A138148 -id:A138148 - cp's OEIS frontend

A332140 a(n) = 4(10^(2n+1)-1)/9 - 410^n.

0, 404, 44044, 4440444, 444404444, 44444044444, 4444440444444, 444444404444444, 44444444044444444, 4444444440444444444, 444444444404444444444, 44444444444044444444444, 4444444444440444444444444, 444444444444404444444444444, 44444444444444044444444444444, 4444444444444440444444444444444

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. A332120 .. A332190 (variants with different repeated digit 2, ..., 9).
Cf. A332141 .. A332149 (variants with different middle digit 1, ..., 9).

A332140 := n -> 4*((10^(2*n+1)-1)/9-10^n);

Array[4 ((10^(2 # + 1)-1)/9 - 10^#) &, 15, 0]
LinearRecurrence[{111,-1110,1000},{0,404,44044},20] (* Harvey P. Dale, Jul 06 2021 *)

apply( {A332140(n)=(10^(n*2+1)\9-10^n)*4}, [0..15])

def A332140(n): return (10**(n*2+1)//9-10**n)*4

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

A332149 a(n) = 4(10^(2n+1)-1)/9 + 5*10^n.

Original entry on oeis.org

9, 494, 44944, 4449444, 444494444, 44444944444, 4444449444444, 444444494444444, 44444444944444444, 4444444449444444444, 444444444494444444444, 44444444444944444444444, 4444444444449444444444444, 444444444444494444444444444, 44444444444444944444444444444, 4444444444444449444444444444444

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. A332119 .. A332189 (variants with different repeated digit 1, ..., 8).
Cf. A332140 .. A332148 (variants with different middle digit 0, ..., 8).

A332149 := n -> 4*(10^(2*n+1)-1)/9+5*10^n;

Array[4 (10^(2 # + 1)-1)/9 + 5*10^# &, 15, 0]

apply( {A332149(n)=10^(n*2+1)\9*4+5*10^n}, [0..15])

def A332149(n): return 10**(n*2+1)//9*4+5*10**n

a(n) = 4*A138148(n) + 9*10^n = A002278(2n+1) + 5*10^n.
G.f.: (9 - 505*x + 100*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.

A332150 a(n) = 5(10^(2n+1)-1)/9 - 510^n.

Original entry on oeis.org

0, 505, 55055, 5550555, 555505555, 55555055555, 5555550555555, 555555505555555, 55555555055555555, 5555555550555555555, 555555555505555555555, 55555555555055555555555, 5555555555550555555555555, 555555555555505555555555555, 55555555555555055555555555555, 5555555555555550555555555555555

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), A002279 (5*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332120 .. A332190 (variants with different repeated digit 2, ..., 9).
Cf. A332151 .. A332159 (variants with different middle digit 1, ..., 9).

A332150 := n -> 5*((10^(2*n+1)-1)/9-10^n);

Array[5 ((10^(2 # + 1)-1)/9 - 10^#) &, 15, 0]
Table[With[{c=PadRight[{},n,5]},FromDigits[Join[c,{0},c]]],{n,0,15}] (* or *) LinearRecurrence[{111,-1110,1000},{0,505,55055},20] (* Harvey P. Dale, Jun 30 2025 *)

apply( {A332150(n)=(10^(n*2+1)\9-10^n)*5}, [0..15])

def A332150(n): return (10**(n*2+1)//9-10**n)*5

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

A332159 a(n) = 5(10^(2n+1)-1)/9 + 4*10^n.

Original entry on oeis.org

9, 595, 55955, 5559555, 555595555, 55555955555, 5555559555555, 555555595555555, 55555555955555555, 5555555559555555555, 555555555595555555555, 55555555555955555555555, 5555555555559555555555555, 555555555555595555555555555, 55555555555555955555555555555, 5555555555555559555555555555555

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), A002279 (5*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332119 .. A332189 (variants with different repeated digit 1, ..., 8).
Cf. A332150 .. A332159 (variants with different middle digit 0, ..., 9).

A332159 := n -> 5*(10^(2*n+1)-1)/9+4*10^n;

Array[5 (10^(2 # + 1)-1)/9 + 4*10^# &, 15, 0]
Table[FromDigits[Join[PadRight[{},n,5],PadRight[{9},n+1,5]]],{n,0,20}] (* or *) LinearRecurrence[ {111,-1110,1000},{9,595,55955},20] (* Harvey P. Dale, May 31 2023 *)

apply( {A332159(n)=10^(n*2+1)\9*5+4*10^n}, [0..15])

def A332159(n): return 10**(n*2+1)//9*5+4*10**n

a(n) = 5*A138148(n) + 9*10^n = A002279(2n+1) + 4*10^n.
G.f.: (9 - 404*x - 100*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.

A332160 a(n) = 6(10^(2n+1)-1)/9 - 610^n.

Original entry on oeis.org

0, 606, 66066, 6660666, 666606666, 66666066666, 6666660666666, 666666606666666, 66666666066666666, 6666666660666666666, 666666666606666666666, 66666666666066666666666, 6666666666660666666666666, 666666666666606666666666666, 66666666666666066666666666666, 6666666666666660666666666666666

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. A332161 .. A332169 (variants with different middle digit 1, ..., 9).
Cf. A332120 .. A332190 (variants with different repeated digit 2, ..., 9).

A332160 := n -> 6*((10^(2*n+1)-1)/9-10^n);

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

apply( {A332160(n)=(10^(n*2+1)\9-10^n)*6}, [0..15])

def A332160(n): return (10**(n*2+1)//9-10**n)*6

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

A332169 a(n) = 6(10^(2n+1)-1)/9 + 3*10^n.

Original entry on oeis.org

9, 696, 66966, 6669666, 666696666, 66666966666, 6666669666666, 666666696666666, 66666666966666666, 6666666669666666666, 666666666696666666666, 66666666666966666666666, 6666666666669666666666666, 666666666666696666666666666, 66666666666666966666666666666, 6666666666666669666666666666666

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. A332119 .. A332189 (variants with different repeated digit 1, ..., 8).
Cf. A332160 .. A332169 (variants with different middle digit 0, ..., 9).

A332169 := n -> 6*(10^(2*n+1)-1)/9+3*10^n;

Array[6 (10^(2 # + 1)-1)/9 + 3*10^# &, 15, 0]

apply( {A332169(n)=10^(n*2+1)\9*6+3*10^n}, [0..15])

def A332169(n): return 10**(n*2+1)//9*6+3*10**n

a(n) = 6*A138148(n) + 9*10^n = A002280(2n+1) + 3*10^n = 3*A332123(n).
G.f.: (9 - 303*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.

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

A332179 a(n) = 7(10^(2n+1)-1)/9 + 210^n.

Original entry on oeis.org

9, 797, 77977, 7779777, 777797777, 77777977777, 7777779777777, 777777797777777, 77777777977777777, 7777777779777777777, 777777777797777777777, 77777777777977777777777, 7777777777779777777777777, 777777777777797777777777777, 77777777777777977777777777777, 7777777777777779777777777777777

Offset: 0

M. F. Hasler, Feb 08 2020

See A183183 = {1, 2, 8, 19, 20, 212, 280, ...} for the indices of primes.

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

Cf. A138148 (cyclops numbers with binary digits only).
Cf. (A077796-1)/2 = A183183: indices of primes.
Cf. A002275 (repunits R_n = [10^n/9]), A002281 (7*R_n), A011557 (10^n).
Cf. A332171 .. A332178 (variants with different middle digit 1, ..., 8).

A332179 := n -> 7*(10^(n*2+1)-1)/9 + 2*10^n;

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

apply( {A332179(n)=10^(n*2+1)\9*7+2*10^n}, [0..15])

def A332179(n): return 10**(n*2+1)//9*7+2*10^n

a(n) = 7*A138148(n) + 9*10^n.
G.f.: (9 - 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.

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.

A332178 a(n) = 7*(10^(2n+1)-1)/9 + 10^n.

Original entry on oeis.org

8, 787, 77877, 7778777, 777787777, 77777877777, 7777778777777, 777777787777777, 77777777877777777, 7777777778777777777, 777777777787777777777, 77777777777877777777777, 7777777777778777777777777, 777777777777787777777777777, 77777777777777877777777777777, 7777777777777778777777777777777

Offset: 0

M. F. Hasler, Feb 08 2020

See A183182 = {1, 3, 39, 54, 168, 240, ...} for the indices of primes.

Makoto Kamada, Factorization of 77...77877...77, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A138148 (cyclops numbers with binary digits only).
Cf. (A077793-1)/2 = A183182: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A002281 (7*R_n), A011557 (10^n).
Cf. A332171 .. A332179 (variants with different middle digit 1, ..., 9).

A332178 := n -> 7*(10^(n*2+1)-1)/9 + 10^n;

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

apply( {A332178(n)=10^(n*2+1)\9*7+10^n}, [0..15])

def A332178(n): return 10**(n*2+1)//9*7+10^n

a(n) = 7*A138148(n) + 8*10^n.
G.f.: (8 - 101*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

A332140 a(n) = 4*(10^(2n+1)-1)/9 - 4*10^n.

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A332149 a(n) = 4*(10^(2*n+1)-1)/9 + 5*10^n.

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A332150 a(n) = 5*(10^(2n+1)-1)/9 - 5*10^n.

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A332159 a(n) = 5*(10^(2*n+1)-1)/9 + 4*10^n.

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A332160 a(n) = 6*(10^(2n+1)-1)/9 - 6*10^n.

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A332169 a(n) = 6*(10^(2*n+1)-1)/9 + 3*10^n.

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Maple

Mathematica

PARI

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

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A332140 a(n) = 4(10^(2n+1)-1)/9 - 410^n.

A332149 a(n) = 4(10^(2n+1)-1)/9 + 5*10^n.

A332150 a(n) = 5(10^(2n+1)-1)/9 - 510^n.

A332159 a(n) = 5(10^(2n+1)-1)/9 + 4*10^n.

A332160 a(n) = 6(10^(2n+1)-1)/9 - 610^n.

A332169 a(n) = 6(10^(2n+1)-1)/9 + 3*10^n.

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

A332179 a(n) = 7(10^(2n+1)-1)/9 + 210^n.