A093137 -id:A093137 - cp's OEIS frontend

A181718 a(n) = (1/9)(10^(2n) + 10^n - 2).

0, 12, 1122, 111222, 11112222, 1111122222, 111111222222, 11111112222222, 1111111122222222, 111111111222222222, 11111111112222222222, 1111111111122222222222, 111111111111222222222222, 11111111111112222222222222, 1111111111111122222222222222

Offset: 0

Paul Curtz, Nov 17 2010

In decimal, n times 1 followed by n times 2.

a(n) = 3 + 3*3, 33 + 33*33, 333 + 333*333, written with 3,6,9,12,... = A008585(n+1) 3's.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002277, A008585, A074992, A093137.

[(1/9)*(10^(2*n) + 10^n - 2): n in [0..20]]; // Vincenzo Librandi, Aug 04 2011

Table[FromDigits[Join[PadRight[{},n,1],PadRight[{},n,2]]],{n,0,20}] (* or *) LinearRecurrence[{111,-1110,1000},{0,12,1122},20] (* Harvey P. Dale, Jul 31 2013 *)

vector(30, n, n--; (10^(2*n) + 10^n - 2)/9) \\ G. C. Greubel, Nov 02 2018

for n in range(30):
    print((10**(2*n)+10**n-2)//9, end=', ')
# Stefano Spezia, Nov 02 2018

[(100^n +10^n -2)//9 for n in range(31)] # G. C. Greubel, Mar 25 2024

a(n) = A002277(n)*A093137(n).
G.f.: 6*x*(2-35*x) / ( (1-x)*(1-10*x)*(1-100*x) ). - R. J. Mathar, Feb 28 2011
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3), a(0)=0, a(1)=12, a(2)=1122. - Harvey P. Dale, Jul 31 2013
a(n) = (A074992(n) - 1)/3. - Michel Marcus, Sep 14 2013
E.g.f.: (1/9)*(-2*exp(x) + exp(10*x) + exp(100*x)). - G. C. Greubel, Mar 25 2024

A322570 Positive integers k such that A270710(k) (= (k+1)(3k-1)) have only 1 or 2 different digits in base 10.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 12, 16, 17, 33, 34, 48, 54, 285, 333, 334, 365, 385, 430, 471, 516, 816, 1049, 3333, 3334, 33333, 33334, 333333, 333334, 483048, 3333333, 3333334, 33333333, 33333334, 333333333, 333333334, 3333333333, 3333333334, 33333333333, 33333333334

Offset: 1

Seiichi Manyama, Aug 29 2019

Giovanni Resta, Table of n, a(n) for n = 1..50

Cf. A002277, A016069, A093137, A213517 (in case of triangular numbers), A270710, A322571.

[k:k in [1..10000000]| #Set(Intseq((k+1)*(3*k-1))) le 2]; // Marius A. Burtea, Aug 29 2019

Select[Range@ 50000, Length@ Union@ IntegerDigits[3 #^2 + 2 # - 1] <= 2 &] (* Giovanni Resta, Sep 04 2019 *)

for(k=1, 1e8, if(#Set(digits(3*k^2+2*k-1))<=2, print1(k", ")))

For k > 0, A002277(k) is a term.

For k >= 0, A002277(k) + 1 (= A093137(k)) is a term.

a(35)-a(36) from Jinyuan Wang, Aug 30 2019

a(37)-a(40) from Giovanni Resta, Sep 04 2019

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

A254398 Final digits of A237424 in decimal representation.

Original entry on oeis.org

1, 4, 7, 4, 7, 7, 4, 7, 7, 7, 4, 7, 7, 7, 7, 4, 7, 7, 7, 7, 7, 4, 7, 7, 7, 7, 7, 7, 4, 7, 7, 7, 7, 7, 7, 7, 4, 7, 7, 7, 7, 7, 7, 7, 7, 4, 7, 7, 7, 7, 7, 7, 7, 7, 7, 4, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 4, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 4, 7, 7, 7, 7, 7, 7, 7

Offset: 1

Reinhard Zumkeller, Feb 23 2015

a(n) = A237424(n) mod 10;
n with a(n) = 4: A237424(n) = (10^a+10^b+1)/3 with b = 0, see also A093137, A133384;
n with a(n) = 7: A237424(n) = (10^a+10^b+1)/3 with 0 < b <= a;
length of k-th run of consecutive 7s = k;
digits 0, 2, 3, 5, 6, 8 and 9 do not occur.

Index entries for sequences related to final digits of numbers

Cf. A237424, A010879, A254339, A093137, A133384.

Haskell
```
a254398 = flip mod 10 . a237424
```

A308900 An explicit example of an infinite sequence with a(1)=1 and, for n >= 2, a(n) and S(n) = Sum_{i=1..n} a(i) have no digit in common.

Original entry on oeis.org

1, 6, 4, 66, 34, 666, 334, 6666, 3334, 66666, 33334, 666666, 333334, 6666666, 3333334, 66666666, 33333334, 666666666, 333333334, 6666666666, 3333333334, 66666666666, 33333333334, 666666666666, 333333333334, 6666666666666, 3333333333334, 66666666666666, 33333333333334

Offset: 1

N. J. A. Sloane, Jul 15 2019

Used in a proof that the initial terms of A309151 are correct.
The S(n) sequence is 1, 7, 11, 77, 111, 777, 1111, 7777, 11111, 77777, ...
A093137 interleaved with positive terms of A002280. - Felix Fröhlich, Jul 15 2019

Robert Israel, Table of n, a(n) for n = 1..1999
R. Israel, Re: Help with a(n) and a cumulative sum, Seqfan (Jul 15 2019).
Index entries for linear recurrences with constant coefficients, signature (-1,10,10).

Cf. A002280, A093137, A309151.

Cf. A289404.

I:=[1,6,4]; [n le 3 select I[n] else - Self(n-1) + 10*Self(n-2) + 10*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jul 20 2019

1, seq(op([6*(10^i-1)/9, 3*(10^i-1)/9+1]), i=1..30); # Robert Israel, Jul 15 2019

CoefficientList[Series[(1 + 7 x)/((1 + x) (1 - 10 x^2)), {x, 0, 26}], x] (* Michael De Vlieger, Jul 18 2019 *)
LinearRecurrence[{-1,10,10},{1,6,4},30] (* Harvey P. Dale, Jan 02 2022 *)

Vec((1+7*x)/((1+x)*(1-10*x^2)) + O(x^20)) \\ Felix Fröhlich, Jul 15 2019

a(n) = if(n==1, 1, if(n%2==0, 6*(10^(n/2)-1)/9, 3*(10^((n-1)/2)-1)/9+1)) \\ Felix Fröhlich, Jul 15 2019

For even n >= 2, a(n) = 6666...66 (with n/2 6's). For odd n >= 5, a(n) = 3333...334 (with (n-3)/2 3's and a single 4).
From Robert Israel, Jul 15 2019: (Start)
G.f. (1+7*x)/((1+x)*(1-10*x^2)).
a(n) = -a(n - 1) + 10*a(n - 2) + 10*a(n - 3). (End)
a(-n) = a(n+1). - Paul Curtz, Jul 18 2019
a(n) = (1/60)*(-40*(-1)^n + (1 + (-1)^n)*(2^(2+n/2)*5^(1+n/2)) + (1 + (-1)^(n+1))*10^((1+n)/2)). - Stefano Spezia, Jul 20 2019

A374258 Square array: T(n,k) = ((3^(n+1) + 1)^(k-1) + 2)/3, read by descending antidiagonals.

Original entry on oeis.org

1, 4, 1, 34, 10, 1, 334, 262, 28, 1, 3334, 7318, 2242, 82, 1, 33334, 204886, 183790, 19846, 244, 1, 333334, 5736790, 15070726, 4842262, 177634, 730, 1, 3333334, 160630102, 1235799478, 1181511766, 129672334, 1595782, 2188, 1, 33333334, 4497642838, 101335557142, 288288870742, 94660803334, 3491569558, 14353282, 6562, 1

Offset: 1

Ahmad J. Masad, Jul 01 2024

This sequence gives the matrix M in the definition of A365450. Similar to A266577.
Conjecture: For each natural number n, the digits of the product of any (n+1) not necessarily distinct terms of the n-th row in the base (3^(n+1)+1) numeral system appear in nondecreasing order.
Proof of the conjecture. Let b := 3^(n+1)+1. The product of any (n+1) terms of the n-th row has the form p/(b-1), where p is the product of (n+1) numbers of the form b^k+2. Let p = (dm, ..., d1, d0)b, and we have d0+d1+...+dm = 3^(n+1) = b-1. Then p/(b-1) = (dm, ..., d2+...+dm, 1+d1+...+dm)_b, which do form a nondecreasing sequence. - _Max Alekseyev, Jul 03 2024
The preceding result is similar to the property of the nondecreasing products mentioned in A237424. Specifically; the first row of this array is A093137, which is a subsequence of A237424.  - Ahmad J. Masad, Jul 30 2024
More generally: Let r and n be positive integers and S be a sequence of all numbers of the form (b^c(1)+b^c(2)+...+b^c(r)+1)/(r+1), where c(1),...,c(r) are nonnegative integers. Then in the numeral system base b := (r+1)^(n+1)+1, the digits of the product of any n+1 (possibly equal) terms of S appear in nondecreasing order. Proof is similar. - Ahmad J. Masad, Jul 30 2024; edited by Max Alekseyev, Aug 01 2024

			The array begins:
  1    4    34   334   3334
  1   10   262  7318
  1   28  2242
  1   82
  1
Example of the conjecture: Take 5 terms from the 4th row and find their product in base 244 numeral system (since 3^(4+1)+1=244) as follows: 82,19846 twice and 4842262 twice, the product is equal to 82*19846*19846*4842262*4842262 = 757279838666167487626528 = (1, 3, 7, 19, 31, 55, 91, 115, 163, 195, 212)_244 which is in agreement with the conjecture since the digits in 244 base numeral system are in nondecreasing order.
Example of the general property: Take r=3 and n=4, then b=4^5+1=1025. The sequence S is the sequence of the numbers of the form (1025^b(1)+1025^b(2)+1025^b(3)+1)/4. Let's multiply 5 terms of the sequence S, say ((1025^0+1025^0+1025^1+1)/4)*(((1025^0+1025^1+1025^1+1)/4)^2)*(((1025^3+1025^4+1025^4+1)/4)^2) = 257*513^2*552175667969^2 = 20621601208620337073958261562113 = (16,112,308,488,580,680,832,936,964,984,1013)_1025. The digits of the product in base 1025 are in nondecreasing order.

Michel Marcus, Table of n, a(n) for n = 1..820

Cf. A093137, A266577, A365450, A237424.

T[n_, k_] := ((3^(n+1) + 1)^(k-1) + 2)/3; Table[T[k, n-k+1], {n, 1, 9}, {k, 1, n}] // Flatten (* Amiram Eldar, Jul 02 2024 *)

T(n,k) = ((3^(n+1) + 1)^(k-1) + 2)/3 \\ Andrew Howroyd, Jul 01 2024

A274986 Numbers k such that (10^k + 23)/3 is prime.

Original entry on oeis.org

1, 2, 6, 146, 326, 380, 1116, 1866, 4808, 5528, 5730, 21836, 24804, 38724

Offset: 1

Vincenzo Librandi, Sep 25 2016

Also numbers k for which A093137(k) + 7 or A002277(k) + 8 is prime.

Cf. numbers k such that (10^k+m)/3 is prime: A099411 (m=11), this sequence (m=23).

Cf. A002277, A093137.

[n: n in [0..400] | IsPrime((10^n+23) div 3)];

Select[Range[1000], PrimeQ[(10^# + 23) / 3] &]

is(n)=ispseudoprime((10^n+23)/3) \\ Charles R Greathouse IV, Jun 13 2017

a(9)-a(11) from Michael S. Branicky, Aug 16 2021
a(12)-a(13) from Michael S. Branicky, May 14 2023
a(14) from Kamada data by Tyler Busby, May 05 2024

cp's OEIS Frontend

A181718 a(n) = (1/9)*(10^(2*n) + 10^n - 2).

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A322570 Positive integers k such that A270710(k) (= (k+1)*(3*k-1)) have only 1 or 2 different digits in base 10.

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

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A254398 Final digits of A237424 in decimal representation.

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A308900 An explicit example of an infinite sequence with a(1)=1 and, for n >= 2, a(n) and S(n) = Sum_{i=1..n} a(i) have no digit in common.

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A374258 Square array: T(n,k) = ((3^(n+1) + 1)^(k-1) + 2)/3, read by descending antidiagonals.

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Mathematica

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A274986 Numbers k such that (10^k + 23)/3 is prime.

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A181718 a(n) = (1/9)(10^(2n) + 10^n - 2).

A322570 Positive integers k such that A270710(k) (= (k+1)(3k-1)) have only 1 or 2 different digits in base 10.