A246057 -id:A246057 - cp's OEIS frontend

A002277 a(n) = 3*(10^n - 1)/9.

0, 3, 33, 333, 3333, 33333, 333333, 3333333, 33333333, 333333333, 3333333333, 33333333333, 333333333333, 3333333333333, 33333333333333, 333333333333333, 3333333333333333, 33333333333333333, 333333333333333333, 3333333333333333333, 33333333333333333333, 333333333333333333333

Offset: 0

N. J. A. Sloane

From Wolfdieter Lang, Feb 08 2017: (Start)
This sequence (for n >= 1) appears in n-families satisfying so-called curious cubic identities based on the Armstrong numbers 153, 370 and 371, A005188(10) - A005188(12).
153 also involves A246057(n-1) and A093143(n). See a comment in A246057 with the van Poorten et al. reference, and A281857.
370 and 371 also involve A067275(n+1). See the comment there, and A281858 and A281860. (End)

			From _Wolfdieter Lang_, Feb 08 2017: (Start)
Curious cubic identities (see a comment above):
1^3 + 5^3 + 3^3 = 153, 16^3 + 50^3 + 33^3 = 165033, 166^3 + 500^3 + 333^3 = 166500333, ...
3^3 + 7^3 + 0^3 = 370; 336700 = 33^3 + 67^3 + (00)^3 = 336700,  333^3 + 667^3 + (000)^3 = 333667000, ...
3^3 + 7^3 + 1^3 = 371, 33^3 + 67^3 + (01)^3 = 336701, 333^3 + 667^3 + (001)^3 = 333667001, ... (End)

Ivan Panchenko, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Repdigit.
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A002275, A002276, A002278, A002279, A002280, A002281, A002282, A002283, A010785, A017197.

Cf. A005188, A067275, A075412, A093143, A135702, A178631, A178633, A246057, A281857, A281858, A281860.

[(10^n - 1)/3 : n in [0..30]]; // Wesley Ivan Hurt, Apr 01 2016

A002277:=n->(10^n-1)/3: seq(A002277(n), n=0..30); # Wesley Ivan Hurt, Apr 01 2016

LinearRecurrence[{11, -10}, {0, 3}, 20] (* Robert G. Wilson v, Jul 06 2013 *)
(10^Range[0, 30] - 1)/3 (* Wesley Ivan Hurt, Apr 01 2016 *)

A002277(n):=(10^n - 1)/3$
makelist(A002277(n),n,0,20); /* Martin Ettl, Nov 12 2012 */

a(n)=(10^n-1)/3 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = 3*A002275(n).
a(n) = A178631(n)/A002283(n). - Reinhard Zumkeller, May 31 2010
From Vincenzo Librandi, Jul 22 2010: (Start)
a(n) = a(n-1) + 3*10^(n-1) with a(0)=0;
a(n) = 11*a(n-1) - 10*a(n-2) with a(0)=0, a(1)=3. (End)
G.f.: 3*x/((1 - x)*(1 - 10*x)). - Ilya Gutkovskiy, Feb 24 2017
Sum_{n>=1} 1/a(n) = A135702. - Amiram Eldar, Nov 13 2020
E.g.f.: exp(x)*(exp(9*x) - 1)/3. - Stefano Spezia, Sep 13 2023
From Elmo R. Oliveira, Jul 20 2025: (Start)
a(n) = (A246057(n) - 1)/5.
a(n) = A010785(A017197(n-1)) for n >= 1. (End)

A093137 Expansion of (1-7x)/((1-x)(1-10*x)).

Original entry on oeis.org

1, 4, 34, 334, 3334, 33334, 333334, 3333334, 33333334, 333333334, 3333333334, 33333333334, 333333333334, 3333333333334, 33333333333334, 333333333333334, 3333333333333334, 33333333333333334, 333333333333333334, 3333333333333333334, 33333333333333333334

Offset: 0

Paul Barry, Mar 24 2004

Second binomial transform of 3*A001045(3n)/3+(-1)^n. Partial sums of A093138. A convex combination of 10^n and 1. In general the second binomial transform of k*Jacobsthal(3n)/3+(-1)^n is 1,1+k,1+11k,1+111k,... This is the case for k=3.
a(n) is the number of n-length sequences of decimal digits whose sum is divisible by 3. - Geoffrey Critzer, Jan 18 2014
This sequence appears in a family of curious cubic identities based on the Armstrong number 407 = A005188(13). See the formula section. For the analog identities based on 153 = A005188(10) see a comment on A246057 with the van der Poorten et al. reference and A281857. For those based on 370 = A005188(11) see A067275, A002277 and A281858. - Wolfdieter Lang, Feb 08 2017

			a(1)^2 = 16
a(2)^2 = 1156
a(3)^2 = 111556
a(4)^2 = 11115556
a(5)^2 = 1111155556
a(6)^2 = 111111555556
a(7)^2 = 11111115555556
a(8)^2 = 1111111155555556
a(9)^2 = 111111111555555556, etc... (see A102807). - _Philippe Deléham_, Oct 03 2011
Curious cubic identities: 407 = 4^3 + 0^3 + 7^3, 340067 = 34^3 + (00)^3 + 67^3, 334000677 = 334^3 + (000)^3 + 677^3, ... - _Wolfdieter Lang_, Feb 08 2017

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See entry 3334 at p. 168.

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

Cf. A005188, A067275, A246057, A281857, A281859.
Cf. A001045, A002277, A093138, A281858.
Cf. A102807 (squared).

nn=20; r=Solve[{s==4x s+3 x a+3x b+1,a==4x a+3x s+3x b,b==4x b+3x s+3x a},{s,a,b}]; CoefficientList[Series[s/.r,{x,0,nn}],x] (* Geoffrey Critzer, Jan 18 2014 *)
Table[3*10^n/9 + 6/9, {n, 0, 20}] (* or *) NestList[10 # - 6 &, 1, 20] (* Michael De Vlieger, Feb 08 2017 *)
LinearRecurrence[{11,-10},{1,4},20] (* Harvey P. Dale, Oct 07 2017 *)

Vec((1-7*x)/((1-x)*(1-10*x)) + O (x^30)) \\ Michel Marcus, Feb 09 2017

a(n) = 3*10^n/9 + 6/9.
a(n) = 10*a(n-1)-6 with a(0)=1. - Vincenzo Librandi, Aug 02 2010
a(n)^3 + 0(n)^3 + A067275(n+1)^3 = concatenation(a(n), 0(n), A067275(n+1)) = A281859(n), where 0(n) denotes n 0's, n >= 1. - Wolfdieter Lang, Feb 08 2017
From Elmo R. Oliveira, Aug 17 2024: (Start)
E.g.f.: exp(x)*(exp(9*x) + 2)/3.
a(n) = 11*a(n-1) - 10*a(n-2) for n > 1. (End)

A093143 Expansion of (1-5x)/(1-10x).

Original entry on oeis.org

1, 5, 50, 500, 5000, 50000, 500000, 5000000, 50000000, 500000000, 5000000000, 50000000000, 500000000000, 5000000000000, 50000000000000, 500000000000000, 5000000000000000, 50000000000000000, 500000000000000000, 5000000000000000000, 50000000000000000000, 500000000000000000000

Offset: 0

Paul Barry, Mar 24 2004

Partial sums are A093142. A convex combination of 10^n and 0^n.
a(n) is the number of compositions of even natural numbers in n parts <= 9 (0 is counted as a part); also the number of ways of placing of an even number of indistinguishable objects into n distinguishable boxes with the condition that at most 9 objects can be placed in each box. - Adi Dani, May 17 2011
See an A246057 comment with a reference for the k-family satisfying a so-called curious cubic identity involving A246057(k-1), a(k) and A002277(k). - Wolfdieter Lang, Feb 07 2017

			From _Adi Dani_, May 17 2011: (Start)
a(2)=50: there are 50 compositions of even numbers into 2 parts <= 9:
(0,0);
(0,2),(2,0),(1,1);
(0,4),(4,0),(1,3),(3,1),(2,2);
(0,6),(6,0),(1,5),(5,1),(2,4),(4,2),(3,3);
(0,8),(8,0),(1,7),(7,1),(2,6),(6,2),(3,5),(5,3),(4,4);
(1,9),(9,1),(2,8),(8,2),(3,7),(7,3),(4,6),(6,4),(5,5);
(3,9),(9,3),(4,8),(8,4),(5,7),(7,5),(6,6);
(5,9),(9,5),(6,8),(8,6),(7,7);
(7,9),(9,7),(8,8);
(9,9).
(End)
Curious cubic identities (see a comment above): 1^3 + 5^3 + 3^3 = 153, 16^3 + 50^3 + 33^3 = 165033, 166^3 + 500^3 + 333^3 = 166500333, ... - _Wolfdieter Lang_, Feb 07 2017

Adi Dani, Restricted compositions of natural numbers
Index entries for linear recurrences with constant coefficients, signature (10).

Cf. A002277, A055372, A093142, A134309, A246057.

Table[Ceiling[1/2*10^n],{n,0,30}] (* Adi Dani, Jun 20 2011 *)
Join[{1},NestList[10#&,5,20]] (* Harvey P. Dale, Apr 10 2021 *)

Vec((1-5*x)/(1-10*x) + O(x^100)) \\ Altug Alkan, Nov 01 2015

a(n) = 5*10^n/10 for n > 0.
a(n) = Sum_{k=0..n} A134309(n,k)*5^k = Sum_{k=0..n} A055372(n,k)*4^k. - Philippe Deléham, Feb 04 2012
From Elmo R. Oliveira, Aug 21 2024: (Start)
E.g.f.: (exp(10*x) + 1)/2.
a(n) = 10*a(n-1) for n > 1. (End)

a(19)-a(21) from Elmo R. Oliveira, Aug 21 2024

A067275 Number of Fibonacci numbers A000045(k), k <= 10^n, which end in 4.

Original entry on oeis.org

0, 1, 7, 67, 667, 6667, 66667, 666667, 6666667, 66666667, 666666667, 6666666667, 66666666667, 666666666667, 6666666666667, 66666666666667, 666666666666667, 6666666666666667, 66666666666666667, 666666666666666667, 6666666666666666667, 66666666666666666667, 666666666666666666667

Offset: 0

Joseph L. Pe, Feb 21 2002

Numbers n such that the digits of A000326(n), the n-th pentagonal number, begin with n. [original definition of this sequence]
The sequence 1,7,67.... has a(n) = 6*10^n/9+3/9. It is the second binomial transform of 6*A001045(3n)/3 + (-1)^n. In general the second binomial transform of k*Jacobsthal(3n)/3+(-1)^n is k*10^n/9 + (1-k/9) = 1, 1+k, 1+11k, 1+111k, ... - Paul Barry, Mar 24 2004
Except for the first two terms, these are the 3-automorphic numbers ending in 7. - Eric M. Schmidt, Aug 28 2012
From Wolfdieter Lang, Feb 08 2017: (Start)
This sequence appears in curious identities based on the Armstrong numbers 370 = A005188(11), 371 = A005188(12) and 407 = A005188(13).
For such identities based on 153 = A005188(10) see a comment in A246057 with the van der Poorten et al. reference.
For 370 these identities are A002277(n)^3 + a(n+1)^3  + 0(n)^3 = A002277(n)*10^(2*n) + a(n+1)*10^n + 0(n) = A281858(n), where 0(n) means n 0's.
For 371 these identities are A002277(n)^3 + a(n+1)^3 + (0(n-1)1)^3 = A002277(n)*10^(2*n) + a(n+1)*10^n + 0(n-1)1 = A281860(n), where 0(n-1)1 means n-1 0's followed by a 1.
For 407 these identities are A093137(n)^3 + 0(n)^3 + a(n+1)^3 = concatenation(A093137(n), 0(n), a(n+1)) = A281859(n). (End)

			a(2) = 7 because 7 of the first 10^2 Fibonacci numbers end in 4.
From _Wolfdieter Lang_, Feb 08 2017: (Start)
Curious cubic identities:
3^3 + 7^3 + 0^3 = 370, 33^3 + 67^3 + (00)^3 = 336700, 333^3 + 667^3 + (000)^3 = 333667000, ...
3^3 + 7^3 + 1^3 = 371, 33^3 + 67^3 + (01)^3 = 336701, 333^3 + 667^3 + (001)^3 = 333667001, ...
4^3 + 0^3 + 7^3 = 407, 34^3 + (00)^3 + 67^3 = 340067 , 334^3 + (000)^3 + 677^3 = 334000677, ... (End)

Seiichi Manyama, Table of n, a(n) for n = 0..1001
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A072702, A093137, A109344, A246057, A281858, A281859, A281860.

Cf. A000045, A000326, A001045, A002277, A005188, A073553, A199682.

s = Fibonacci@ Range[10^5]; Table[Count[Take[s, 10^n], m_ /; Mod[m, 10] == 4], {n, 0, Floor@ Log10@ Length@ s}] (* or *) Table[Boole[n > 0] Ceiling[10^n/15], {n, 0, 20}] (* or *) CoefficientList[Series[x (1 - 4 x)/((1 - x) (1 - 10 x)), {x, 0, 20}], x] (* Michael De Vlieger, Feb 08 2017 *)

a(n)=(10^n+13)\15 \\ Charles R Greathouse IV, Jun 05 2011

a(n) = ceiling((2/30)*10^n) - Benoit Cloitre, Aug 27 2002
From Paul Barry, Mar 24 2004: (Start)
G.f.: x*(1 - 4*x)/((1 - x)*(1 - 10*x)).
a(n) = 10^n/15 + 1/3 for n>0. (End)
a(n) = 10*a(n-1) - 3 for n>1. - Vincenzo Librandi, Dec 07 2010 [Immediate consequence of the previous formula, R. J. Mathar]
From Eric M. Schmidt, Oct 28 2012: (Start)
For n>0, a(n) = A199682(n-1)/3 = (2*10^(n-1) + 1)/3.
For n>=2, a(n+1) = a(n) + 6*10^n. (End)
From Elmo R. Oliveira, Jul 22 2025: (Start)
E.g.f.: (-6 + 5*exp(x) + exp(10*x))/15.
a(n) = 11*a(n-1) - 10*a(n-2) for n >= 3.
a(n) = A073553(n)/2 for n >= 1. (End)

A073552 merged into this sequence by Eric M. Schmidt, Oct 28 2012

A281858 Curious cubic identities based on the Armstrong number 370.

Original entry on oeis.org

370, 336700, 333667000, 333366670000, 333336666700000, 333333666667000000, 333333366666670000000, 333333336666666700000000, 333333333666666667000000000, 333333333366666666670000000000, 333333333336666666666700000000000, 333333333333666666666667000000000000

Offset: 1

Wolfdieter Lang, Feb 08 2017

See a comment in A067275, and the analog to the Armstrong number 153 = A005188(10) treated in A281857, 370 = A005188(11).

			n=1: 370 =  3^3 + 7^3 + 0^3; n=2: 336700 = 33^3 + 67^3 + (00)^3; n=3: 333667000 = 333^3 + 667^3 + (000)^3.

Colin Barker, Table of n, a(n) for n = 1..333
Index entries for linear recurrences with constant coefficients, signature (1110,-111000,1000000).

Cf. A002277, A005188, A067275, A246057, A281857.

Table[FromDigits@ Join[ConstantArray[3, n], ReplacePart[ConstantArray[6, n], -1 -> 7], ConstantArray[0, n]], {n, 12}] (* Michael De Vlieger, Feb 08 2017 *)

Vec(10*x*(37 - 7400*x + 100000*x^2) / ((1 - 10*x)*(1 - 100*x)*(1 - 1000*x)) + O(x^30)) \\ Colin Barker, Feb 08 2017

a(n) = A002277(n)^3 + A067275(n+1)^3 + 0(n)^3, n >= 1, with 0(n) standing for n 0's.
From Colin Barker, Feb 08 2017: (Start)
G.f.: 10*x*(37 - 7400*x + 100000*x^2) / ((1 - 10*x)*(1 - 100*x)*(1 - 1000*x)).
a(n) = 10^n*(1 + 10^n + 100^n) / 3.
a(n) = 1110*a(n-1) - 111000*a(n-2) + 1000000*a(n-3) for n>3. (End)

A281857 Numbers occurring in a curious cubic identity.

Original entry on oeis.org

153, 165033, 166500333, 166650003333, 166665000033333, 166666500000333333, 166666650000003333333, 166666665000000033333333, 166666666500000000333333333, 166666666650000000003333333333, 166666666665000000000033333333333, 166666666666500000000000333333333333

Offset: 1

Wolfdieter Lang, Feb 07 2017

See A246057 for the van der Poorten et al. reference and a comment.

153 is the Armstrong number A005188(10). [Typo corrected by Jeremy Tan, Feb 25 2023]

			1^3 + 5^3 + 3^3 = 153, 16^3 + 50^3 + 33^3 = 165033, 166^3 + 500^3 + 333^3 = 166500333, ...

Colin Barker, Table of n, a(n) for n = 1..333
Index entries for linear recurrences with constant coefficients, signature (1111,-112110,1111000,-1000000).

Cf. A002277, A005188, A093143, A246057, A281858, A281859, A281860.

Table[FromDigits@ Join[ReplacePart[ConstantArray[6, n], 1 -> 1], ReplacePart[ConstantArray[0, n], 1 -> 5], ConstantArray[3, n]], {n, 12}] (* Michael De Vlieger, Feb 08 2017 *)

Vec(9*x*(17 - 550*x + 33500*x^2) / ((1 - x)*(1 - 10*x)*(1 - 100*x)*(1 - 1000*x)) + O(x^15)) \\ Colin Barker, Feb 08 2017

a(n) = (((10^n - 4)/6)^3) + ((10^n/2)^3) + (((10^n - 1)/3)^3) \\ Jean-Jacques Vaudroz, Aug 11 2024

a(n) = A246057(n-1)^3 + A093143(n)^3 + A002277(n)^3, n >= 1.
From Colin Barker, Feb 08 2017: (Start)
G.f.: 9*x*(17 - 550*x + 33500*x^2) / ((1 - x)*(1 - 10*x)*(1 - 100*x)*(1 - 1000*x)).
a(n) = (-2 + 2^(1+n)*5^n - 100^n + 1000^n) / 6.
a(n) = 1111*a(n-1) - 112110*a(n-2) + 1111000*a(n-3) - 1000000*a(n-4) for n>4. (End)

A086948 a(n) = k where R(k+8) = 2.

Original entry on oeis.org

12, 192, 1992, 19992, 199992, 1999992, 19999992, 199999992, 1999999992, 19999999992, 199999999992, 1999999999992, 19999999999992, 199999999999992, 1999999999999992, 19999999999999992, 199999999999999992, 1999999999999999992, 19999999999999999992, 199999999999999999992

Offset: 1

Ray Chandler, Jul 24 2003

Vincenzo Librandi, Table of n, a(n) for n = 1..300
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A004086, A085331, A086947, A246057, A323639.

[2*(10^n-4): n in [1..25] ]; // Vincenzo Librandi, Aug 22 2011

Table[10*FromDigits[PadRight[{1},n,9]]+2,{n,20}] (* Harvey P. Dale, Dec 15 2017 *)

a(n) = 2*(10^n - 4).
R(a(n)) = A086947(n).
From Chai Wah Wu, Aug 01 2020: (Start)
a(n) = 11*a(n-1) - 10*a(n-2) for n > 2.
G.f.: x*(60*x + 12)/((x - 1)*(10*x - 1)). (End)
From Elmo R. Oliveira, May 01 2025: (Start)
E.g.f.: 2*(3 - 4*exp(x) + exp(10*x)).
a(n) = 12*A246057(n-1) = 6*A323639(n). (End)

A246058 a(n) = (16*10^n - 7)/9.

Original entry on oeis.org

1, 17, 177, 1777, 17777, 177777, 1777777, 17777777, 177777777, 1777777777, 17777777777, 177777777777, 1777777777777, 17777777777777, 177777777777777, 1777777777777777, 17777777777777777, 177777777777777777, 1777777777777777777, 17777777777777777777, 177777777777777777777

Offset: 0

Vincenzo Librandi, Aug 13 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. similar sequences listed in A246057.

Magma
```
[(16*10^n-7)/9: n in [0..20]];
```
Mathematica
```
Table[(16 10^n - 7)/9, {n, 0, 20}]
```

G.f.: (1+6*x)/((1-x)*(1-10*x)).
a(n) = 11*a(n-1) - 10*a(n-2).
E.g.f.: exp(x)*(16*exp(9*x) - 7)/9. - Elmo R. Oliveira, May 06 2025

A323639 a(n) = 3*(10^n - 4)/9.

Original entry on oeis.org

-1, 2, 32, 332, 3332, 33332, 333332, 3333332, 33333332, 333333332, 3333333332, 33333333332, 333333333332, 3333333333332, 33333333333332, 333333333333332, 3333333333333332, 33333333333333332, 333333333333333332, 3333333333333333332, 33333333333333333332

Offset: 0

Seiichi Manyama, Aug 31 2019

			        (0+1) * (3*0-1) = -1.
        (3+1) * (3*3-1) = 32.
      (33+1) * (3*33-1) = 3332.
    (333+1) * (3*333-1) = 333332.
  (3333+1) * (3*3333-1) = 33333332.
(33333+1) * (3*33333-1) = 3333333332.
-------------------------------------
        8 * 4 = 32.
      68 * 49 = 3332.
    668 * 499 = 333332.
  6668 * 4999 = 33333332.
66668 * 49999 = 3333333332.

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

Cf. A002277, A073555, A086948, A093137, A198971, A246057.

Table[(10^n-4)/3,{n,0,20}] (* or *) LinearRecurrence[{11,-10},{-1,2},21] (* Harvey P. Dale, Jan 09 2021 *)

PARI
```
{a(n) = 3*(10^n-4)/9}
```

N=40; x='x+O('x^N); Vec((-1+13*x)/((1-x)*(1-10*x)))

G.f.: (-1+13*x)/((1-x)*(1-10*x)).
a(n) = 11*a(n-1) - 10*a(n-2).
a(n) = A002277(n) - 1.
a(n) = 2*A246057(n-1) for n > 0.
a(2*n) = (A002277(n)+1) * (3*A002277(n)-1).
a(2*n) = A073555(n+1) * A198971(n-1) for n > 0.
E.g.f.: exp(x)*(exp(9*x) - 4)/3. - Stefano Spezia, May 02 2025
a(n) = A086948(n)/6 for n >= 1. - Elmo R. Oliveira, May 06 2025

A246059 a(n) = (17*10^n - 8)/9.

Original entry on oeis.org

1, 18, 188, 1888, 18888, 188888, 1888888, 18888888, 188888888, 1888888888, 18888888888, 188888888888, 1888888888888, 18888888888888, 188888888888888, 1888888888888888, 18888888888888888, 188888888888888888, 1888888888888888888, 18888888888888888888, 188888888888888888888

Offset: 0

Vincenzo Librandi, Aug 13 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. similar sequences listed in A246057.

Magma
```
[(17*10^n-8)/9: n in [0..20]];
```
Mathematica
```
Table[(17 10^n - 8)/9, {n, 0, 20}]
```

G.f.: (1 + 7*x)/((1 - x)*(1 - 10*x)).
a(n) = 11*a(n-1) - 10*a(n-2).
E.g.f.: exp(x)*(17*exp(9*x) - 8)/9. - Elmo R. Oliveira, May 06 2025

cp's OEIS Frontend

A002277 a(n) = 3*(10^n - 1)/9.

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A093137 Expansion of (1-7*x)/((1-x)*(1-10*x)).

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A093143 Expansion of (1-5*x)/(1-10*x).

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A067275 Number of Fibonacci numbers A000045(k), k <= 10^n, which end in 4.

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A281858 Curious cubic identities based on the Armstrong number 370.

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A281857 Numbers occurring in a curious cubic identity.

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A093137 Expansion of (1-7x)/((1-x)(1-10*x)).

A093143 Expansion of (1-5x)/(1-10x).