A178634 -id:A178634 - cp's OEIS frontend

A002283 a(n) = 10^n - 1.

0, 9, 99, 999, 9999, 99999, 999999, 9999999, 99999999, 999999999, 9999999999, 99999999999, 999999999999, 9999999999999, 99999999999999, 999999999999999, 9999999999999999, 99999999999999999, 999999999999999999, 9999999999999999999, 99999999999999999999, 999999999999999999999, 9999999999999999999999

Offset: 0

N. J. A. Sloane

A friend from Germany remarks that the sequence 9, 99, 999, 9999, 99999, 999999, ... might be called the grumpy German sequence: nein!, nein! nein!, nein! nein! nein!, ...
The Regan link shows that integers of the form 10^n -1 have binary representations with exactly n trailing 1 bits. Also those integers have quinary expressions with exactly n trailing 4's. For example, 10^4 -1 = (304444)5. The first digits in quinary correspond to the number 2^n -1, in our example (30)5 = 2^4 -1. A similar pattern occurs in the binary case. Consider 9 = (1001)2. - Washington Bomfim Dec 23 2010
a(n) is the number of positive integers with less than n+1 digits. - Bui Quang Tuan, Mar 09 2015
From Peter Bala, Sep 27 2015: (Start)
For n >= 1, the simple continued fraction expansion of sqrt(a(2*n)) = [10^n - 1; 1, 2*(10^n - 1), 1, 2*(10^n - 1), ...] has period 2. The simple continued fraction expansion of sqrt(a(2*n))/a(n) = [1; 10^n - 1, 2,  10^n - 1, 2, ...] also has period 2. Note the occurrence of large partial quotients in both expansions.
A theorem of Kuzmin in the measure theory of continued fractions says that large partial quotients are the exception in continued fraction expansions.
Empirically, we also see the presence of unexpectedly large partial quotients early in the continued fraction expansions of the m-th roots of the numbers a(m*n) for m >= 3. Some typical examples are given below. (End)
For n > 0, numbers whose smallest decimal digit is 9. - Stefano Spezia, Nov 16 2023

			From _Peter Bala_, Sep 27 2015: (Start)
Continued fraction expansions showing large partial quotients:
a(12)^(1/3) = [9999; 1, 299999998, 1, 9998, 1, 449999998, 1, 7998, 1, 535714284, 1, 2, 2, 142, 2, 2, 1, 599999999, 3, 1, 1,...].
Compare with a(30)^(1/3) = [9999999999; 1, 299999999999999999998, 1, 9999999998, 1, 449999999999999999998, 1, 7999999998, 1, 535714285714285714284, 1, 2, 2, 142857142, 2, 2, 1, 599999999999999999999, 3, 1, 1,...].
a(24)^(1/4) = [999999; 1, 3999999999999999998, 1, 666665, 1, 1, 1, 799999999999999999, 3, 476190, 7, 190476190476190476, 21, 43289, 1, 229, 1, 1864801864801863, 1, 4, 6,...].
Compare with a(48)^(1/4) = [999999999999; 1, 3999999999999999999999999999999999998, 1, 666666666665, 1, 1, 1, 799999999999999999999999999999999999, 3, 476190476190, 7, 190476190476190476190476190476190476, 21, 43290043289, 1, 229, 1, 1864801864801864801864801864801863, 1, 4, 6,...].
a(25)^(1/5) = [99999, 1, 499999999999999999998, 1, 49998, 1, 999999999999999999998, 1, 33332, 3, 151515151515151515151, 5, 1, 1, 1947, 1, 1, 38, 3787878787878787878, 1, 3, 5,...].
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Peter Bala, A002283 and some continued fraction expansions.
Rick Regan, Nines in quinary, September 8th, 2009.
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A000533, A003020, A007138, A007953, A008591, A010888, A048379, A066138, A073668, A168624, A276352.
Cf. A002275, A002276, A002277, A002278, A002279, A002280, A002281, A002282.
Cf. A075412, A075415, A178630, A178631, A178632, A178633, A178634, A178635.
Cf. A010785.

a002283 = subtract 1 . (10 ^)  -- Reinhard Zumkeller, Feb 21 2014

[(10^n-1): n in [0..20]]; // Vincenzo Librandi, Apr 26 2011

Table[10^n - 1, {n, 0, 22}] (* Michael De Vlieger, Sep 27 2015 *)

A002283(n):=10^n-1$
makelist(A002283(n),n,0,20); /* Martin Ettl, Nov 08 2012 */

a(n)=10^n-1; \\ Charles R Greathouse IV, Jan 30 2012

def a(n): return 10**n-1 # Michael S. Branicky, Feb 25 2023

From Mohammad K. Azarian, Jan 14 2009: (Start)
G.f.: 1/(1-10*x)-1/(1-x).
E.g.f.: e^(10*x)-e^x. (End)
a(n) = A075412(n)/A002275(n) = A178630(n)/A002276(n) = A178631(n)/A002277(n) = A075415(n)/A002278(n) = A178632(n)/A002279(n) = A178633(n)/A002280(n) = A178634(n)/A002281(n) = A178635(n)/A002282(n). - Reinhard Zumkeller, May 31 2010
a(n) = a(n-1) + 9*10^(n-1) with a(0)=0; Also: a(n) = 11*a(n-1) - 10*a(n-2) with a(0)=0, a(1)=9. - Vincenzo Librandi, Jul 22 2010
For n>0, A007953(a(n)) = A008591(n) and A010888(a(n)) = 9. - Reinhard Zumkeller, Aug 06 2010
A048379(a(n)) = 0. - Reinhard Zumkeller, Feb 21 2014
a(n) = Sum_{k=1..n} 9*10^k. - Carauleanu Marc, Sep 03 2016
Sum_{n>=1} 1/a(n) = A073668. - Amiram Eldar, Nov 13 2020
From Elmo R. Oliveira, Jul 19 2025: (Start)
a(n) = 9*A002275(n).
a(n) = A010785(A008591(n)). (End)

More terms from Michael De Vlieger, Sep 27 2015

A002276 a(n) = 2*(10^n - 1)/9.

Original entry on oeis.org

0, 2, 22, 222, 2222, 22222, 222222, 2222222, 22222222, 222222222, 2222222222, 22222222222, 222222222222, 2222222222222, 22222222222222, 222222222222222, 2222222222222222, 22222222222222222, 222222222222222222, 2222222222222222222, 22222222222222222222, 222222222222222222222

Offset: 0

N. J. A. Sloane

a(n) is also the total number of holes in a variation of a box fractal as in illustration. - Kival Ngaokrajang, May 23 2014 [As observed by Hans Havermann, this seems to be incorrect: e.g., for n = 2 the illustration shows 28 small holes plus two larger holes. - M. F. Hasler, Oct 05 2020]

Ivan Panchenko, Table of n, a(n) for n = 0..200
Kival Ngaokrajang, Illustration for n = 1..4.
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A002275, A002277, A002278, A002279, A002280, A002281, A002282, A002283, A178630, A178634.

Cf. A010785, A017185.

LinearRecurrence[{11, -10}, {0, 2}, 50] (* Jinyuan Wang, Feb 27 2020 *)

A002276(n):=2*(10^n - 1)/9$
makelist(A002276(n),n,0,20); /* Martin Ettl, Nov 12 2012 */

a(n)=10^n\9*2 \\ M. F. Hasler, Mar 27 2015

a(n) = A178630(n)/A002283(n). - Reinhard Zumkeller, May 31 2010
From Vincenzo Librandi, Jul 22 2010: (Start)
a(n) = a(n-1) + 2*10^(n-1) with a(0) = 0.
a(n) = 11*a(n-1) - 10*a(n-2) with a(0) = 0, a(1) = 2. (End)
G.f.: 2*x/((1 - x)*(1 - 10*x)). - Ilya Gutkovskiy, Feb 24 2017
E.g.f.: 2*exp(x)*(exp(9*x) - 1)/9. - Stefano Spezia, Sep 13 2023
From Elmo R. Oliveira, Jul 19 2025: (Start)
a(n) = 2*A002275(n).
a(n) = A010785(A017185(n-1)) for n >= 1. (End)

A002281 a(n) = 7*(10^n - 1)/9.

Original entry on oeis.org

0, 7, 77, 777, 7777, 77777, 777777, 7777777, 77777777, 777777777, 7777777777, 77777777777, 777777777777, 7777777777777, 77777777777777, 777777777777777, 7777777777777777, 77777777777777777, 777777777777777777, 7777777777777777777, 77777777777777777777, 777777777777777777777

Offset: 0

N. J. A. Sloane

Ivan Panchenko, 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, A002282, A002283, A178630, A178634.

Cf. A010785, A017245, A099915.

[7*(10^n-1)/9 : n in [0..30]]; // Wesley Ivan Hurt, Mar 24 2015

A002281:=n->7*(10^n-1)/9: seq(A002281(n), n=0..30); # Wesley Ivan Hurt, Mar 24 2015

LinearRecurrence[{11,-10},{0,7},25] (* Robert G. Wilson v, Jul 06 2013 *)

a(n)=7*(10^n-1)/9 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = A178634(n)/A002283(n). - Reinhard Zumkeller, May 31 2010
From Vincenzo Librandi, Jul 22 2010: (Start)
a(n) = a(n-1) + 7*10^(n-1) with n>0, a(0)=0.
a(n) = 11*a(n-1) - 10*a(n-2) with n>1, a(0)=0, a(1)=7. (End)
G.f.: 7*x/((x-1)*(10*x-1)). - Colin Barker, Jan 24 2013
a(n) = 7*A002275(n). - Wesley Ivan Hurt, Mar 24 2015
E.g.f.: 7*exp(x)*(exp(9*x) - 1)/9. - Stefano Spezia, Sep 13 2023
From Elmo R. Oliveira, Jul 20 2025: (Start)
a(n) = (A099915(n) - 1)/2.
a(n) = A010785(A017245(n-1)) for n >= 1. (End)

A002477 Wonderful Demlo numbers: a(n) = ((10^n - 1)/9)^2.

Original entry on oeis.org

1, 121, 12321, 1234321, 123454321, 12345654321, 1234567654321, 123456787654321, 12345678987654321, 1234567900987654321, 123456790120987654321, 12345679012320987654321, 1234567901234320987654321

Offset: 1

N. J. A. Sloane

Only the first nine terms of this sequence are palindromes. - Bui Quang Tuan, Mar 30 2015

Not all of the terms are Demlo numbers as defined by Kaprekar, i.e., concat(L,M,R) with M and L+R repdigits using the same digit. For example, a(10), a(19), a(28) are not, but a(k) for k = 11, 12, ..., 18 are. - M. F. Hasler, Nov 18 2017

			From _José de Jesús Camacho Medina_, Apr 01 2016: (Start)
n=1: ....................... 1 = 9 / 9;
n=2: ..................... 121 = 1089 / 9;
n=3: ................... 12321 = 110889 / 9;
n=4: ................. 1234321 = 11108889 / 9;
n=5: ............... 123454321 = 1111088889 / 9;
n=6: ............. 12345654321 = 111110888889 / 9;
n=7: ........... 1234567654321 = 11111108888889 / 9;
n=8: ......... 123456787654321 = 1111111088888889 / 9;
n=9: ....... 12345678987654321 = 111111110888888889 / 9.        (End)
a(11) = concat(L = 1234567901, R = 20987654321), with L + R = 22222222222 = 2*(10^11-1)/9, of same length as R. - _M. F. Hasler_, Nov 23 2017

D. R. Kaprekar, On Wonderful Demlo numbers, Math. Stud., 6 (1938), 68.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 29.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..300
Lubomira Dvorakova, Stanislav Kruml, and David Ryzak, Antipalindromic numbers, arXiv:2008.06864 [math.CO], 2020. [Mentions this sequence.]
K. R. Gunjikar and D. R. Kaprekar, Theory of Demlo numbers, J. Univ. Bombay, Vol. VIII, Part 3, Nov. 1939, pp. 3-9. [Annotated scanned copy]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Eric Weisstein's World of Mathematics, Demlo Number
Eric Weisstein's World of Mathematics, Repunit
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002275, A080151.

[((10^n - 1)/9)^2: n in [1..20]]; // Vincenzo Librandi, Jul 26 2011

A002477 := proc(n)
    (10^n-1)^2/81 ;
end proc:
seq(A002477(n),n=1..12) ; # R. J. Mathar, Aug 06 2019

Table[FromDigits[PadRight[{},n,1]]^2,{n,15}] (* Harvey P. Dale, Oct 16 2012 *)
(10^Range[20] - 1)^2/81 (* Paolo Xausa, Aug 03 2025 *)

A002477(n):=((10^n - 1)/9)^2$
makelist(A002477(n),n,1,10); /* Martin Ettl, Nov 12 2012 */

a(n) = (10^n\9)^2 \\ Charles R Greathouse IV, Jul 25 2011

G.f.: x*(1+10*x) / ((1-x)*(1-10*x)*(1-100*x)). - Simon Plouffe in his 1992 dissertation
a(n+1) = 100*a(n) + 20*A000042(n) + 1; a(1) = 1. - Reinhard Zumkeller, May 31 2010
a(n) = A000042(n)^2.
a(n) = A075412(n)/9 = A178630(n)/18 = A178631(n)/27 = A075415(n)/36 = A178632(n)/45 = A178633(n)/54 = A178634(n)/63 = A178635(n)/72 = A059988(n)/81. - Reinhard Zumkeller, May 31 2010
a(n+2) = -1000*a(n)+110*a(n+1)+11. - Alexander R. Povolotsky, Jun 06 2014
E.g.f.: exp(x)*(1 - 2*exp(9*x) + exp(99*x))/81. - Stefano Spezia, May 23 2025

Minor edits from N. J. A. Sloane, Aug 18 2009

Further edits from Reinhard Zumkeller, May 12 2010

A075415 Squares of A002280 or numbers (666...6)^2.

Original entry on oeis.org

0, 36, 4356, 443556, 44435556, 4444355556, 444443555556, 44444435555556, 4444444355555556, 444444443555555556, 44444444435555555556, 4444444444355555555556, 444444444443555555555556, 44444444444435555555555556, 4444444444444355555555555556, 444444444444443555555555555556

Offset: 0

Michael Taylor (michael.taylor(AT)vf.vodafone.co.uk), Sep 14 2002

A transformation of the Wonderful Demlo numbers (A002477).

			a(2) = 66^2 = 4356.
From _Reinhard Zumkeller_, May 31 2010: (Start)
n=1: ..................... 36 = 9 * 4;
n=2: ................... 4356 = 99 * 44;
n=3: ................. 443556 = 999 * 444;
n=4: ............... 44435556 = 9999 * 4444;
n=5: ............. 4444355556 = 99999 * 44444;
n=6: ........... 444443555556 = 999999 * 444444;
n=7: ......... 44444435555556 = 9999999 * 4444444;
n=8: ....... 4444444355555556 = 99999999 * 44444444;
n=9: ..... 444444443555555556 = 999999999 * 444444444. (End)

Seiichi Manyama, Table of n, a(n) for n = 0..500
Gérard Villemin, Variations sur les carrés.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A075411, A075412, A075413, A075414, A075415, A075416, A075417.
Cf. A059988, A178630, A178631, A178632, A178633, A178634, A178635. [From Reinhard Zumkeller, May 31 2010]
Cf. A002275, A002278, A002279,  A002280, A002283, A002477, A052041, A102807, A309827.

Table[FromDigits[PadRight[{},n,6]]^2,{n,0,20}] (* or *) LinearRecurrence[ {111,-1110,1000},{0,36,4356},20] (* Harvey P. Dale, May 20 2021 *)

a(n) = A002280(n)^2 = (6*A002275(n))^2 = 36*A002275(n)^2.
a(n) = (6*(10^n-1)/9)^2 = (4/9)*(10^(2*n) - 2*10^n + 1), which is n-1 4's, followed by a 3, n-1 5's and a 6. - Ignacio Larrosa Cañestro, Feb 26 2005
From Reinhard Zumkeller, May 31 2010: (Start)
a(n) = ((A002278(n-1)*10 + 3)*10^(n-1) + A002279(n-1))*10 + 6 for n>0.
a(n) = A002283(n)*A002278(n). (End)
G.f.: 36*x*(1 + 10*x)/((1 - x)*(1 - 10*x)*(1 - 100*x)). - Arkadiusz Wesolowski, Dec 26 2011
From Elmo R. Oliveira, Jul 27 2025: (Start)
E.g.f.: 4*exp(x)*(1 - 2*exp(9*x) + exp(99*x))/9.
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3).
a(n) = 36*A002477(n). (End)

Edited by Alois P. Heinz, Aug 21 2019 (merged with A102794, submitted by Richard C. Schroeppel, Feb 26 2005)

A059988 a(n) = (10^n - 1)^2.

Original entry on oeis.org

0, 81, 9801, 998001, 99980001, 9999800001, 999998000001, 99999980000001, 9999999800000001, 999999998000000001, 99999999980000000001, 9999999999800000000001, 999999999998000000000001, 99999999999980000000000001, 9999999999999800000000000001, 999999999999998000000000000001

Offset: 0

Henry Bottomley, Mar 07 2001

From James D. Klein, Feb 05 2012: (Start)
The periods of the reciprocals of a(n) are the consecutive integers from 0 to 10^n-1, omitting the one integer 10^n-2, right-justified in field widths of size n.
E.g.:
  1/81 = 0.012345679...
  1/9801 = 0.000102030405060708091011...9799000102...
  1/998001 = 0.000001002003004005...997999000001002... (End)
Sum of first 10^n - 1 odd numbers. - Arkadiusz Wesolowski, Jun 12 2013

			From _Reinhard Zumkeller_, May 31 2010: (Start)
n=1: ..................... 81 = 9^2;
n=2: ................... 9801 = 99^2;
n=3: ................. 998001 = 999^2;
n=4: ............... 99980001 = 9999^2;
n=5: ............. 9999800001 = 99999^2;
n=6: ........... 999998000001 = 999999^2;
n=7: ......... 99999980000001 = 9999999^2;
n=8: ....... 9999999800000001 = 99999999^2;
n=9: ..... 999999998000000001 = 999999999^2. (End)

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 32 at p. 61.
Walther Lietzmann, Lustiges und Merkwuerdiges von Zahlen und Formen, (F. Hirt, Breslau 1921-43), p. 149.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 34.

Harry J. Smith, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A075411, A075412, A075413, A075414, A075415, A075416, A075417, A178630, A178631, A178632, A178633, A178634, A178635, A272066, A272067, A272068.
Subsequence of A238237.
Cf. A002275, A002283, A002477.

A059988:=n->(10^n-1)^2; seq(A059988(n), n=0..20); # Wesley Ivan Hurt, Nov 21 2013

Table[(10^n-1)^2, {n,0,20}] (* Wesley Ivan Hurt, Nov 21 2013 *)

a(n) = { (10^n - 1)^2 } \\ Harry J. Smith, Jul 01 2009

def a(n): return (10**n - 1)**2  # Martin Gergov, Sep 10 2022

a(n) = 81*A002477(n) = A002283(n)^2 = (9*A002275(n))^2.
a(n) = {999... (n times)}^2 = {999... (n times), 000... (n times)} - {999... (n times)}. For example, 999^2 = 999000 - 999 = 998001. - Kyle D. Balliet, Mar 07 2009
a(n) = (A002283(n-1)*10 + 8) * 10^(n-1) + 1, for n>0. - Reinhard Zumkeller, May 31 2010
From Ilya Gutkovskiy, Apr 19 2016: (Start)
O.g.f.: 81*x*(1 + 10*x)/((1 - x)*(1 - 10*x)*(1 - 100*x)).
E.g.f.: (1 - 2*exp(9*x) + exp(99*x))*exp(x). (End)
Sum_{n>=1} 1/a(n) = (log(10)*(QPolyGamma(0, 1, 1/10) - log(10/9)) + QPolyGamma(1, 1, 1/10))/log(10)^2 = 0.012448721523422795191... . - Stefano Spezia, Jul 31 2024
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3). - Elmo R. Oliveira, Aug 02 2025

A075412 Squares of A002277.

Original entry on oeis.org

0, 9, 1089, 110889, 11108889, 1111088889, 111110888889, 11111108888889, 1111111088888889, 111111110888888889, 11111111108888888889, 1111111111088888888889, 111111111110888888888889, 11111111111108888888888889, 1111111111111088888888888889, 111111111111110888888888888889

Offset: 0

Michael Taylor (michael.taylor(AT)vf.vodafone.co.uk), Sep 14 2002

A transformation of the Wonderful Demlo numbers (A002477).

			a(2) = 33^2 = 1089.
Contribution from _Reinhard Zumkeller_, May 31 2010: (Start)
n=1: ...................... 9 = 9 * 1;
n=2: ................... 1089 = 99 * 11;
n=3: ................. 110889 = 999 * 111;
n=4: ............... 11108889 = 9999 * 1111;
n=5: ............. 1111088889 = 99999 * 11111;
n=6: ........... 111110888889 = 999999 * 111111;
n=7: ......... 11111108888889 = 9999999 * 1111111;
n=8: ....... 1111111088888889 = 99999999 * 11111111;
n=9: ..... 111111110888888889 = 999999999 * 111111111. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Gérard Villemin, Variations sur les carrés.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A075411, A075412, A075413, A075414, A075415, A075416, A075417, A002283, A178630, A178631, A178632, A178633, A178634, A178635.

Cf. A000042, A002277, A002275, A002282, A002283, A002477, A059988.

LinearRecurrence[{11, -10}, {0, 3}, 20]^2 (* Vincenzo Librandi, Mar 20 2014 *)
Table[FromDigits[PadRight[{},n,9]]FromDigits[PadRight[{},n,1]],{n,0,15}] (* Harvey P. Dale, Feb 12 2023 *)

a(n) = A002277(n)^2 = (3*A002275(n))^2 = 9*A002275(n)^2.
a(n) = {111111... (2n times)} - 2*{ 111... (n times)} a(n) = A000042(2*n) - 2*A000042(n). - Amarnath Murthy, Jul 21 2003
a(n) = {333... (n times)}^2 = {111...(n times)}{000... (n times)} - {111... (n times)}. For example, 333^2 = 111000 - 111 = 110889. - Kyle D. Balliet, Mar 07 2009
From Reinhard Zumkeller, May 31 2010: (Start)
a(n) = A002283(n)*A002275(n).
For n>0, a(n) = (A002275(n-1)*10^n + A002282(n-1))*10 + 9. (End)
a(n) = (10^(n+1)-10)^2/900. - José de Jesús Camacho Medina, Apr 01 2016
From Elmo R. Oliveira, Jul 27 2025: (Start)
G.f.: 9*x*(1+10*x)/((1-x)*(1-10*x)*(1-100*x)).
E.g.f.: exp(x)*(1 - 2*exp(9*x) + exp(99*x))/9.
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3).
a(n) = 9*A002477(n). (End)

A178630 a(n) = 18*((10^n - 1)/9)^2.

Original entry on oeis.org

18, 2178, 221778, 22217778, 2222177778, 222221777778, 22222217777778, 2222222177777778, 222222221777777778, 22222222217777777778, 2222222222177777777778, 222222222221777777777778, 22222222222217777777777778, 2222222222222177777777777778, 222222222222221777777777777778

Offset: 1

Reinhard Zumkeller, May 31 2010

			n=1: ..................... 18 = 9 * 2;
n=2: ................... 2178 = 99 * 22;
n=3: ................. 221778 = 999 * 222;
n=4: ............... 22217778 = 9999 * 2222;
n=5: ............. 2222177778 = 99999 * 22222;
n=6: ........... 222221777778 = 999999 * 222222;
n=7: ......... 22222217777778 = 9999999 * 2222222;
n=8: ....... 2222222177777778 = 99999999 * 22222222;
n=9: ..... 222222221777777778 = 999999999 * 222222222.

G. C. Greubel, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A059988, A075412, A075415, A178631, A178632, A178633, A178634, A178635.

Cf. A002276, A002281, A002283, A002477.

List([1..20], n -> 18*((10^n-1)/9)^2); # G. C. Greubel, Jan 28 2019

[18*((10^n - 1)/9)^2: n in [1..20]]; // Vincenzo Librandi, Dec 28 2010

Table[18*((10^n-1)/9)^2, {n, 1, 20}] (* G. C. Greubel, Jan 28 2019 *)

a(n)=18*(10^n\9)^2 \\ Charles R Greathouse IV, Aug 21 2011

[18*((10^n-1)/9)^2 for n in (1..20)] # G. C. Greubel, Jan 28 2019

a(n) = 18*A002477(n) = A002283(n)*A002276(n).
a(n)=((A002276(n-1)*10 + 1)*10^(n-1) + A002281(n-1))*10 + 8.
G.f.: 18*x*(1 + 10*x)/((1 - x)*(1 - 10*x)*(1 - 100*x)). - Ilya Gutkovskiy, Feb 24 2017
From Elmo R. Oliveira, Jul 30 2025: (Start)
E.g.f.: 2*exp(x)*(1 - 2*exp(9*x) + exp(99*x))/9.
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 3. (End)

A178631 a(n) = 27*((10^n - 1)/9)^2.

Original entry on oeis.org

27, 3267, 332667, 33326667, 3333266667, 333332666667, 33333326666667, 3333333266666667, 333333332666666667, 33333333326666666667, 3333333333266666666667, 333333333332666666666667, 33333333333326666666666667, 3333333333333266666666666667, 333333333333332666666666666667

Offset: 1

Reinhard Zumkeller, May 31 2010

			n=1: ..................... 27 = 9 * 3;
n=2: ................... 3267 = 99 * 33;
n=3: ................. 332667 = 999 * 333;
n=4: ............... 33326667 = 9999 * 3333;
n=5: ............. 3333266667 = 99999 * 33333;
n=6: ........... 333332666667 = 999999 * 333333;
n=7: ......... 33333326666667 = 9999999 * 3333333;
n=8: ....... 3333333266666667 = 99999999 * 33333333;
n=9: ..... 333333332666666667 = 999999999 * 333333333.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A059988, A075412, A075415, A178630, A178632, A178633, A178634, A178635.

Cf. A002277, A002280, A002283, A002477.

[27*((10^n-1)/9)^2: n in [1..50]]; // Vincenzo Librandi, Dec 28 2010

27*(FromDigits/@Table[PadRight[{},n,1],{n,20}])^2 (* or *) LinearRecurrence[ {111,-1110,1000},{27,3267,332667},20] (* Harvey P. Dale, Oct 11 2012 *)

A178631(n):=27*((10^n-1)/9)^2$ makelist(A178631(n),n,1,10); /* Martin Ettl, Nov 12 2012 */

a(n)=27*(10^n\9)^2 \\ Charles R Greathouse IV, Jul 02 2013

a(n) = 27*A002477(n) = A002283(n)*A002277(n).
a(n) = ((A002277(n-1)*10 + 2)*10^(n-1) + A002280(n-1))*10 + 7.
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n>3, a(1)=27, a(2)=3267, a(3)=332667. - Harvey P. Dale, Oct 11 2012
G.f.: 27*x*(1 + 10*x)/((1 - x)*(1 - 10*x)*(1 - 100*x)). - Ilya Gutkovskiy, Feb 24 2017
E.g.f.: exp(x)*(1 - 2*exp(9*x) + exp(99*x))/3. - Elmo R. Oliveira, Aug 01 2025

A178632 a(n) = 45*((10^n - 1)/9)^2.

Original entry on oeis.org

45, 5445, 554445, 55544445, 5555444445, 555554444445, 55555544444445, 5555555444444445, 555555554444444445, 55555555544444444445, 5555555555444444444445, 555555555554444444444445, 55555555555544444444444445, 5555555555555444444444444445, 555555555555554444444444444445

Offset: 1

Reinhard Zumkeller, May 31 2010

			n=1: ..................... 45 = 9 * 5;
n=2: ................... 5445 = 99 * 55;
n=3: ................. 554445 = 999 * 555;
n=4: ............... 55544445 = 9999 * 5555;
n=5: ............. 5555444445 = 99999 * 55555;
n=6: ........... 555554444445 = 999999 * 555555;
n=7: ......... 55555544444445 = 9999999 * 5555555;
n=8: ....... 5555555444444445 = 99999999 * 55555555;
n=9: ..... 555555554444444445 = 999999999 * 555555555.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A059988, A075412, A075415, A178630, A178631, A178633, A178634, A178635.

Cf. A002278, A002279, A002283, A002477.

[45*((10^n-1)/9)^2: n in [1..50]]; // Vincenzo Librandi, Dec 28 2010

45 (FromDigits/@Table[PadRight[{}, n, 1], {n, 20}])^2 (* Vincenzo Librandi, Mar 20 2014 *)
LinearRecurrence[{111,-1110,1000},{45,5445,554445},20] (* Harvey P. Dale, Jan 23 2019 *)

A178632(n):=45*((10^n-1)/9)^2$ makelist(A178632(n),n,1,12); /* Martin Ettl, Nov 08 2012 */

a(n)=45*(10^n\9)^2 \\ Charles R Greathouse IV, Jul 02 2013

a(n) = 45*A002477(n) = A002283(n)*A002279(n).
a(n) = (A002279(n-1)*10^n + A002278(n))*10 + 5.
G.f.: 45*x*(1 + 10*x)/((1 - x)*(1 - 10*x)*(1 - 100*x)). - Ilya Gutkovskiy, Feb 24 2017
From Elmo R. Oliveira, Aug 01 2025: (Start)
E.g.f.: 5*exp(x)*(1 - 2*exp(9*x) + exp(99*x))/9.
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 3. (End)

cp's OEIS Frontend

A002283 a(n) = 10^n - 1.

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A002276 a(n) = 2*(10^n - 1)/9.

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A002281 a(n) = 7*(10^n - 1)/9.

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A002477 Wonderful Demlo numbers: a(n) = ((10^n - 1)/9)^2.

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A075415 Squares of A002280 or numbers (666...6)^2.

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A059988 a(n) = (10^n - 1)^2.

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