888888, 8888888888888888888, 88888888888888888888, 888888888888888888888

Offset: 0

N. J. A. Sloane

If the initial term is omitted, might be called eightful (or hateful) numbers!

			Curious multiplications:
9*9 + 7 = 88;
98*9 + 6 = 888;
987*9 + 5 = 8888;
9876*9 + 4 = 88888;
98765*9 + 3 = 888888;
987654*9 + 2 = 8888888;
9876543*9 + 1 = 88888888;
98765432*9 + 0 = 888888888;
987654321*9 - 1 = 8888888888;
9876543210*9 - 2 = 88888888888. - _Philippe Deléham_, Mar 09 2014

Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 32.

Harry J. Smith, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A002275, A002276, A002277, A002278, A002279, A002280, A002281, A002283, A059988, A075412, A178635.

Cf. A010785, A017257, A051003, A059482, A246058.

A002282:=n->8*(10^n - 1)/9; seq(A002282(n), n=0..20); # Wesley Ivan Hurt, Mar 10 2014

LinearRecurrence[{11,-10}, {0,8}, 20] (* Harvey P. Dale, May 30 2013 *)

{ a=-4/5; for (n = 0, 200, a+=8*10^(n - 1); write("b002282.txt", n, " ", a); ) } \\ Harry J. Smith, Jun 27 2009

def a(n): return 8*(10**n - 1)//9 # Martin Gergov, Oct 19 2022

From Jaume Oliver Lafont, Feb 03 2009: (Start)
a(n) = 11*a(n-1) - 10*a(n-2), with a(0)=0, a(1)=8.
G.f.: 8*x/((1-x)*(1-10*x)). (End)
a(n) = A178635(n)/A002283(n). - Reinhard Zumkeller, May 31 2010
a(n) = a(n-1) + 8*10^(n-1), with a(0)=0. - Vincenzo Librandi, Jul 22 2010
a(n) = 8*A002275(n) = A002283(n) - A002275(n). - Carauleanu Marc, Sep 03 2016
From Ilya Gutkovskiy, Sep 03 2016: (Start)
E.g.f.: 8*(exp(9*x) - 1)*exp(x)/9.
a(n) = floor(8*10^n/9). (End)
From Elmo R. Oliveira, Jul 20 2025: (Start)
a(n) = (A246058(n) - 1)/2.
a(n) = A010785(A017257(n-1)) for n >= 1. (End)

A083064 Square number array T(n,k) = (k*(k+2)^n+1)/(k+1) read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 11, 14, 1, 1, 5, 19, 43, 41, 1, 1, 6, 29, 94, 171, 122, 1, 1, 7, 41, 173, 469, 683, 365, 1, 1, 8, 55, 286, 1037, 2344, 2731, 1094, 1, 1, 9, 71, 439, 2001, 6221, 11719, 10923, 3281, 1, 1, 10, 89, 638, 3511, 14006, 37325, 58594, 43691, 9842, 1

Offset: 0

Paul Barry, Apr 21 2003

			Rows begin:
1  1   1    1     1      1       1        1         1 ...
1  2   5   14    41    122     365     1094      3281 ...  A007051
1  3  11   43   171    683    2731    10923     43691 ...  A007583
1  4  19   94   469   2344   11719    58594    292969 ...  A083065
1  5  29  173  1037   6221   37325   223949   1343693 ...  A083066
1  6  41  286  2001  14006   98041   686286   4804001 ...  A083067
1  7  55  439  3511  28087  224695  1797559  14380471 ...  A083068
1  8  71  638  5741  51668  465011  4185098  37665881 ...  A187709
1  9  89  889  8889  88889  888889  8888889  88888889 ...  A059482
1 10 109 1198 13177 144946 1594405 17538454 192922993 ...  A199760, etc.
Column 2: A000027;
column 3: A028387;
column 4: A083074;
column 5: A125082;
column 6: A125083.
Diagonals:
1,  2,  11,   94,  1037,  14006, ... A083069;
1,  3,  19,  173,  2001,  28087, ... A083071;
1,  4,  29,  286,  3511,  51668, ... A083072;
1,  5,  41,  439,  5741,  88889, ... A083073;
1,  5,  43,  469,  6221,  98041, ... A083070;
1, 14, 171, 2344, 37325, 686286, ... A191690.
Triangle begins:
1;
1, 1;
1, 2, 1;
1, 3, 5, 1;
1, 4, 11, 14, 1;
1, 5, 19, 43, 41, 1;
1, 6, 29, 94, 171, 122, 1; etc.

Cf. rows: A007051, A007583, A059482, A083065 - A083068, A187709, A199760; columns: A000027, A028387, A083074, A125082, A125083; diagonals: A083069 - A083073, A191690.

Edited by Bruno Berselli, Jun 21 2013

A187709 a(n) = (7*9^n + 1)/8.

Original entry on oeis.org

1, 8, 71, 638, 5741, 51668, 465011, 4185098, 37665881, 338992928, 3050936351, 27458427158, 247125844421, 2224132599788, 20017193398091, 180154740582818, 1621392665245361, 14592533987208248, 131332805884874231, 1181995252963868078, 10637957276674812701, 95741615490073314308, 861674539410659828771

Offset: 0

Sture Sjöstedt, Mar 30 2011

Case r=9 in a(n)=((r-2)*r^n+1)/(r-1).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Cf. A007051, A083068, A270472 (first differences), A059482: cases r=3,8,10 in ((r-2)*r^n+1)/(r-1), respectively.

[(7*9^n+1)/8: n in [0..25]]; // Vincenzo Librandi, Mar 30 2011

(7*9^Range[0,30]+1)/8 (* or *) LinearRecurrence[{10,-9},{1,8},30] (* Harvey P. Dale, Jul 20 2012 *)

a(n)=(7*9^n+1)/8 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (7*9^n + 1)/8.
a(n) = +10*a(n-1) -9*a(n-2).
a(n) = 8*Sum_{i=0..n-1} a(i) -n + 1.
G.f.: (1-2*x)/((1-x)*(1-9*x)).
a(n) = 9^n - Sum_{i=0..n-1} 9^i for n>0. - Bruno Berselli, Jun 20 2013
E.g.f.: (7*exp(9*x) + exp(x))/8. - G. C. Greubel, Nov 06 2018

Additional formulas from Bruno Berselli

A199688 a(n) = (8*10^n + 1)/3.

Original entry on oeis.org

3, 27, 267, 2667, 26667, 266667, 2666667, 26666667, 266666667, 2666666667, 26666666667, 266666666667, 2666666666667, 26666666666667, 266666666666667, 2666666666666667, 26666666666666667, 266666666666666667, 2666666666666666667, 26666666666666666667, 266666666666666666667

Offset: 0

Vincenzo Librandi, Nov 09 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A059482, A199689.

Magma
```
[(8*10^n+1)/3: n in [0..30]];
```

LinearRecurrence[{11,-10},{3,27},20] (* Harvey P. Dale, Jul 24 2016 *)

a(n) = 3*A059482(n).
a(n) = 10*a(n-1) - 3.
a(n) = 11*a(n-1) - 10*a(n-2).
G.f.: 3*(1-2*x)/((1-x)*(1-10*x)).
From Elmo R. Oliveira, May 04 2025: (Start)
E.g.f.: exp(x)*(1 + 8*exp(9*x))/3.
a(n) = A199689(n)/3. (End)

A199689 a(n) = 8*10^n + 1.

Original entry on oeis.org

9, 81, 801, 8001, 80001, 800001, 8000001, 80000001, 800000001, 8000000001, 80000000001, 800000000001, 8000000000001, 80000000000001, 800000000000001, 8000000000000001, 80000000000000001, 800000000000000001, 8000000000000000001, 80000000000000000001, 800000000000000000001

Offset: 0

Vincenzo Librandi, Nov 09 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A059482, A199688.

Magma
```
[8*10^n+1: n in [0..30]];
```

8*10^Range[0,20]+1 (* or *) LinearRecurrence[{11,-10},{9,81},20] (* Harvey P. Dale, Jun 11 2022 *)

a(n) = 9*A059482(n).
a(n) = 10*a(n-1) - 9.
a(n) = 11*a(n-1) - 10*a(n-2).
G.f.: 9*(1-2*x)/((1-x)*(1-10*x)).
From Elmo R. Oliveira, May 04 2025: (Start)
E.g.f.: exp(x)*(1 + 8*exp(9*x)).
a(n) = 3*A199688(n). (End)

A138342 First differences of A007088.

Original entry on oeis.org

1, 9, 1, 89, 1, 9, 1, 889, 1, 9, 1, 89, 1, 9, 1, 8889, 1, 9, 1, 89, 1, 9, 1, 889, 1, 9, 1, 89, 1, 9, 1, 88889, 1, 9, 1, 89, 1, 9, 1, 889, 1, 9, 1, 89, 1, 9, 1, 8889, 1, 9, 1, 89, 1, 9, 1, 889, 1, 9, 1, 89, 1, 9, 1, 888889, 1, 9, 1, 89, 1, 9, 1, 889, 1, 9, 1, 89, 1, 9, 1, 8889, 1, 9, 1, 89, 1, 9

Offset: 1

Jaume Simon Gispert (jaume(AT)nuem.com), May 17 2008

			1-0 = 1, 10-1 = 9, 11-10 = 1, 100-11 = 89, ...

Antti Karttunen, Table of n, a(n) for n = 1..16383

Cf. A007088, A007814, A059482.

Differences[Table[FromDigits[IntegerDigits[n,2]],{n,0,90}]] (* Harvey P. Dale, Feb 26 2012 *)

A007088(n) = fromdigits(binary(n), 10); \\ From A007088.
A138342(n) = (A007088(n) - A007088(n-1)); \\ Antti Karttunen, Nov 06 2018

A059482(n) = ((10^n)*(1000/1125) + (1/9));
A138342(n) = { my(f=factor(n)); prod(i=1,#f~,if(2==f[i,1],A059482(f[i,2]),1)); }; \\ Antti Karttunen, Nov 06 2018

a(n) = A059482(A007814(n)).
From Antti Karttunen, Nov 06 2018: (Start)
a(n) = A007088(n) - A007088(n-1).
Multiplicative with a(2^e) = A059482(e), a(p^e) = 1 for odd primes p.
(End)
G.f.: Sum_{k>=0} 10^k * x^(2^k) / (1 + x^(2^k)). - Ilya Gutkovskiy, Dec 14 2020

Offset corrected and keyword:mult added by Antti Karttunen, Nov 06 2018

A184337 a(n) is the integer whose decimal representation consists of n 8's followed by n 1's.

Original entry on oeis.org

0, 81, 8811, 888111, 88881111, 8888811111, 888888111111, 88888881111111, 8888888811111111, 888888888111111111, 88888888881111111111, 8888888888811111111111, 888888888888111111111111, 88888888888881111111111111

Offset: 0

Paul Curtz, Feb 13 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Richard Blecksmith and Charles Nicol, Monotonic Numbers, Mathematics Magazine, 66, (1993), 257-262.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A098210 (with 1 and 5 instead of 8 and 1).
Cf. A002283, A059482.
Cf. A008591 (digits sums).

[(8*100^n-7*10^n-1)/9: n in [0..20]]; // Vincenzo Librandi, Aug 04 2011

Table[(8*100^n - 7*10^n - 1)/9, {n,0,30}] (* G. C. Greubel, Nov 02 2018 *)
FromDigits/@Table[Join[PadRight[{},n,8],PadRight[{},n,1]],{n,0,15}] (* or *) LinearRecurrence[ {111,-1110,1000},{0,81,8811},15] (* Harvey P. Dale, Jul 03 2023 *)

vector(30, n, n--; (8*100^n - 7*10^n - 1)/9) \\ G. C. Greubel, Nov 02 2018

for n in range(30):
    print((8*100**n-7*10**n-1)//9, end=', ')
# Stefano Spezia, Nov 02 2018

a(n) = (8*100^n - 7*10^n - 1)/9.
a(n) = A059482(n)*A002283(n).
G.f.: 9*x*(-9+20*x) / ( (x-1)*(100*x-1)*(10*x-1) ). - R. J. Mathar, Feb 28 2011

A105255 Number of distinct prime divisors of 88...889 (with n 8's).

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 2, 5, 3, 2, 5, 3, 1, 6, 4, 1, 4, 4, 3, 4, 4, 2, 3, 3, 3, 4, 4, 2, 6, 5, 4, 6, 3, 1, 5, 6, 2, 6, 4, 3, 7, 5, 2, 8, 7, 4, 7, 5, 4, 7, 3, 7, 5, 3, 3, 9, 6, 5, 3, 3, 3, 7, 3, 4, 6, 4, 2, 6, 7, 5, 7, 7, 4, 7, 6, 4, 5, 3, 4, 6, 2, 2, 7, 5, 3, 9, 6, 4, 8, 6, 7, 5, 2, 5, 6, 5, 5, 7, 7

Offset: 0

Parthasarathy Nambi, Apr 29 2005

			The number of distinct prime divisors of 89 is 1 (prime).
The number of distinct prime divisors of 889 is 2.
The number of distinct prime divisors of 8889 is 2.

Amiram Eldar, Table of n, a(n) for n = 0..225

Cf. A104543.

Table[PrimeNu[(8*10^(n + 1) + 1)/9], {n, 0,50}] (* G. C. Greubel, May 16 2017 *)

a(n) = omega((8*10^(n+1)+1)/9); \\ Michel Marcus, Jan 27 2014

a(n) = A001221(A059482(n+1)). - Michel Marcus, Jan 27 2014

A108904 a(n) has two outer digits 9 and n inner digits 7.

Original entry on oeis.org

99, 979, 9779, 97779, 977779, 9777779, 97777779, 977777779, 9777777779, 97777777779, 977777777779, 9777777777779, 97777777777779, 977777777777779, 9777777777777779, 97777777777777779, 977777777777777779, 9777777777777777779, 97777777777777777779, 977777777777777777779

Offset: 0

Cino Hilliard, Jul 16 2005

These numbers are all composite. They are divisible by 3,7 or 11.

All terms are divisible by 11. - Jason Yuen, Sep 02 2024

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

Cf. A059482.

g(n,r=7,m=9) = /* Repeat rrr.. surrounded by 2 m's */ { for(x=0,n, y=m*10^(x+1)+m+r*10*(10^x-1)/9; print1(y",") ) }

a(n) = 9*10^(n+1) + 9 + 70*(10^n-1)/9.
a(n) = 11*(80*10^n + 1)/9. - Jason Yuen, Sep 02 2024
From Alois P. Heinz, Sep 02 2024: (Start)
a(n) = 11*A059482(n+1).
G.f.: 11*(9-10*x)/((10*x-1)*(x-1)). (End)

Minor edits by N. J. A. Sloane, Aug 01 2010

a(0) prepended by Alois P. Heinz, Sep 02 2024

cp's OEIS Frontend

A002282 a(n) = 8*(10^n - 1)/9.

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A083064 Square number array T(n,k) = (k*(k+2)^n+1)/(k+1) read by antidiagonals.

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A187709 a(n) = (7*9^n + 1)/8.

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Magma

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A199688 a(n) = (8*10^n + 1)/3.

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Magma

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A199689 a(n) = 8*10^n + 1.

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Magma

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A138342 First differences of A007088.

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A184337 a(n) is the integer whose decimal representation consists of n 8's followed by n 1's.

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