A332121 -id:A332121 - cp's OEIS frontend

A332120 a(n) = 2(10^(2n+1)-1)/9 - 210^n.

0, 202, 22022, 2220222, 222202222, 22222022222, 2222220222222, 222222202222222, 22222222022222222, 2222222220222222222, 222222222202222222222, 22222222222022222222222, 2222222222220222222222222, 222222222222202222222222222, 22222222222222022222222222222, 2222222222222220222222222222222

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. A332130 .. A332190 (variants with different repeated digit 3, ..., 9).
Cf. A332121 .. A332129 (variants with different middle digit 1, ..., 9).

A332120 := n -> 2*((10^(2*n+1)-1)/9-10^n);

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

apply( {A332120(n)=(10^(n*2+1)\9-10^n)*2}, [0..15])

def A332120(n): return (10**(n*2+1)//9-10**n)*2

a(n) = 2*A138148(n) = A002276(2n+1) - 2*10^n.
G.f.: 2*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.
E.g.f.: 2*exp(x)*(10*exp(99*x) - 9*exp(9*x) - 1)/9. - Stefano Spezia, Jul 13 2024

A332171 a(n) = 7(10^(2n+1)-1)/9 - 610^n.

Original entry on oeis.org

1, 717, 77177, 7771777, 777717777, 77777177777, 7777771777777, 777777717777777, 77777777177777777, 7777777771777777777, 777777777717777777777, 77777777777177777777777, 7777777777771777777777777, 777777777777717777777777777, 77777777777777177777777777777, 7777777777777771777777777777777

Offset: 0

M. F. Hasler, Feb 06 2020

For n == 0 or n == 2 (mod 6), there is no obvious divisibility pattern.

According to M. Kamada, n = 116 is the only index of a prime up to n = 10^5.

Makoto Kamada, Factorization of 77...77177...77.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A138148 (cyclops numbers with binary digits only), A002113 (palindromes).
Cf. A002275 (repunits R_n = [10^n/9]), A002281 (7*R_n), A011557 (10^n).
Cf. A332121 .. A332191 (variants with different repeated digit 2, ..., 9).
Cf. A332170 .. A332179 (variants with different middle digit 2, ..., 9).

Array[7 (10^(2 # + 1) - 1)/9 - 6*10^# &, 15, 0] (* or *)
CoefficientList[Series[(1 + 606 x - 1300 x^2)/((1 - x) (1 - 10 x) (1 - 100 x)), {x, 0, 15}], x] (* Michael De Vlieger, Feb 08 2020 *)
Table[FromDigits[Join[PadRight[{},n,7],{1},PadRight[{},n,7]]],{n,0,20}] (* or *) LinearRecurrence[ {111,-1110,1000},{1,717,77177},20] (* Harvey P. Dale, Apr 04 2024 *)

apply( {A332171(n)=10^(n*2+1)\9*7-6*10^n}, [0..15])

Vec((1 + 606*x - 1300*x^2) / ((1 - x)*(1 - 10*x)*(1 - 100*x)) + O(x^15)) \\ Colin Barker, Feb 07 2020

def A332171(n): return 10**(n*2+1)//9*7-6*10^n

a(n) = 7*A138148(n) + 10^n.
For n == 1 (mod 3), 3 | a(n) and a(n)/3 = 259*(10^(2n+1)-1)/999 - 2*10^n;
for n == 3 or 5 (mod 6), 13 | a(n) and a(n)/13 = (A(n)-1)*10^n + B(n), where A(n) (resp. B(n)) are the n leftmost (resp. rightmost) digits of 59829*(10^(ceiling(n/6)*6)-1)/(10^6-1).
From Colin Barker, Feb 07 2020: (Start)
G.f.: (1 + 606*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.
(End)
E.g.f.: (1/9)*exp(x)*(70*exp(99*x) - 54*exp(9*x) - 7). - Stefano Spezia, Feb 08 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.

A332131 a(n) = (10^(2n+1)-1)/3 - 2*10^n.

Original entry on oeis.org

1, 313, 33133, 3331333, 333313333, 33333133333, 3333331333333, 333333313333333, 33333333133333333, 3333333331333333333, 333333333313333333333, 33333333333133333333333, 3333333333331333333333333, 333333333333313333333333333, 33333333333333133333333333333, 3333333333333331333333333333333

Offset: 0

M. F. Hasler, Feb 09 2020

See A183174 = {1, 3, 7, 61, 90, 92, 269, ...} for the indices of primes.

Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).
Patrick De Geest, Palindromic Wing Primes: (3)1(3), updated: June 25, 2017.
Makoto Kamada, Factorization of 33...33133...33, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. (A077775-1)/2 = A183174: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A002277 (3*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332121 .. A332191 (variants with different repeated digit 2, ..., 9).
Cf. A332130 .. A332139 (variants with different middle digit 0, ..., 9).

A332131 := n -> (10^(2*n+1)-1)/3-2*10^n;

Array[3 (10^(2 # + 1)-1)/9 - 2*10^# &, 15, 0]

apply( {A332131(n)=10^(n*2+1)\3-2*10^n}, [0..15])

def A332131(n): return 10**(n*2+1)//3-2*10**n

a(n) = 3*A138148(n) + 1*10^n = A002277(2n+1) - 2*10^n.
G.f.: (1 + 202*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.

A332181 a(n) = 8(10^(2n+1)-1)/9 - 710^n.

Original entry on oeis.org

1, 818, 88188, 8881888, 888818888, 88888188888, 8888881888888, 888888818888888, 88888888188888888, 8888888881888888888, 888888888818888888888, 88888888888188888888888, 8888888888881888888888888, 888888888888818888888888888, 88888888888888188888888888888, 8888888888888881888888888888888

Offset: 0

M. F. Hasler, Feb 08 2020

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), A002282 (8*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits only).
Cf. A332121 .. A332191 (variants with different repeated digit 2, ..., 9).
Cf. A332180 .. A332189 (variants with different middle digit 0, ..., 9).

A332181 := n -> 8*(10^(2*n+1)-1)/9-7*10^n;

Array[8 (10^(2 # + 1)-1)/9 - 7*10^# &, 15, 0]

apply( {A332181(n)=10^(n*2+1)\9*8-7*10^n}, [0..15])

def A332181(n): return 10**(n*2+1)//9*8-7*10**n

a(n) = 8*A138148(n) + 10^n = A002282(2n+1) - 7*10^n.
G.f.: (1 + 707*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.

A332141 a(n) = 4(10^(2n+1)-1)/9 - 3*10^n.

Original entry on oeis.org

1, 414, 44144, 4441444, 444414444, 44444144444, 4444441444444, 444444414444444, 44444444144444444, 4444444441444444444, 444444444414444444444, 44444444444144444444444, 4444444444441444444444444, 444444444444414444444444444, 44444444444444144444444444444, 4444444444444441444444444444444

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), A002278 (4*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332121 .. A332191 (variants with different repeated digit 2, ..., 9).
Cf. A332140 .. A332149 (variants with different middle digit 0, ..., 9).

A332141 := n -> 4*(10^(2*n+1)-1)/9-3*10^n;

Array[4 (10^(2 # + 1)-1)/9 - 3*10^# &, 15, 0]
LinearRecurrence[{111,-1110,1000},{1,414,44144},20] (* or *) Table[ FromDigits[Join[PadRight[{},n,4],{1},PadRight[{},n,4]]],{n,0,20}](* Harvey P. Dale, Aug 17 2020 *)

apply( {A332141(n)=10^(n*2+1)\9*4-3*10^n}, [0..15])

def A332141(n): return 10**(n*2+1)//9*4-3*10**n

a(n) = 4*A138148(n) + 1*10^n = A002278(2n+1) - 3*10^n.
G.f.: (1 + 303*x - 700*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.

A332151 a(n) = 5(10^(2n+1)-1)/9 - 4*10^n.

Original entry on oeis.org

1, 515, 55155, 5551555, 555515555, 55555155555, 5555551555555, 555555515555555, 55555555155555555, 5555555551555555555, 555555555515555555555, 55555555555155555555555, 5555555555551555555555555, 555555555555515555555555555, 55555555555555155555555555555, 5555555555555551555555555555555

Offset: 0

M. F. Hasler, Feb 09 2020

Harvey P. Dale, Table of n, a(n) for n = 0..499
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002275 (repunits R_n = (10^n-1)/9), A002279 (5*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332121 .. A332191 (variants with different repeated digit 2, ..., 9).
Cf. A332150 .. A332159 (variants with different middle digit 0, ..., 9).

A332151 := n -> 5*(10^(2*n+1)-1)/9-4*10^n;

Array[5 (10^(2 # + 1)-1)/9 - 4*10^# &, 15, 0]
Table[With[{c=PadRight[{},n,5]},FromDigits[Join[c,{1},c]]],{n,0,20}] (* Harvey P. Dale, Mar 16 2021 *)

apply( {A332151(n)=10^(n*2+1)\9*5-4*10^n}, [0..15])

def A332151(n): return 10**(n*2+1)//9*5-4*10**n

a(n) = 5*A138148(n) + 10^n = A002279(2n+1) - 4*10^n.
G.f.: (1 + 404*x - 900*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.

A332161 a(n) = 6(10^(2n+1)-1)/9 - 5*10^n.

Original entry on oeis.org

1, 616, 66166, 6661666, 666616666, 66666166666, 6666661666666, 666666616666666, 66666666166666666, 6666666661666666666, 666666666616666666666, 66666666666166666666666, 6666666666661666666666666, 666666666666616666666666666, 66666666666666166666666666666, 6666666666666661666666666666666

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), A002280 (6*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332121 .. A332191 (variants with different repeated digit 2, ..., 9).
Cf. A332160 .. A332169 (variants with different middle digit 0, ..., 9).

A332161 := n -> 6*(10^(2*n+1)-1)/9-5*10^n;

Array[6 (10^(2 # + 1)-1)/9 - 5*10^# &, 15, 0]

apply( {A332161(n)=10^(n*2+1)\9*6-5*10^n}, [0..15])

def A332161(n): return 10**(n*2+1)//9*6-5*10**n

a(n) = 6*A138148(n) + 1*10^n = A002280(2n+1) - 5*10^n.
G.f.: (1 + 505*x - 1100*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.

A332184 a(n) = 8(10^(2n+1)-1)/9 - 410^n.

Original entry on oeis.org

4, 848, 88488, 8884888, 888848888, 88888488888, 8888884888888, 888888848888888, 88888888488888888, 8888888884888888888, 888888888848888888888, 88888888888488888888888, 8888888888884888888888888, 888888888888848888888888888, 88888888888888488888888888888, 8888888888888884888888888888888

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), A002282 (8*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits only).
Cf. A332180 .. A332189 (variants with different middle digit 0, ..., 9).

A332184 := n -> 8*(10^(2*n+1)-1)/9-4*10^n;

Array[8 (10^(2 # + 1)-1)/9- 4*10^# &, 15, 0]

apply( {A332184(n)=10^(n*2+1)\9*8-4*10^n}, [0..15])

def A332184(n): return 10**(n*2+1)//9*8-4*10**n

a(n) = 8*A138148(n) + 4*10^n = A002282(2n+1)- 4*10^n = 4*A332121(n).
G.f.: (4 + 404*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.

A332142 a(n) = 4(10^(2n+1)-1)/9 - 2*10^n.

Original entry on oeis.org

2, 424, 44244, 4442444, 444424444, 44444244444, 4444442444444, 444444424444444, 44444444244444444, 4444444442444444444, 444444444424444444444, 44444444444244444444444, 4444444444442444444444444, 444444444444424444444444444, 44444444444444244444444444444, 4444444444444442444444444444444

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), A002278 (4*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332112 .. A332192 (variants with different repeated digit 1, ..., 9).
Cf. A332140 .. A332149 (variants with different middle digit 0, ..., 9).

A332142 := n -> 4*(10^(2*n+1)-1)/9-2*10^n;

Array[4 (10^(2 # + 1)-1)/9 - 2*10^# &, 15, 0]

apply( {A332142(n)=10^(n*2+1)\9*4-2*10^n}, [0..15])

def A332142(n): return 10**(n*2+1)//9*4-2*10**n

a(n) = 4*A138148(n) + 2*10^n = A002278(2n+1) - 2*10^n = 2*A332121(n).
G.f.: (2 + 202*x - 600*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

A332120 a(n) = 2*(10^(2n+1)-1)/9 - 2*10^n.

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A332171 a(n) = 7*(10^(2n+1)-1)/9 - 6*10^n.

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

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Maple

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A332131 a(n) = (10^(2n+1)-1)/3 - 2*10^n.

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Maple

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A332181 a(n) = 8*(10^(2n+1)-1)/9 - 7*10^n.

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Maple

Mathematica

PARI

Python

Formula

A332141 a(n) = 4*(10^(2*n+1)-1)/9 - 3*10^n.

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Programs

Maple

Mathematica

PARI

Python

Formula

A332151 a(n) = 5*(10^(2*n+1)-1)/9 - 4*10^n.

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Keywords

A332120 a(n) = 2(10^(2n+1)-1)/9 - 210^n.

A332171 a(n) = 7(10^(2n+1)-1)/9 - 610^n.

A332181 a(n) = 8(10^(2n+1)-1)/9 - 710^n.

A332141 a(n) = 4(10^(2n+1)-1)/9 - 3*10^n.

A332151 a(n) = 5(10^(2n+1)-1)/9 - 4*10^n.

A332161 a(n) = 6(10^(2n+1)-1)/9 - 5*10^n.

A332184 a(n) = 8(10^(2n+1)-1)/9 - 410^n.

A332142 a(n) = 4(10^(2n+1)-1)/9 - 2*10^n.