cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 15 results. Next

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

Original entry on oeis.org

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

Views

Author

M. F. Hasler, Feb 09 2020

Keywords

Links

Crossrefs

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).

Programs

  • Maple
    A332120 := n -> 2*((10^(2*n+1)-1)/9-10^n);
  • Mathematica
    Array[2 ((10^(2 # + 1)-1)/9 - 10^#) &, 15, 0]
  • PARI
    apply( {A332120(n)=(10^(n*2+1)\9-10^n)*2}, [0..15])
  • Python
    def A332120(n): return (10**(n*2+1)//9-10**n)*2

Formula

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

A332197 a(n) = 10^(2n+1) - 1 - 2*10^n.

Original entry on oeis.org

7, 979, 99799, 9997999, 999979999, 99999799999, 9999997999999, 999999979999999, 99999999799999999, 9999999997999999999, 999999999979999999999, 99999999999799999999999, 9999999999997999999999999, 999999999999979999999999999, 99999999999999799999999999999, 9999999999999997999999999999999
Offset: 0

Views

Author

M. F. Hasler, Feb 08 2020

Keywords

Comments

According to Kamada, n = 118 and n = 145126 are the only known indices of primes (the so-called palindromic near-repdigit or wing primes).

Links

Crossrefs

Cf. A002275 (repunits R_n = (10^n-1)/9), A002283 (9*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits only).
Cf. A332190 .. A332196, A181965 (variants with different middle digit 0, ..., 8).
Cf. A332117 .. A332187 (variants with different repeated digit 1, ..., 9).

Programs

  • Maple
    A332197 := n -> 10^(n*2+1)-1-2*10^n;
  • Mathematica
    Array[ 10^(2 # + 1) -1 -2*10^# &, 15, 0]
Table[FromDigits[Join[PadRight[{},n,9],{7},PadRight[{},n,9]]],{n,0,20}] (* or *) LinearRecurrence[{111,-1110,1000},{7,979,99799},20] (* Harvey P. Dale, Mar 03 2023 *)
  • PARI
    apply( {A332197(n)=10^(n*2+1)-1-2*10^n}, [0..15])
  • Python
    def A332197(n): return 10**(n*2+1)-1-2*10^n

Formula

a(n) = 9*A138148(n) + 7*10^n.
G.f.: (7 + 202*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.

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

Views

Author

Ivan Panchenko, Apr 04 2012

Keywords

Comments

n 9's followed by an 8 followed by n 9's.
See A183187 = {26, 378, 1246, 1798, 2917, ...} for the indices of primes.

Links

Crossrefs

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).

Programs

  • Maple
    A181965 := n -> 10^(2*n+1)-1-10^n; # M. F. Hasler, Feb 08 2020
  • Mathematica
    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 *)
  • PARI
    apply( {A181965(n)=10^(n*2+1)-1-10^n}, [0..15]) \\ M. F. Hasler, Feb 08 2020
  • Python
    def A181965(n): return 10**(n*2+1)-1-10^n # M. F. Hasler, Feb 08 2020

Formula

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)

Extensions

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

A332180 a(n) = 8*(10^(2n+1)-1)/9 - 8*10^n.

Original entry on oeis.org

0, 808, 88088, 8880888, 888808888, 88888088888, 8888880888888, 888888808888888, 88888888088888888, 8888888880888888888, 888888888808888888888, 88888888888088888888888, 8888888888880888888888888, 888888888888808888888888888, 88888888888888088888888888888, 8888888888888880888888888888888
Offset: 0

Views

Author

M. F. Hasler, Feb 08 2020

Keywords

Links

Crossrefs

Cf. A002275 (repunits R_n = (10^n-1)/9), A002282 (8*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits only), A002113 (palindromes).
Cf. A332120 .. A332190 (variants with different repeated digit 2, ..., 9).
Cf. A332181 .. A332189 (variants with different middle digit 1, ..., 9).
Subsequence of A006072 (numbers with mirror symmetry about middle), A153806 (strobogrammatic cyclops numbers), and A204095 (numbers whose decimal digits are in {0,8}).

Programs

  • Maple
    A332180 := n -> 8*((10^(2*n+1)-1)/9-10^n);
  • Mathematica
    Array[8 ((10^(2 # + 1)-1)/9 - 10^#) &, 15, 0]
  • PARI
    apply( {A332180(n)=(10^(n*2+1)\9-10^n)*8}, [0..15])
  • Python
    def A332180(n): return (10**(n*2+1)//9-10**n)*8

Formula

a(n) = 8*A138148(n) = A002282(2n+1) - 8*10^n.
G.f.: 8*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.: 8*exp(x)*(10*exp(99*x) - 9*exp(9*x) - 1)/9. - Stefano Spezia, Jul 13 2024

A332130 a(n) = (10^(2n+1)-1)/3 - 3*10^n.

Original entry on oeis.org

0, 303, 33033, 3330333, 333303333, 33333033333, 3333330333333, 333333303333333, 33333333033333333, 3333333330333333333, 333333333303333333333, 33333333333033333333333, 3333333333330333333333333, 333333333333303333333333333, 33333333333333033333333333333, 3333333333333330333333333333333
Offset: 0

Views

Author

M. F. Hasler, Feb 09 2020

Keywords

Links

Crossrefs

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. A332120 .. A332190 (variants with different repeated digit 2, ..., 9).
Cf. A332131 .. A332139 (variants with different middle digit 1, ..., 9).

Programs

  • Maple
    A332130 := n -> (10^(2*n+1)-1)/3-3*10^n;
  • Mathematica
    Array[ ((10^(2 # + 1)-1)/3 - 3*10^#) &, 15, 0]
  • PARI
    apply( {A332130(n)=10^(n*2+1)\3-3*10^n}, [0..15])
  • Python
    def A332130(n): return 10**(n*2+1)//3-3*10**n

Formula

a(n) = 3*A138148(n) = A002277(2n+1) - 3*10^n.
G.f.: 3*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.: exp(x)*(10*exp(99*x) - 9*exp(9*x) - 1)/3. - Stefano Spezia, Jul 13 2024

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

Original entry on oeis.org

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

Views

Author

M. F. Hasler, Feb 09 2020

Keywords

Links

Crossrefs

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).

Programs

  • Maple
    A332140 := n -> 4*((10^(2*n+1)-1)/9-10^n);
  • Mathematica
    Array[4 ((10^(2 # + 1)-1)/9 - 10^#) &, 15, 0]
LinearRecurrence[{111,-1110,1000},{0,404,44044},20] (* Harvey P. Dale, Jul 06 2021 *)
  • PARI
    apply( {A332140(n)=(10^(n*2+1)\9-10^n)*4}, [0..15])
  • Python
    def A332140(n): return (10**(n*2+1)//9-10**n)*4

Formula

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.

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

Original entry on oeis.org

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

Views

Author

M. F. Hasler, Feb 09 2020

Keywords

Links

Crossrefs

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).

Programs

  • Maple
    A332150 := n -> 5*((10^(2*n+1)-1)/9-10^n);
  • Mathematica
    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 *)
  • PARI
    apply( {A332150(n)=(10^(n*2+1)\9-10^n)*5}, [0..15])
  • Python
    def A332150(n): return (10**(n*2+1)//9-10**n)*5

Formula

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.

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

Original entry on oeis.org

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

Views

Author

M. F. Hasler, Feb 09 2020

Keywords

Links

Crossrefs

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).

Programs

  • Maple
    A332160 := n -> 6*((10^(2*n+1)-1)/9-10^n);
  • Mathematica
    Array[6 ((10^(2 # + 1)-1)/9 - 10^#) &, 15, 0]
  • PARI
    apply( {A332160(n)=(10^(n*2+1)\9-10^n)*6}, [0..15])
  • Python
    def A332160(n): return (10**(n*2+1)//9-10**n)*6

Formula

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.

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

Views

Author

M. F. Hasler, Feb 08 2020

Keywords

Comments

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

Links

Crossrefs

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).

Programs

  • Maple
    A332191 := n -> 10^(n*2+1)-1-8*10^n;
  • Mathematica
    Array[ 10^(2 # + 1)-1-8*10^# &, 15, 0]
  • PARI
    apply( {A332191(n)=10^(n*2+1)-1-8*10^n}, [0..15])
  • Python
    def A332191(n): return 10**(n*2+1)-1-8*10^n

Formula

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.

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

Views

Author

M. F. Hasler, Feb 08 2020

Keywords

Comments

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

Links

Crossrefs

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).

Programs

  • Maple
    A332192 := n -> 10^(n*2+1)-1-7*10^n;
  • Mathematica
    Array[ 10^(2 # +1) -1 -7*10^# &, 15, 0]
LinearRecurrence[{111,-1110,1000},{2,929,99299},20] (* Harvey P. Dale, Nov 07 2022 *)
  • PARI
    apply( {A332192(n)=10^(n*2+1)-1-7*10^n}, [0..15])
  • Python
    def A332192(n): return 10**(n*2+1)-1-7*10^n

Formula

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.
Showing 1-10 of 15 results. Next

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