A093143 -id:A093143 - cp's OEIS frontend

A093142 Expansion of g.f. (1-5x)/((1-x)(1-10*x)).

1, 6, 56, 556, 5556, 55556, 555556, 5555556, 55555556, 555555556, 5555555556, 55555555556, 555555555556, 5555555555556, 55555555555556, 555555555555556, 5555555555555556, 55555555555555556, 555555555555555556, 5555555555555555556, 55555555555555555556, 555555555555555555556

Offset: 0

Paul Barry, Mar 24 2004

Second binomial transform of 5*A001045(3n)/3+(-1)^n.
Partial sums of A093143.
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=5.

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

Cf. A001045, A002275, A062397, A093143.

CoefficientList[Series[(1-5x)/((1-x)(1-10x)),{x,0,20}],x] (* or *) LinearRecurrence[{11,-10},{1,6},20] (* Harvey P. Dale, Aug 23 2014 *)

{a(n) = (5*10^n+4)/9} \\ Seiichi Manyama, Sep 14 2019

a(n) = 5*10^n/9 + 4/9.
a(n) = 10*a(n-1) - 4 with a(0)=1. - Vincenzo Librandi, Aug 02 2010
a(n) = 11*a(n-1) - 10*a(n-2), n > 1. - Harvey P. Dale, Aug 23 2014
From Elmo R. Oliveira, Apr 29 2025: (Start)
E.g.f.: exp(x)*(5*exp(9*x) + 4)/9.
a(n) = (A062397(n) + A002275(n))/2. (End)

A178769 a(n) = (5*10^n + 13)/9.

Original entry on oeis.org

2, 7, 57, 557, 5557, 55557, 555557, 5555557, 55555557, 555555557, 5555555557, 55555555557, 555555555557, 5555555555557, 55555555555557, 555555555555557, 5555555555555557, 55555555555555557, 555555555555555557, 5555555555555555557, 55555555555555555557, 555555555555555555557

Offset: 0

Bruno Berselli, Jun 13 2010

Bruno Berselli, Table of n, a(n) for n = 0..1000.
Bruno Berselli, A property that includes the numbers of the form 5..57.
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A165246 (..17, 117, 1117,..), A173193 (..27, 227, 2227,..), A173766 (..37, 337, 3337,..), A173772 (..47, 447, 4447,..), A067275 (..67, 667, 6667,..), A002281 (..77, 777, 7777,..), A173812 (..87, 887, 8887,..), A173833 (..97, 997, 9997,..).

Cf. A093143.

List([0..20], n -> (5*10^n+13)/9); # G. C. Greubel, Jan 24 2019

[(5*10^n+13)/9: n in [0..20]]; // Vincenzo Librandi, Jun 06 2013

CoefficientList[Series[(2 - 15 x) / ((1 - x) (1 - 10 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Jun 06 2013 *)
LinearRecurrence[{11,-10},{2,7},20] (* Harvey P. Dale, Feb 28 2017 *)

vector(20, n, n--; (5*10^n+13)/9) \\ G. C. Greubel, Jan 24 2019

[(5*10^n+13)/9 for n in (0..20)] # G. C. Greubel, Jan 24 2019

a(n)^(4*k+2) + 1 == 0 (mod 250) for n > 1, k >= 0.
G.f.: (2-15*x)/((1-x)*(1-10*x)).
a(n) - 11*a(n-1) + 10*a(n-2) = 0 (n > 1).
a(n) = a(n-1) + 5*10^(n-1) = 10*a(n-1) - 13 for n > 0.
a(n) = 1 + Sum_{i=0..n} A093143(i). - Bruno Berselli, Feb 16 2015
E.g.f.: exp(x)*(5*exp(9*x) + 13)/9. - Elmo R. Oliveira, Sep 09 2024

A136025 Sum of distinct proper prime divisors of odd integers below 10^n.

Original entry on oeis.org

3, 373, 24307, 1691682, 127867801, 10233538789, 850896280551, 72812857079241, 6363727756215813, 565232434009370012, 50843507342073211151, 4620323131256374760046, 423405369424475640435621, 39074878176445767411791424

Offset: 1

Enoch Haga, Dec 12 2007

Through 10^5 about 37.5% of total sums for all integers N comprise sums of odd N and the remaining 62.5% of even N.

			a(0)=3 because the only odd N <=10^1-1 having a prime factor is 9 and its factor is 3 and sum is 3.

Cf. A136021, A136024, A143851.

A105221 := proc(n) local a,ifs,p; ifs := ifactors(n)[2] ; a := 0 ; for p in ifs do if op(1,p) <> 1 and op(1,p) <> n then a := a+op(1,p) ; fi ; od: RETURN(a) ; end: A136025 := proc(n) local a,k ; a := 0 ; for k from 5 to 10^n-1 by 2 do a := a+A105221(k) ; od: RETURN(a) ; end: for n from 1 do print(A136025(n)); od: # R. J. Mathar, Jan 29 2008

a(n) = sum_{k=1,2,...,A093143(n)} A105221(2k-1). - R. J. Mathar, Jan 29 2008

a(n) = sum_{prime p, 3<=p<10^n} p*floor((10^n-p)/(2p)). - Max Alekseyev, Jan 30 2012

a(6) from R. J. Mathar, Jan 29 2008
a(7)-a(11) from Max Alekseyev, Jan 30 2012
a(12)-a(14) from Hiroaki Yamanouchi, Jul 06 2014

A216156 Period of powers of 11 mod 10^n.

Original entry on oeis.org

10, 50, 500, 5000, 50000, 500000, 5000000, 50000000, 500000000, 5000000000, 50000000000, 500000000000, 5000000000000, 50000000000000, 500000000000000, 5000000000000000, 50000000000000000, 500000000000000000

Offset: 1

V. Raman, Sep 02 2012

Essentially the same as A093143. - R. J. Mathar, Sep 06 2012

Also period of squares mod 10^n. - Mohammed Yaseen, Apr 18 2023

			a(2) = 50 because 11^50 = 11739085287969531650666649599035831993898213898723001 = 1 mod 1000.

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

Cf. A001020, A216099.

Join[{10}, NestList[10*# &, 50, 20]] (* Paolo Xausa, Feb 20 2024 *)

a(n) = 5*10^(n-1), n >= 2.

A326811 Numbers in A326806 whose sum of digits is not a power of 10 and are not of the form 510^k or 610^k.

Original entry on oeis.org

0, 6667, 58824, 8823529412, 5263157894737, 19607843137255, 65217391304348, 98360655737705, 746268656716418, 4761904761904762, 8955223880597015, 58823529411764706, 1369863013698630137, 60240963855421686747, 3061224489795918367347, 34090909090909090909091

Offset: 1

Chai Wah Wu, Oct 19 2019

A326806 = A326811 UNION A326833 UNION A090019 UNION A093143. Note that several terms (e.g. a(10), a(13), a(17)-a(19)) look like the rounding off of a periodic sequence, i.e. yxxxa9xxxa9xxxa9... rounded off to yxxxa9xxxa9xxxb, where b = a+1. Perhaps these can be considered near-cyclic numbers? - Chai Wah Wu, Oct 21 2019

Giovanni Resta, Table of n, a(n) for n = 1..70 (first 19 terms from Chai Wah Wu)

Cf. A326806, A326833, A090019, A093143.

More terms from Chai Wah Wu, Oct 21 2019

A346629 Number of n-digit positive integers that are the product of two integers ending with 2.

Original entry on oeis.org

1, 4, 45, 450, 4500, 45000, 450000, 4500000, 45000000, 450000000, 4500000000, 45000000000, 450000000000, 4500000000000, 45000000000000, 450000000000000, 4500000000000000, 45000000000000000, 450000000000000000, 4500000000000000000, 45000000000000000000, 450000000000000000000

Offset: 1

Stefano Spezia, Jul 25 2021

a(n) is the number of n-digit numbers in A139245.

After initial 1 or 2 values the same as A137233. - R. J. Mathar, Aug 23 2021

Index entries for linear recurrences with constant coefficients, signature (10).
Index entries for sequences related to digits.

Cf. A011557 (powers of 10), A017293 (positive integers ending with 2), A052268 (number of n-digit integers), A139245 (product of two integers ending with 2), A093143, A337855, A337856.

Cf. A137233.

Mathematica
```
LinearRecurrence[{10},{1,4,45},25]
```

O.g.f.: x*(1 - 6*x + 5*x^2)/(1 - 10*x).
E.g.f.: (9*exp(10*x) - 9 + 110*x - 50*x^2)/200.
a(n) = 10*a(n-1) for n > 3, with a(1) = 1, a(2) = 4 and a(3) = 45.
a(n) = 45*10^(n-3) for n > 2.
a(n) = 45*A011557(n-3) for n > 2.
Sum_{i=1..n} a(n) = A093143(n-1).

A191687 Table T(n,k) = ceiling((1/2)*((k+1)^n+(1+(-1)^k)/2)) read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 1, 4, 5, 2, 1, 1, 8, 14, 8, 3, 1, 1, 16, 41, 32, 13, 3, 1, 1, 32, 122, 128, 63, 18, 4, 1, 1, 64, 365, 512, 313, 108, 25, 4, 1, 1, 128, 1094, 2048, 1563, 648, 172, 32, 5, 1

Offset: 1

Adi Dani, Jun 11 2011

T(n,k) is the number of compositions of even natural numbers into n parts <= k.

			Top left corner:
  1, 1, 1,  1,  1,...
  1, 1, 2,  2,  3,...
  1, 2, 5,  8, 13,...
  1, 4,14, 32, 63,...
  1, 8,41,128,313,...
T(2,4)=13: there are 13 compositions of even natural numbers into 2 parts <=4
0: (0,0);
2: (0,2), (2,0), (1,1);
4: (0,4), (4,0), (1,3), (3,1), (2,2);
6: (2,4), (4,2), (3,3);
8: (4,4).

Adi Dani, Restricted compositions of natural numbers

Rows sums gives: A000012, A004526, A000982, A036486, A171714, A191484, A191489, A191494, A191495, A191496

Columns sums gives: A000012, A000079, A007051, A004171, A034478, A081341, A034494, A092811, A083884, A093143

Table[Table[Ceiling[1/2*((k+1)^n+(1+(-1)^k)/2)],{n,0,9},{k,0,9}]]

A346178 Expansion of (1-2x)/(1-10x).

Original entry on oeis.org

1, 8, 80, 800, 8000, 80000, 800000, 8000000, 80000000, 800000000, 8000000000, 80000000000, 800000000000, 8000000000000, 80000000000000, 800000000000000, 8000000000000000, 80000000000000000, 800000000000000000, 8000000000000000000, 80000000000000000000

Offset: 0

Felix Fröhlich, Jul 09 2021

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

Cf. expansion of (1-k*x)/(1-10*x) A011557 (k=0), A196662 (k=3), A090019 (k=4), A093143 (k=5), A093141 (k=6), A093138 (k=7), A093136 (k=8).

PARI
```
Vec((1-2*x)/(1-10*x) + O(x^20))
```

a(n) = 8*10^(n-1), n>0.

E.g.f.: (1 + 4*exp(10*x))/5. - Stefano Spezia, Jul 09 2021

cp's OEIS Frontend

A093142 Expansion of g.f. (1-5*x)/((1-x)*(1-10*x)).

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A178769 a(n) = (5*10^n + 13)/9.

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A136025 Sum of distinct proper prime divisors of odd integers below 10^n.

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A216156 Period of powers of 11 mod 10^n.

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A326811 Numbers in A326806 whose sum of digits is not a power of 10 and are not of the form 5*10^k or 6*10^k.

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A346629 Number of n-digit positive integers that are the product of two integers ending with 2.

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A191687 Table T(n,k) = ceiling((1/2)*((k+1)^n+(1+(-1)^k)/2)) read by antidiagonals.

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A346178 Expansion of (1-2*x)/(1-10*x).

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A093142 Expansion of g.f. (1-5x)/((1-x)(1-10*x)).

A326811 Numbers in A326806 whose sum of digits is not a power of 10 and are not of the form 510^k or 610^k.

A346178 Expansion of (1-2x)/(1-10x).