A097166 -id:A097166 - cp's OEIS frontend

A198970 a(n) = 4*10^n - 1.

3, 39, 399, 3999, 39999, 399999, 3999999, 39999999, 399999999, 3999999999, 39999999999, 399999999999, 3999999999999, 39999999999999, 399999999999999, 3999999999999999, 39999999999999999, 399999999999999999, 3999999999999999999, 39999999999999999999, 399999999999999999999

Offset: 0

Vincenzo Librandi, Nov 01 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A097166.

Magma
```
[4*10^n-1 : n in [0..20]];
```

CoefficientList[Series[3*(1 + 2*x)/(1 - 11*x + 10*x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 03 2013 *)

a(n) = 3*A097166(n).
a(n) = 10*a(n-1) + 9, n > 0.
a(n) = 11*a(n-1) - 10*a(n-2), n > 1.
G.f.: 3*(1 + 2*x)/(1 - 11*x + 10*x^2). - Vincenzo Librandi, Jan 03 2013
E.g.f.: exp(x)*(4*exp(9*x) - 1). - Elmo R. Oliveira, Sep 10 2024

A073556 Number of Fibonacci numbers F(k), k <= 10^n, which end in 9.

Original entry on oeis.org

0, 13, 133, 1333, 13333, 133333, 1333333, 13333333, 133333333, 1333333333, 13333333333, 133333333333, 1333333333333, 13333333333333, 133333333333333, 1333333333333333, 13333333333333333, 133333333333333333, 1333333333333333333, 13333333333333333333, 133333333333333333333, 1333333333333333333333, 13333333333333333333333

Offset: 1

Shyam Sunder Gupta, Aug 15 2002

Same as A097166(n) = (4*10^n - 1)/3, for n>0. - Lekraj Beedassy, Apr 08 2005

			a(2) = 13 because there are 13 Fibonacci numbers up to 10^2 which end in 9.

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

Cf. A000045, A097166.

If n>1 then a(n) = (2*10^n - 5)/15. - Robert Gerbicz, Sep 06 2002
From Elmo R. Oliveira, Jul 22 2025: (Start)
G.f.: x^2*(13 - 10*x)/((1-x)*(1-10*x)).
E.g.f.: (3 - 15*x - 5*exp(x) + 2*exp(10*x))/15.
a(n) = 11*a(n-1) - 10*a(n-2) for n > 3. (End)

More terms from Robert Gerbicz, Sep 06 2002

A137215 a(n) = 3(10^n) + (n^2 + 1)(10^n - 1)/9.

Original entry on oeis.org

3, 32, 355, 4110, 48887, 588886, 7111107, 85555550, 1022222215, 12111111102, 142222222211, 1655555555542, 19111111111095, 218888888888870, 2488888888888867, 28111111111111086, 315555555555555527, 3522222222222222190, 39111111111111111075, 432222222222222222182

Offset: 0

Ctibor O. Zizka, Mar 06 2008

Sequence generalized: a(n) = a(0)*(B^n) + F(n)* ((B^n)-1)/(B-1); a(0), B integers, F(n) arithmetic function.
Examples:
a(0) = 1, B = 10, F(n) = 1 gives A002275, F(n) = 2 gives A090843, F(n) = 3 gives A097166, F(n) = 4 gives A099914, F(n) = 5 gives A099915.
a(0) = 1, B = 2, F(n) = 1 gives A000225, F(n) = 2 gives A033484, F(n) = 3 gives A036563, F(n) = 4 gives A048487, F(n) = 5 gives A048488, F(n) = 6 gives A048489.
a(0) = 1, B = 3, F(n) = 1 gives A003462, F(n) = 2 gives A048473, F(n) = 3 gives A134931, F(n) = 4 gives A058481, F(n) = 5 gives A116952.
a(0) = 1, B = 4, F(n) = 1 gives A002450, F(n) = 2 gives A020989, F(n) = 3 gives A083420, F(n) = 4 gives A083597, F(n) = 5 gives A083584.
a(0) = 1, B = 5, F(n) = 1 gives A003463, F(n) = 2 gives A057651, F(n) = 3 gives A117617, F(n) = 4 gives A081655.
a(0) = 2, B = 10, F(n) = 1 gives A037559, F(n) = 2 gives A002276.

			a(3) = 3*10^3 + (3*3 + 1)*(10^3 - 1)/9 = 4110.

G. C. Greubel, Table of n, a(n) for n = 0..985
Index entries for linear recurrences with constant coefficients, signature (33,-393,1991,-3930,3300,-1000).

Cf. A002275, A011557.

Table[3*10^n +(n^2 +1)*(10^n -1)/9, {n,0,30}] (* G. C. Greubel, Jan 05 2022 *)

a(n) = 3*(10^n) + (n*n+1)*((10^n)-1)/9; \\ Jinyuan Wang, Feb 27 2020

[3*10^n +(1+n^2)*(10^n -1)/9 for n in (0..30)] # G. C. Greubel, Jan 05 2022

a(n) = 3*(10^n) + (n^2 + 1)*(10^n - 1)/9.

O.g.f.: (3 - 67*x + 478*x^2 - 1002*x^3 + 850*x^4 - 100*x^5)/((1-x)^3 * (1-10*x)^3). - R. J. Mathar, Mar 16 2008

More terms from R. J. Mathar, Mar 16 2008

More terms from Jinyuan Wang, Feb 27 2020

A336019 a(n) is the smallest integer k (k>=2) such that 13...3 (1 followed by n 3's) mod k is even.

Original entry on oeis.org

7, 7, 11, 9, 7, 17, 7, 7, 11, 9, 7, 17, 7, 7, 11, 9, 7, 17, 7, 7, 11, 9, 7, 17, 7, 7, 11, 9, 7, 17, 7, 7, 11, 9, 7, 23, 7, 7, 11, 9, 7, 17, 7, 7, 11, 9, 7, 23, 7, 7, 11, 9, 7, 17, 7, 7, 11, 9, 7, 17, 7, 7, 11, 9, 7, 17, 7, 7, 11, 9, 7, 17, 7, 7, 11, 9, 7, 17, 7, 7

Offset: 1

Yuan-Hao Huang, Jul 05 2020

More sequences can be generated by replacing the digit 1 by any integers of the form 3x+1. However, the sequence won't be as interesting if the following digits (the 3's) are replaced by any other digits.

			a(5)=7 because
133333 mod 2 = 1
133333 mod 3 = 1
133333 mod 4 = 1
133333 mod 5 = 3
133333 mod 6 = 1
133333 mod 7 = 4, which is the first time the result is even.

Antti Karttunen, Table of n, a(n) for n = 1..20004

Cf. A097166.

f(n) = (4*10^n-1)/3; \\ A097166
a(n) = my(k=2); while ((f(n) % k) % 2, k++); k; \\ Michel Marcus, Jul 05 2020

n=1
a=13
while n<=1000:
    c=2
    while True:
        if (a%c)%2==1:
            c=c+1
        else:
            print(c,end=", ")
            break
    n=n+1
    a=10*a+3

I have proved the following properties:
For n=12x+1, a(n)=7.
For n=12x+2, a(n)=7.
For n=12x+3, a(n)=11.
For n=12x+4, a(n)=9.
For n=12x+5, a(n)=7.
For n=12x+6, a(n)=17.
For n=12x+7, a(n)=7.
For n=12x+8, a(n)=7.
For n=12x+9, a(n)=11.
For n=12x+10, a(n)=9.
For n=12x+11, a(n)=7.
For n=12x, a(n) can be 17, 19, 23 or 25.

cp's OEIS Frontend

A198970 a(n) = 4*10^n - 1.

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A073556 Number of Fibonacci numbers F(k), k <= 10^n, which end in 9.

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A137215 a(n) = 3*(10^n) + (n^2 + 1)*(10^n - 1)/9.

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A336019 a(n) is the smallest integer k (k>=2) such that 13...3 (1 followed by n 3's) mod k is even.

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Programs

PARI

Python

Formula

A137215 a(n) = 3(10^n) + (n^2 + 1)(10^n - 1)/9.