A093143 -id:A093143 - 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)

A169964 Numbers whose decimal expansion contains only 0's and 5's.

Original entry on oeis.org

0, 5, 50, 55, 500, 505, 550, 555, 5000, 5005, 5050, 5055, 5500, 5505, 5550, 5555, 50000, 50005, 50050, 50055, 50500, 50505, 50550, 50555, 55000, 55005, 55050, 55055, 55500, 55505, 55550, 55555, 500000, 500005, 500050, 500055, 500500, 500505, 500550, 500555

Offset: 1

N. J. A. Sloane, Aug 07 2010

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to carryless arithmetic

Cf. A007088, A014263, A169965, A169966, A169967.

Cf. A078244, A204093, A204094, A204095, A097256.

a169964 n = a169964_list !! (n-1)
a169964_list = map (* 5) a007088_list
-- Reinhard Zumkeller, Jan 10 2012

Map[FromDigits,Tuples[{0,5},6]] (* Paolo Xausa, Oct 30 2023 *)

print1(0);for(d=1,5,for(n=2^(d-1),2^d-1,print1(", ");forstep(i=d-1,0,-1,print1((n>>i)%2*5)))) \\ Charles R Greathouse IV, Nov 16 2011

a(n+1) = Sum_{k>=0} A030308(n,k)*A093143(k+1). - Philippe Deléham, Oct 16 2011

a(n) = 5 * A007088(n-1).

A055372 Invert transform of Pascal's triangle A007318.

Original entry on oeis.org

1, 1, 1, 2, 4, 2, 4, 12, 12, 4, 8, 32, 48, 32, 8, 16, 80, 160, 160, 80, 16, 32, 192, 480, 640, 480, 192, 32, 64, 448, 1344, 2240, 2240, 1344, 448, 64, 128, 1024, 3584, 7168, 8960, 7168, 3584, 1024, 128, 256, 2304, 9216, 21504, 32256, 32256, 21504, 9216, 2304, 256

Offset: 0

Christian G. Bower, May 16 2000

Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 1, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 10 2005

T(n,k) is the number of nonempty bit strings with n bits and exactly k 1's over all strings in the sequence. For example, T(2,1)=4 because we have {(01)},{(10)},{(0),(1)},{(1),(0)}. - Geoffrey Critzer, Apr 06 2013

			Triangle begins:
  1;
  1,  1;
  2,  4,  2;
  4, 12, 12,  4;
  8, 32, 48, 32,  8;
  ...

N. J. A. Sloane, Transforms
Index entries for triangles and arrays related to Pascal's triangle

Row sums give A081294. Cf. A000079, A007318, A055373, A055374.
Cf. A134309.
T(2n,n) gives A098402.

nn=10;f[list_]:=Select[list,#>0&];a=(x+y x)/(1-(x+y x));Map[f,CoefficientList[Series[1/(1-a),{x,0,nn}],{x,y}]]//Grid  (* Geoffrey Critzer, Apr 06 2013 *)

a(n,k) = 2^(n-1)*C(n, k), for n>0.
G.f.: A(x, y)=(1-x-xy)/(1-2x-2xy).
As an infinite lower triangular matrix, equals A134309 * A007318. - Gary W. Adamson, Oct 19 2007
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A011782(n), A081294(n), A081341(n), A092811(n), A093143(n), A067419(n) for x = -1, 0, 1, 2, 3, 4, 5 respectively. - Philippe Deléham, Feb 05 2012

A134309 Triangle read by rows, where row n consists of n zeros followed by 2^(n-1).

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 64, 0, 0, 0, 0, 0, 0, 0, 0, 128, 0, 0, 0, 0, 0, 0, 0, 0, 0, 256, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 512, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1024, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2048, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gary W. Adamson, Oct 19 2007

As infinite lower triangular matrices, binomial transform of A134309 = A082137. A134309 * A007318 = A055372. A134309 * [1,2,3,...] = A057711: (1, 2, 6, 16, 40, 96, 224,...).

Triangle read by rows given by [0,0,0,0,0,0,0,0,...] DELTA [1,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 20 2007

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  0, 1;
  0, 0, 2;
  0, 0, 0, 4;
  0, 0, 0, 0, 8;
  0, 0, 0, 0, 0, 16;
  ...

Cf. A011782 (diagonal elements: 1 followed by 1, 2, 4, 8, ... = A000079: 2^n).

Cf. A055372, A057711, A082137.

Join[{1},Flatten[Table[Join[{PadRight[{},n],2^(n-1)}],{n,20}]]] (* Harvey P. Dale, Jan 04 2024 *)

A134309(r,c)=if(r==c,2^max(r-1,0),0) \\ M. F. Hasler, Mar 29 2022

Triangle, T(0,0) = 1, then for n > 0, n zeros followed by 2^(n-1). Infinite lower triangular matrix with (1, 1, 2, 4, 8, 16, ...) in the main diagonal and the rest zeros.
G.f.: (1 - y*x)/(1 - 2*y*x). - Philippe Deléham, Feb 04 2012
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A011782(n), A081294(n), A081341(n), A092811(n), A093143(n), A067419(n) for x = 0, 1, 2, 3, 4, 5, 6 respectively. - Philippe Deléham, Feb 04 2012
Diagonal is A011782, other elements are 0. - M. F. Hasler, Mar 29 2022

A246057 a(n) = (5*10^n - 2)/3.

Original entry on oeis.org

1, 16, 166, 1666, 16666, 166666, 1666666, 16666666, 166666666, 1666666666, 16666666666, 166666666666, 1666666666666, 16666666666666, 166666666666666, 1666666666666666, 16666666666666666, 166666666666666666, 1666666666666666666, 16666666666666666666, 166666666666666666666

Offset: 0

Vincenzo Librandi, Aug 13 2014

a(k-1) = (10^k - 4)/6, together with b(k) = 3*a(k-1) + 2 = A093143(k) and c(k) = 2*a(k-1) + 1 = A002277(k) are k-digit numbers for k >= 1 satisfying the so-called curious cubic identity a(k-1)^3 + b(k)^3 + c(k)^3 = a(k)*10^(2*k) + b(k)*10^k + c(k) (concatenated a(k)b(k)c(k)). This k-family and the proof of the identity has been given in the introduction of the van der Poorten reference. Thanks go to S. Heinemeyer for bringing these identities to my attention. - Wolfdieter Lang, Feb 07 2017

			Curious cubic identities (see a comment and reference 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

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Alf van der Poorten, Kurt Thomsen, and Mark Wiebe, A curious cubic identity and self-similar sums of squares, The Mathematical Intelligencer, Vol. 29(2), pp. 39-41, March 2007.
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. sequences with terms of the form 1k..k where the digit k is repeated n times: A000042 (k=1), A090843 (k=2), A097166 (k=3), A099914 (k=4), A099915 (k=5), this sequence (k=6), A246058 (k=7), A246059 (k=8), A067272 (k=9).

Cf. A002277, A086948, A093143, A323639.

Magma
```
[(5*10^n-2)/3: n in [0..20]];
```
Mathematica
```
Table[(5 10^n - 2)/3, {n, 0, 20}]
```

vector(50, n, (5*10^(n-1)-2)/3) \\ Derek Orr, Aug 13 2014

G.f.: (1 + 5*x)/((1 - x)*(1 - 10*x)).
a(n) = 11*a(n-1) - 10*a(n-2).
E.g.f.: exp(x)*(5*exp(9*x) - 2)/3. - Stefano Spezia, May 02 2025
a(n) = A323639(n+1)/2 = A086948(n+1)/12. - Elmo R. Oliveira, May 07 2025

A341383 Numbers m such that the largest digit in the decimal expansion of 1/m is 2.

Original entry on oeis.org

5, 45, 50, 450, 495, 500, 819, 825, 4500, 4545, 4950, 4995, 5000, 8190, 8250, 8325, 45000, 45045, 45450, 47619, 49500, 49950, 49995, 50000, 81819, 81900, 82500, 83250, 83325, 89109, 450000, 450045, 450450, 454500, 454545, 476190, 495000, 499500, 499950, 499995, 500000

Offset: 1

Bernard Schott, Feb 10 2021

If m is a term, 10*m is also a term.
5 is the only prime up to 2.6*10^8 (comments in A333237).
Some subsequences: {45, 4545, 454545, ...}, {45045, 45045045, 45045045045, ...}, {45, 495, 4995, 49995, ...}, {819, 81819, 8181819, ...}, {825, 8325, 83325, 833325...}, ...
The subsequence of terms where 1/m has only digits {0,2} is m = 5*A333402 = 5, 45, 50, etc.  A333402 is those t where 1/t has only digits {0,1}, so that 1/(5*t) = 2*(1/t)*(1/10) has digits {0,2}, starting from 1/5 = 0.2.  These m are also A333402/2 of the even terms from A333402, since A333402 (like here) is self-similar in that the multiples of 10, divided by 10, are the sequence itself. - Kevin Ryde, Feb 13 2021

			As 1/45 = 0.0202020202..., 45 is a term.
As 1/825 = 0.0012121212121212...., 825 is a term.
As 1/47619 = 0.000021000021000021..., 47619 is a term.
As 1/4545045 = 0.000000220019824..., 4545045 is not a term.

Index entries for sequences related to decimal expansion of 1/n

Cf. A333236.
Similar with largest digit k: A333402 (k=1), A333237 (k=9).
Subsequence: A093143 \ {1}.
Decimal expansion: A021499 (1/495), A021823 (1/819).

Select[Range[10^5], Max[RealDigits[1/#][[1]]] == 2 &] (* Amiram Eldar, Feb 10 2021 *)

from itertools import count, islice
from sympy import n_order, multiplicity
def A341383_gen(startvalue=1): # generator of terms >= startvalue
    for m in count(max(startvalue,1)):
        m2, m5 = multiplicity(2,m), multiplicity(5,m)
        if max(str(10**(max(m2,m5)+n_order(10,m//2**m2//5**m5))//m)) == '2':
            yield m
A341383_list = list(islice(A341383_gen(),10)) # Chai Wah Wu, Feb 07 2022

Missing terms added by Amiram Eldar, Feb 10 2021

A037156 a(n) = 10^n*(10^n+1)/2.

Original entry on oeis.org

1, 55, 5050, 500500, 50005000, 5000050000, 500000500000, 50000005000000, 5000000050000000, 500000000500000000, 50000000005000000000, 5000000000050000000000, 500000000000500000000000, 50000000000005000000000000, 5000000000000050000000000000

Offset: 0

Marvin Ray Burns

Sum of first 10^n positive integers. - Omar E. Pol, May 03 2015

			From _Omar E. Pol_, May 03 2015: (Start)
For n = 0; a(0) = 1                       =    1 * 1   = 1
For n = 1; a(1) = 1 + 2 + ...... + 9 + 10 =   11 * 5   = 55
For n = 2; a(2) = 1 + 2 + .... + 99 + 100 =  101 * 50  = 5050
For n = 3; a(3) = 1 + 2 + .. + 999 + 1000 = 1001 * 500 = 500500
...
(End)

C. Pickover, Wonders of Numbers, Oxford University Press, NY, 2001, p. 328.

Brian Hayes, Gauss's Day of Reckoning, American Scientist
Bill Johnson, The great Gauss summation trick
C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review
Index entries for linear recurrences with constant coefficients, signature (110, -1000).

A subsequence of the triangular numbers A000217.

Cf. A038544.

LinearRecurrence[{110,-1000},{1,55},20] (* Harvey P. Dale, Oct 11 2023 *)

a(n) = A000533(n) * A093143(n). - Omar E. Pol, May 03 2015
From Chai Wah Wu, May 28 2016: (Start)
a(n) = 110*a(n-1) - 1000*a(n-2).
G.f.: (1 - 55*x)/((10*x - 1)*(100*x - 1)).
(End)
a(n) = sqrt(A038544(n)). - Bernard Schott, Jan 20 2022

Corrected by T. D. Noe, Nov 07 2006

A352156 Numbers m such that the smallest digit in the decimal expansion of 1/m is 2, ignoring leading and trailing 0's.

Original entry on oeis.org

4, 5, 16, 36, 40, 44, 45, 50, 108, 160, 216, 252, 288, 292, 308, 360, 364, 375, 396, 400, 404, 440, 444, 450, 500, 1024, 1080, 1375, 1600, 2072, 2160, 2368, 2520, 2880, 2920, 3080, 3125, 3375, 3600, 3640, 3750, 3848, 3960, 4000, 4040, 4125, 4224, 4368, 4400, 4440, 4500, 5000

Offset: 1

Bernard Schott and Robert G. Wilson v, Mar 19 2022

Leading 0's are not considered, otherwise every integer >= 11 would be a term (see examples).
Trailing 0's are also not considered, otherwise numbers of the form 2^i*5^j with i, j >= 0, apart from 1 (A003592) would be terms.
If k is a term, 10*k is also a term; so, terms with no trailing zeros are all primitive terms.

			m = 16 is a term since 1/16 = 0.0625 and the smallest term after the leading 0 is 2.
m = 216 is a term since 1/216 = 0.004629629629... and the smallest term after the leading 0's is 2.
m = 4444 is not a term since 1/4444 = 0.00022502250225... and the smallest term after the leading 0's is 0.

Cf. A341383.
Subsequences: A093141 \ {1}, A093143 \ {1}.
Similar with smallest digit k: A352154 (k=0), A352155 (k=1), this sequence (k=2), A352157 (k=3), A352158 (k=4), A352159 (k=5), A352160 (k=6), A352153 (no known term for k=7), A352161 (k=8), no term (k=9).

f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 1100, Min@ f@# == 2 &]

from itertools import count, islice
from sympy import multiplicity, n_order
def A352156_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        m2, m5 = multiplicity(2,n), multiplicity(5,n)
        k, m = 10**max(m2,m5), 10**(t := n_order(10,n//2**m2//5**m5))-1
        c = k//n
        s = str(m*k//n-c*m).zfill(t)
        if s == '0' and min(str(c)) == '2':
            yield n
        elif '0' not in s and min(str(c).lstrip('0')+s) == '2':
                yield n
A352156_list = list(islice(A352156_gen(),20)) # Chai Wah Wu, Mar 28 2022

A352153(a(n)) = 2.

A216099 Period of powers of 3 mod 10^n.

Original entry on oeis.org

4, 20, 100, 500, 5000, 50000, 500000, 5000000, 50000000, 500000000, 5000000000, 50000000000, 500000000000, 5000000000000, 50000000000000, 500000000000000, 5000000000000000, 50000000000000000, 500000000000000000

Offset: 1

V. Raman, Sep 01 2012

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

Cf. A000244, A001148, A093143, A001218.

Table[If[n<5,(4*5^n)/5,10^n/20],{n,20}] (* or *) Join[{4,20,100},NestList[ 10#&,500,20]] (* Harvey P. Dale, May 31 2017 *)

a(n)=if(n<5,4*5^n/5,10^n/20) \\ Charles R Greathouse IV, Mar 26 2016

a(n) = 4*5^(n-1) for n <= 4.

a(n) = 5*10^(n-2) for n >= 5.

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)

cp's OEIS Frontend

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

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A169964 Numbers whose decimal expansion contains only 0's and 5's.

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A055372 Invert transform of Pascal's triangle A007318.

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A134309 Triangle read by rows, where row n consists of n zeros followed by 2^(n-1).

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A246057 a(n) = (5*10^n - 2)/3.

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A341383 Numbers m such that the largest digit in the decimal expansion of 1/m is 2.

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A037156 a(n) = 10^n*(10^n+1)/2.

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