A059988 -id:A059988 - cp's OEIS frontend

A178634 a(n) = 63*((10^n - 1)/9)^2.

63, 7623, 776223, 77762223, 7777622223, 777776222223, 77777762222223, 7777777622222223, 777777776222222223, 77777777762222222223, 7777777777622222222223, 777777777776222222222223, 77777777777762222222222223, 7777777777777622222222222223, 777777777777776222222222222223

Offset: 1

Reinhard Zumkeller, May 31 2010

			n=1: ..................... 63 = 9 * 7;
n=2: ................... 7623 = 99 * 77;
n=3: ................. 776223 = 999 * 777;
n=4: ............... 77762223 = 9999 * 7777;
n=5: ............. 7777622223 = 99999 * 77777;
n=6: ........... 777776222223 = 999999 * 777777;
n=7: ......... 77777762222223 = 9999999 * 7777777;
n=8: ....... 7777777622222223 = 99999999 * 77777777;
n=9: ..... 777777776222222223 = 999999999 * 777777777.

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 33 at p. 62.
Walther Lietzmann, Lustiges und Merkwuerdiges von Zahlen und Formen, (F. Hirt, Breslau 1921-43), p. 149.

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, A178630, A178631, A178632, A178633, A178635.

Cf. A002276, A002281, A002283, A002477.

List([1..20], n -> 63*((10^n - 1)/9)^2); # G. C. Greubel, Jan 28 2019

[63*((10^n - 1)/9)^2: n in [1..20]]; // Vincenzo Librandi, Dec 28 2010

63((10^Range[15]-1)/9)^2 (* or *) Table[FromDigits[Join[PadRight[{},n,7],{6},PadRight[{},n,2],{3}]],{n,0,15}] (* Harvey P. Dale, Apr 23 2012 *)

a(n)=63*(10^n\9)^2 \\ Charles R Greathouse IV, Jul 02 2013

[63*((10^n - 1)/9)^2 for n in (1..20)] # G. C. Greubel, Jan 28 2019

a(n) = 63*A002477(n) = A002283(n)*A002281(n).
a(n) = ((A002281(n-1)*10 + 6)*10^(n-1) + A002276(n-1))*10 + 3.
G.f.: 63*x*(1 + 10*x)/((1 - x)*(1 - 10*x)*(1 - 100*x)). - Ilya Gutkovskiy, Feb 24 2017
E.g.f.: 7*exp(x)*(1 - 2*exp(9*x) + exp(99*x))/9. - Stefano Spezia, Jul 31 2024
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 3. - Elmo R. Oliveira, Aug 01 2025

A178635 a(n) = 72*((10^n - 1)/9)^2.

Original entry on oeis.org

72, 8712, 887112, 88871112, 8888711112, 888887111112, 88888871111112, 8888888711111112, 888888887111111112, 88888888871111111112, 8888888888711111111112, 888888888887111111111112, 88888888888871111111111112, 8888888888888711111111111112, 888888888888887111111111111112

Offset: 1

Reinhard Zumkeller, May 31 2010

			n=1: ..................... 72 = 9 * 8;
n=2: ................... 8712 = 99 * 88;
n=3: ................. 887112 = 999 * 888;
n=4: ............... 88871112 = 9999 * 8888;
n=5: ............. 8888711112 = 99999 * 88888;
n=6: ........... 888887111112 = 999999 * 888888;
n=7: ......... 88888871111112 = 9999999 * 8888888;
n=8: ....... 8888888711111112 = 99999999 * 88888888;
n=9: ..... 888888887111111112 = 999999999 * 888888888.

Walther Lietzmann, Lustiges und Merkwuerdiges von Zahlen und Formen, (F. Hirt, Breslau 1921-43), p. 149.

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, A178632, A178633, A178634.

Cf. A002275, A002282, A002283, A002477.

[72*((10^n-1)/9)^2: n in [1..50]]; // Vincenzo Librandi, Dec 28 2010

72 (FromDigits/@Table[PadRight[{}, n, 1], {n, 40}])^2 (* Vincenzo Librandi, Mar 21 2014 *)

a(n)=72*(10^n\9)^2 \\ Charles R Greathouse IV, Jul 02 2013

a(n) = 72*A002477(n) = A002283(n)*A002282(n).
a(n) = ((A002282(n-1)*10 + 7)*10^(n-1) + A002275(n-1))*10 + 2.
G.f.: 72*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.: 8*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)

A238237 Numbers which when chopped into two parts with equal length, added and squared result in the same number.

Original entry on oeis.org

81, 2025, 3025, 9801, 494209, 998001, 24502500, 25502500, 52881984, 60481729, 99980001, 6049417284, 6832014336, 9048004641, 9999800001, 101558217124, 108878221089, 123448227904, 127194229449, 152344237969, 213018248521, 217930248900, 249500250000, 250500250000

Offset: 1

Arkadiusz Wesolowski, Feb 20 2014

Yet another variant of the Kaprekar numbers A006886. - N. J. A. Sloane, Aug 06 2017
From Bernard Schott, Jan 21 2022: (Start)
Three subsequences:
-> {(10^m-1)^2, m >= 1} = A059988 \ {0}; see example 9801.
-> {(10^m-1)^2 * 10^(2*m) / 4, m >= 1} = A350869 \ {0}; see example 2025.
-> {(10^m+1)^2 * 10^(2*m) / 4, m >= 1} = A038544 \ {1}, see example 3025. (End)

			2025 = (20 + 25)^2, so 2025 is in the sequence.
3025 = (30 + 25)^2, so 3025 is in the sequence.
9801 = (98 + 01)^2, so 9801 is in the sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..25000

Subsequence of A102766.
Subsequence: A350870.
Cf. A006886, A038544, A059988, A350869.
For square roots see A290449.

Select[Range[600000]^2, EvenQ[len=IntegerLength[#]] && # == (Mod[#,10^(len/2)] + Floor[#/10^(len/2)])^2 &] (* Stefano Spezia, Jan 01 2025 *)

forstep(m=1, 7, 2, p=10^((m+1)/2); for(n=10^m, 10^(m+1)-1, d=lift(Mod(n, p)); if(((n-d)/p+d)^2==n, print1(n, ", "))));

a(n) = A290449(n)^2. - Bernard Schott, Jan 20 2022

a(12)-a(24) from Donovan Johnson, Feb 22 2014

A272066 a(n) = (10^n-1)^3.

Original entry on oeis.org

0, 729, 970299, 997002999, 999700029999, 999970000299999, 999997000002999999, 999999700000029999999, 999999970000000299999999, 999999997000000002999999999, 999999999700000000029999999999, 999999999970000000000299999999999, 999999999997000000000002999999999999

Offset: 0

Seiichi Manyama, Apr 19 2016

The sum of the digits of a(n) is divisible by 18. For example, 9^3 = 729 and 7 + 2 + 9 = 18 * 1.

Number of 9 in a(n) is 2*n-1 for n > 0. - Seiichi Manyama, Sep 18 2018

			From _Seiichi Manyama_, Sep 18 2018: (Start)
n| a(n) can be divided into 3 parts for n > 1.
-+--------------------------------------------
1|        72    9
2|   9   702   99
3|  99  7002  999
4| 999 70002 9999
(End)

Index entries for linear recurrences with constant coefficients, signature (1111,-112110,1111000,-1000000).

Cf. A002283, A059988, A272067, A272068, A319358.

[(10^n-1)^3 : n in [0..10]]; // Wesley Ivan Hurt, Apr 19 2016

A272066:=n->(10^n-1)^3: seq(A272066(n), n=0..15); # Wesley Ivan Hurt, Apr 19 2016

(10^Range[0, 10] - 1)^3 (* Wesley Ivan Hurt, Apr 19 2016 *)

a(n) = (10^n-1)^3; \\ Michel Marcus, Apr 19 2016

Ruby
```
(0..n).each{|i| p ('9' * i).to_i ** 3}
```

a(n) = A002283(n)^3.
From Ilya Gutkovskiy, Apr 19 2016: (Start)
O.g.f.: 729*x*(1 + 220*x + 1000*x^2)/((1 - x)*(1 - 10*x)*(1 - 100*x)*(1 - 1000*x)).
E.g.f.: (-1 + 3*exp(9*x) - 3*exp(99*x) + exp(999*x))*exp(x). (End)

A272067 a(n) = (10^n-1)^4.

Original entry on oeis.org

0, 6561, 96059601, 996005996001, 9996000599960001, 99996000059999600001, 999996000005999996000001, 9999996000000599999960000001, 99999996000000059999999600000001, 999999996000000005999999996000000001, 9999999996000000000599999999960000000001, 99999999996000000000059999999999600000000001

Offset: 0

Seiichi Manyama, Apr 19 2016

The sum of the digits of a(n) is divisible by 18. For example, 9^4 = 6561 and 6 + 5 + 6 + 1 = 18 * 1.

Number of 9 in a(n) is 2*n-2 for n > 0. - Seiichi Manyama, Sep 18 2018

			From _Seiichi Manyama_, Sep 18 2018: (Start)
n| a(n) can be divided into 4 parts for n > 1.
-+--------------------------------------------
1|        65        61
2|   9   605   9   601
3|  99  6005  99  6001
4| 999 60005 999 60001
(End)

Index entries for linear recurrences with constant coefficients, signature (11111,-11222110,1122211000,-11111000000,10000000000).

Cf. A002283, A059988, A272066, A272068, A319358.

[(10^n-1)^4 : n in [0..10]]; // Wesley Ivan Hurt, Apr 19 2016

A272067:=n->(10^n-1)^4: seq(A272067(n), n=0..15); # Wesley Ivan Hurt, Apr 19 2016

(10^Range[0, 10] - 1)^4 (* Wesley Ivan Hurt, Apr 19 2016 *)

a(n) = (10^n-1)^4; \\ Michel Marcus, Apr 19 2016

Ruby
```
(0..n).each{|i| p ('9' * i).to_i ** 4}
```

a(n) = A059988(n)^2 = A002283(n)^4.
From Ilya Gutkovskiy, Apr 19 2016: (Start)
O.g.f.: 6561*x*(1 + 100*x)*(1 + 3430*x + 10000*x^2)/((1 - x)*(1 - 10*x)*(1 - 100*x)*(1 - 1000*x)*(1 - 10000*x)).
E.g.f.: (1 - 4*exp(9*x) + 6*exp(99*x) - 4*exp(999*x) + exp(9999*x))*exp(x). (End)

A272068 a(n) = (10^n-1)^5.

Original entry on oeis.org

0, 59049, 9509900499, 995009990004999, 99950009999000049999, 9999500009999900000499999, 999995000009999990000004999999, 99999950000009999999000000049999999, 9999999500000009999999900000000499999999, 999999995000000009999999990000000004999999999, 99999999950000000009999999999000000000049999999999

Offset: 0

Seiichi Manyama, Apr 19 2016

The sum of the digits of a(n) is divisible by 27. For example, 9^5 = 59049 and 5 + 9 + 0 + 4 + 9 = 27 * 1.

Number of 9 in a(n) is 3*n-1 for n > 0. - Seiichi Manyama, Sep 18 2018

			From _Seiichi Manyama_, Sep 18 2018: (Start)
n| a(n) can be divided into 5 parts for n > 1.
-+--------------------------------------------
1|        5    9    04    9
2|   9   50   99   004   99
3|  99  500  999  0004  999
4| 999 5000 9999 00004 9999
(End)

Index entries for linear recurrences with constant coefficients, signature (111111,-1122322110,1123333211000,-112232211000000,1111110000000000,-1000000000000000).

Cf. A002283, A059988, A272066, A272067, A319358.

[(10^n-1)^5 : n in [0..10]]; // Wesley Ivan Hurt, Apr 19 2016

A272068:=n->(10^n-1)^5: seq(A272068(n), n=0..10); # Wesley Ivan Hurt, Apr 19 2016

(10^Range[0, 10] - 1)^5 (* Wesley Ivan Hurt, Apr 19 2016 *)

a(n) = (10^n-1)^5; \\ Michel Marcus, Apr 19 2016

Ruby
```
(0..n).each{|i| p ('9' * i).to_i ** 5}
```

a(n) = A002283(n)^5.
From Ilya Gutkovskiy, Apr 19 2016: (Start)
O.g.f.: 59049*x*(1 + 49940*x + 78366000*x^2 + 4994000000*x^3 + 10000000000*x^4)/((1 - x)*(1 - 10*x)*(1 - 100*x)*(1 - 1000*x)*(1 - 10000*x)*(1 - 100000*x)).
E.g.f.: -exp(x) + 5*exp(10*x) - 10*exp(100*x) + 10*exp(1000*x) - 5*exp(10000*x) + exp(100000*x). (End)

A319358 a(n) = (10^n - 1)^9.

Original entry on oeis.org

0, 387420489, 913517247483640899, 991035916125874083964008999, 999100359916012598740083996400089999, 999910003599916001259987400083999640000899999, 999991000035999916000125999874000083999964000008999999

Offset: 0

Seiichi Manyama, Sep 17 2018

nonn

Number of 9 in a(n) is 5*n-8 for n > 2.

			n|   a(n) can be divided into 9 parts for n > 3.
-+------------------------------------------------------
3|   99   1035   9   16125      874083   9   64008   999
4|  999  10035  99  160125  9  8740083  99  640008  9999
5| 9999 100035 999 1600125 99 87400083 999 6400008 99999

Seiichi Manyama, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (1111111111, -112233445443322110, 1123457901110987543211000, -1123570145779775409653211000000, 112358025801220975197532110000000000, -1123570145779775409653211000000000000000, 1123457901110987543211000000000000000000000, -112233445443322110000000000000000000000000000, 1111111111000000000000000000000000000000000000, -1000000000000000000000000000000000000000000000).

Cf. A001017, A002283, A059988, A272067, A272068.

a[n_]:=(10^n - 1)^9 ; Array[a,  50, 0] (* Stefano Spezia, Sep 17 2018 *)

PARI
```
a(n) = (10^n-1)^9;
```

a(n) = A002283(n)^9.

A271528 a(n) = 2*(10^n - 1)^2/27.

Original entry on oeis.org

0, 6, 726, 73926, 7405926, 740725926, 74073925926, 7407405925926, 740740725925926, 74074073925925926, 7407407405925925926, 740740740725925925926, 74074074073925925925926, 7407407407405925925925926, 740740740740725925925925926, 74074074074073925925925925926

Offset: 0

Ilya Gutkovskiy, Apr 09 2016

All terms are multiple of 6.
Converges in a 10-adic sense to ...925925925926.
A transformation of the Wonderful Demlo numbers (A002477).
More generally, the ordinary generating function for the transformation of the Wonderful Demlo numbers, is k*x*(1 + 10*x)/(1 - 111*x + 1110*x^2 - 1000*x^3).

			n=1:                  6 = 2 * 3;
n=2:                726 = 22 * 33;
n=3:              73926 = 222 * 333;
n=4:            7405926 = 2222 * 3333;
n=5:          740725926 = 22222 * 33333;
n=6:        74073925926 = 222222 * 333333;
n=7:      7407405925926 = 2222222 * 3333333;
n=8:    740740725925926 = 22222222 * 33333333;
n=9:  74074073925925926 = 222222222 * 333333333, etc.

Ilya Gutkovskiy, Transformation of the Wonderful Demlo numbers
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, A002276, A002277, A002477.

Cf. similar sequences of the form k*((10^n - 1)/9)^2: A075411 (k=4), this sequence (k=6), A075412 (k=9), A075413 (k=16), A178630 (k=18), A075414 (k=25), A178631 (k=27), A075415 (k=36), A178632 (k=45), A075416 (k=49), A178633 (k=54), A178634 (k=63), A075417 (k=64), A178635 (k=72), A059988 (k=81).

Table[2 ((10^n - 1)^2/27), {n, 0, 15}]
LinearRecurrence[{111, -1110, 1000}, {0, 6, 726}, 16]

x='x+O('x^99); concat(0, Vec(6*x*(1+10*x)/(1-111*x+1110*x^2-1000*x^3))) \\ Altug Alkan, Apr 09 2016

for n in range(0,10**1):print((int)((2*(10**n-1)**2)/27))
# Soumil Mandal, Apr 10 2016

O.g.f.: 6*x*(1 + 10*x)/(1 - 111*x + 1110*x^2 - 1000*x^3).
E.g.f.: 2 (exp(x) - 2*exp(10*x) + exp(100*x))/27.
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3).
a(n) = 6*A002477(n) = 6*A002275(n)^2 = A002276(n)*A002277(n) = sqrt(A075411(n)*A075412(n)).
Sum_{n>=1} 1/a(n) = 0.1680577405662077350849154881928636039793563...
Lim_{n -> infinity} a(n + 1)/a(n) = 100.

A350918 Numbers k = x.y which when split into two parts x and y of equal length, added and squared result in the same number k, '.' means concatenation, and the second part y starts with 0.

Original entry on oeis.org

9801, 998001, 99980001, 9048004641, 9999800001, 923594037444, 989444005264, 999998000001, 7901234409876544, 8434234407495744, 8934133805179209, 9999999800000001, 999999998000000001, 79012345680987654321, 82644628100826446281, 83407877440792003584, 87138706300620940900

Offset: 1

Bernard Schott, Jan 22 2022

Problem proposed on French site Diophante (see link).

			(998+001)^2 = 999^2 = 998001, as x = 998 and y = 001 starts with 0, 998001 is a term.
(30+25)^2 = 55^2 = 3025, here x = 30 but y = 25 does not start with 0, hence 3025 is not a term.

Diophante, A1945 - Concaténations en tous genres (in French).

Equals A238237 \ A350870.

A059988 \ {0, 81} is a subsequence.

cp's OEIS Frontend

A178634 a(n) = 63*((10^n - 1)/9)^2.

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A178635 a(n) = 72*((10^n - 1)/9)^2.

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A238237 Numbers which when chopped into two parts with equal length, added and squared result in the same number.

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A272066 a(n) = (10^n-1)^3.

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A272067 a(n) = (10^n-1)^4.

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A272068 a(n) = (10^n-1)^5.

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