A332197 -id:A332197 - cp's OEIS frontend

A332190 a(n) = 10^(2n+1) - 1 - 9*10^n.

0, 909, 99099, 9990999, 999909999, 99999099999, 9999990999999, 999999909999999, 99999999099999999, 9999999990999999999, 999999999909999999999, 99999999999099999999999, 9999999999990999999999999, 999999999999909999999999999, 99999999999999099999999999999, 9999999999999990999999999999999

Offset: 0

M. F. Hasler, Feb 08 2020

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002275 (repunits R_n = (10^n-1)/9), A002283 (9*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits only), A002113 (palindromes).
Cf. A332120 .. A332180 (variants with different repeated digit 2, ..., 8).
Cf. A332191 .. A332197, A181965 (variants with different middle digit 1, ..., 8).

Maple
```
A332190 := n -> 10^(2*n+1)-1-9*10^n;
```

Array[10^(2 # + 1)-1-9*10^# &, 15, 0]
LinearRecurrence[{111,-1110,1000},{0,909,99099},20] (* Harvey P. Dale, May 28 2021 *)

apply( {A332190(n)=10^(n*2+1)-1-9*10^n}, [0..15])

def A332190(n): return 10**(n*2+1)-1-9*10^n

a(n) = 9*A138148(n) = A002283(2n+1) - A011557(n).
G.f.: 9*x*(101 - 200*x)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A181965 a(n) = 10^(2n+1) - 10^n - 1.

Original entry on oeis.org

8, 989, 99899, 9998999, 999989999, 99999899999, 9999998999999, 999999989999999, 99999999899999999, 9999999998999999999, 999999999989999999999, 99999999999899999999999, 9999999999998999999999999, 999999999999989999999999999, 99999999999999899999999999999, 9999999999999998999999999999999

Offset: 0

Ivan Panchenko, Apr 04 2012

n 9's followed by an 8 followed by n 9's.

See A183187 = {26, 378, 1246, 1798, 2917, ...} for the indices of primes.

Patrick De Geest, Palindromic Wing Primes: (9)8(9), updated: June 25, 2017.
Makoto Kamada, Factorization of 99...99899...99, updated Dec 11 2018.
Markus Tervooren, Factorizations of (9)w8(9)w, FactorDB.com
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. (A077794-1)/2 = A183187 (indices of primes).
Cf. A002275 (repunits R_n = (10^n-1)/9), A002283 (9*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits only), A002113 (palindromes).
Cf. A332190 .. A332197 (variants with different middle digit 0, ..., 7).

A181965 := n -> 10^(2*n+1)-1-10^n; # M. F. Hasler, Feb 08 2020

Array[10^(2 # + 1) - 1- 10^# &, 15, 0] (*  M. F. Hasler, Feb 08 2020 *)
Table[With[{c=PadRight[{},n,9]},FromDigits[Join[c,{8},c]]],{n,0,20}] (* Harvey P. Dale, Jun 07 2021 *)

apply( {A181965(n)=10^(n*2+1)-1-10^n}, [0..15]) \\ M. F. Hasler, Feb 08 2020

def A181965(n): return 10**(n*2+1)-1-10^n # M. F. Hasler, Feb 08 2020

From M. F. Hasler, Feb 08 2020: (Start)
a(n) = 9*A138148(n) + 8*10^n = A002283(2n+1) - A011557(10^n).
G.f.: (8 + 101*x - 1000*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2. (End)

Edited and extended to a(0) = 8 by M. F. Hasler, Feb 10 2020

A332191 a(n) = 10^(2n+1) - 1 - 8*10^n.

Original entry on oeis.org

1, 919, 99199, 9991999, 999919999, 99999199999, 9999991999999, 999999919999999, 99999999199999999, 9999999991999999999, 999999999919999999999, 99999999999199999999999, 9999999999991999999999999, 999999999999919999999999999, 99999999999999199999999999999, 9999999999999991999999999999999

Offset: 0

M. F. Hasler, Feb 08 2020

See A183184 = {1, 5, 13, 43, 169, 181, ...} for the indices of primes.

Patrick De Geest, Palindromic Wing Primes: (9)1(9), updated: June 25, 2017.
Makoto Kamada, Factorization of 9999199...99, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. (A077776-1)/2 = A183184: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A002283 (9*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits only), A002113 (palindromes).
Cf. A332121 .. A332181 (variants with different repeated digit 2, ..., 8).
Cf. A332190 .. A332197, A181965 (variants with different middle digit 0, ..., 8).

Maple
```
A332191 := n -> 10^(n*2+1)-1-8*10^n;
```
Mathematica
```
Array[ 10^(2 # + 1)-1-8*10^# &, 15, 0]
```

apply( {A332191(n)=10^(n*2+1)-1-8*10^n}, [0..15])

def A332191(n): return 10**(n*2+1)-1-8*10^n

a(n) = 9*A138148(n) + 10^n.
G.f.: (1 + 808*x - 1700*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A332117 a(n) = (10^(2n+1)-1)/9 + 6*10^n.

Original entry on oeis.org

7, 171, 11711, 1117111, 111171111, 11111711111, 1111117111111, 111111171111111, 11111111711111111, 1111111117111111111, 111111111171111111111, 11111111111711111111111, 1111111111117111111111111, 111111111111171111111111111, 11111111111111711111111111111, 1111111111111117111111111111111

Offset: 0

M. F. Hasler, Feb 09 2020

See A107127 = {0, 3, 33, 311, 2933, ...} for the indices of primes.

Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).
Patrick De Geest, Palindromic Wing Primes: (1)7(1), updated: June 25, 2017.
Makoto Kamada, Factorization of 11...11711...11, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. (A077789-1)/2 = A107127: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332127 .. A332197 (variants with different repeated digit 2, ..., 9).
Cf. A332112 .. A332119 (variants with different middle digit 2, ..., 9).

A332117 := n -> (10^(2*n+1)-1)/9+6*10^n;

Array[(10^(2 # + 1)-1)/9 + 6*10^# &, 15, 0]

apply( {A332117(n)=10^(n*2+1)\9+6*10^n}, [0..15])

def A332117(n): return 10**(n*2+1)//9+6*10**n

a(n) = A138148(n) + 7*10^n = A002275(2n+1) + 6*10^n.
G.f.: (7 - 606*x + 500*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A332192 a(n) = 10^(2n+1) - 1 - 7*10^n.

Original entry on oeis.org

2, 929, 99299, 9992999, 999929999, 99999299999, 9999992999999, 999999929999999, 99999999299999999, 9999999992999999999, 999999999929999999999, 99999999999299999999999, 9999999999992999999999999, 999999999999929999999999999, 99999999999999299999999999999, 9999999999999992999999999999999

Offset: 0

M. F. Hasler, Feb 08 2020

See A115073 = {1, 8, 9, 352, 530, 697, ...} for the indices of primes.

Patrick De Geest, Palindromic Wing Primes: (9)2(9), updated: June 25, 2017.
Makoto Kamada, Factorization of 99...99299...99, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. (A077778-1)/2 = A115073: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A002283 (9*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits only), A002113 (palindromes).
Cf. A332190 .. A332197, A181965 (variants with different middle digit 0, ..., 8).
Cf. A332112 .. A332182 (variants with different repeated digit 1, ..., 8).

Maple
```
A332192 := n -> 10^(n*2+1)-1-7*10^n;
```

Array[ 10^(2 # +1) -1 -7*10^# &, 15, 0]
LinearRecurrence[{111,-1110,1000},{2,929,99299},20] (* Harvey P. Dale, Nov 07 2022 *)

apply( {A332192(n)=10^(n*2+1)-1-7*10^n}, [0..15])

def A332192(n): return 10**(n*2+1)-1-7*10^n

a(n) = 9*A138148(n) + 2*10^n = A002283(2n+1) - 7*10^n.
G.f.: (2 + 707*x - 1600*x^2)/((1 - x)*(1 - 10*x)*(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A332193 a(n) = 10^(2n+1) - 1 - 6*10^n.

Original entry on oeis.org

3, 939, 99399, 9993999, 999939999, 99999399999, 9999993999999, 999999939999999, 99999999399999999, 9999999993999999999, 999999999939999999999, 99999999999399999999999, 9999999999993999999999999, 999999999999939999999999999, 99999999999999399999999999999, 9999999999999993999999999999999

Offset: 0

M. F. Hasler, Feb 08 2020

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002275 (repunits R_n = (10^n-1)/9), A002283 (9*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits only), A002113 (palindromes).
Cf. A332113 .. A332183 (variants with different repeated digit 1, ..., 8).
Cf. A332190 .. A332197, A181965 (variants with different middle digit 0, ..., 8).

Maple
```
A332193 := n -> 10^(n*2+1)-1-6*10^n;
```

Array[ 10^(2 # + 1) - 1 - 6*10^# &, 15, 0]
LinearRecurrence[{111,-1110,1000},{3,939,99399},20] (* Harvey P. Dale, Jan 19 2024 *)

apply( {A332193(n)=10^(n*2+1)-1-6*10^n}, [0..15])

def A332193(n): return 10**(n*2+1)-1-6*10^n

a(n) = 9*A138148(n) + 3*10^n = A002283(2n+1) - 6*10^n.
G.f.: (3 + 606*x - 1500*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A332195 a(n) = 10^(2n+1) - 4*10^n - 1.

Original entry on oeis.org

5, 959, 99599, 9995999, 999959999, 99999599999, 9999995999999, 999999959999999, 99999999599999999, 9999999995999999999, 999999999959999999999, 99999999999599999999999, 9999999999995999999999999, 999999999999959999999999999, 99999999999999599999999999999, 9999999999999995999999999999999

Offset: 0

M. F. Hasler, Feb 08 2020

See A183186 = {88, 112, 198, 622, 4228, ...} for the indices of primes.

Patrick De Geest, Palindromic Wing Primes: (9)5(9), updated Jun 25 2017.
Makoto Kamada, Factorization of 99...99599...99, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. (A077786-1)/2 = A183186: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A002283 (9*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits only), A002113 (palindromes).
Cf. A332115 .. A332185 (variants with different repeated digit 1, ..., 8).
Cf. A332190 .. A332197, A181965 (variants with different middle digit 0, ..., 8).

Maple
```
A332195 := n -> 10^(n*2+1)-4*10^n-1;
```

Array[ 10^(2 # + 1) - 1 - 4*10^# &, 15, 0]

apply( {A332195(n)=10^(n*2+1)-1-4*10^n}, [0..15])

def A332195(n): return 10**(n*2+1)-1-4*10^n

a(n) = 9*A138148(n) + 5*10^n.
G.f.: (5 + 404*x - 1300*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A332196 a(n) = 10^(2n+1) - 1 - 3*10^n.

Original entry on oeis.org

6, 969, 99699, 9996999, 999969999, 99999699999, 9999996999999, 999999969999999, 99999999699999999, 9999999996999999999, 999999999969999999999, 99999999999699999999999, 9999999999996999999999999, 999999999999969999999999999, 99999999999999699999999999999

Offset: 0

M. F. Hasler, Feb 08 2020

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002275 (repunits R_n = (10^n-1)/9), A002283 (9*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits only), A002113 (palindromes).
Cf. A332116 .. A332186 (variants with different repeated digit 1, ..., 8).
Cf. A332190 .. A332197, A181965 (variants with different middle digit 0, ..., 8).

Maple
```
A332196 := n -> 10^(n*2+1)-1-3*10^n;
```

Array[ 10^(2 # + 1) - 1 - 3*10^# &, 15, 0]
FromDigits/@Table[Join[PadLeft[{6},n,9],PadRight[{},n-1,9]],{n,30}] (* or *) LinearRecurrence[{111,-1110,1000},{6,969,99699},30] (* Harvey P. Dale, May 03 2021 *)

apply( {A332196(n)=10^(n*2+1)-1-3*10^n}, [0..15])

def A332196(n): return 10**(n*2+1)-1-3*10^n

a(n) = 9*A138148(n) + 6*10^n.
G.f.: (6 + 303*x - 1200*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.
E.g.f.: exp(x)*(10*exp(99*x) - 3*exp(9*x) - 1). - Stefano Spezia, Jul 13 2024

A332194 a(n) = 10^(2n+1) - 1 - 5*10^n.

Original entry on oeis.org

4, 949, 99499, 9994999, 999949999, 99999499999, 9999994999999, 999999949999999, 99999999499999999, 9999999994999999999, 999999999949999999999, 99999999999499999999999, 9999999999994999999999999, 999999999999949999999999999, 99999999999999499999999999999, 9999999999999994999999999999999

Offset: 0

M. F. Hasler, Feb 08 2020

See A183185 = {14, 22, 36, 104, 1136, ...} for the indices of primes.

Patrick De Geest, Palindromic Wing Primes: (9)4(9), updated Jun 25 2017.
Makoto Kamada, Factorization of 99...99499...99, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. (A077782-1)/2 = A183185: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A002283 (9*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits only), A002113 (palindromes).
Cf. A332114 .. A332184 (variants with different repeated digit 1, ..., 8).
Cf. A332190 .. A332197, A181965 (variants with different middle digit 0, ..., 8).

Maple
```
A332194 := n -> 10^(n*2+1)-1-5*10^n;
```

Array[ 10^(2 # + 1) -1 -5*10^# &, 15, 0]

apply( {A332194(n)=10^(n*2+1)-1-5*10^n}, [0..15])

def A332194(n): return 10**(n*2+1)-1-5*10^n

a(n) = 9*A138148(n) + 4*10^n = A002283(2n+1) - 5*A011557(n).
G.f.: (4 + 505*x - 1400*x^2)/((1 - x)*(1 - 10*x)*(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A332127 a(n) = 2(10^(2n+1)-1)/9 + 510^n.

Original entry on oeis.org

7, 272, 22722, 2227222, 222272222, 22222722222, 2222227222222, 222222272222222, 22222222722222222, 2222222227222222222, 222222222272222222222, 22222222222722222222222, 2222222222227222222222222, 222222222222272222222222222, 22222222222222722222222222222, 2222222222222227222222222222222

Offset: 0

M. F. Hasler, Feb 09 2020

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002275 (repunits R_n = (10^n-1)/9), A002276 (2*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332117 .. A332197 (variants with different repeated digit 1, ..., 9).
Cf. A332120 .. A332129 (variants with different middle digit 0, ..., 9).

A332127 := n -> 2*(10^(2*n+1)-1)/9+5*10^n;

Array[2 (10^(2 # + 1)-1)/9 + 5*10^# &, 15, 0]

apply( {A332127(n)=10^(n*2+1)\9*2+5*10^n}, [0..15])

def A332127(n): return 10**(n*2+1)//9*2+5*10**n

a(n) = 2*A138148(n) + 7*10^n = A002276(2n+1) + 5*10^n.
G.f.: (7 - 505*x + 300*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

cp's OEIS Frontend

A332190 a(n) = 10^(2n+1) - 1 - 9*10^n.

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A181965 a(n) = 10^(2n+1) - 10^n - 1.

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A332191 a(n) = 10^(2n+1) - 1 - 8*10^n.

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Maple

Mathematica

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Python

Formula

A332117 a(n) = (10^(2n+1)-1)/9 + 6*10^n.

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Author

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Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A332192 a(n) = 10^(2n+1) - 1 - 7*10^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A332193 a(n) = 10^(2n+1) - 1 - 6*10^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A332195 a(n) = 10^(2n+1) - 4*10^n - 1.

Views

A332127 a(n) = 2(10^(2n+1)-1)/9 + 510^n.