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 14 results. Next

A332119 a(n) = (10^(2n+1)-1)/9 + 8*10^n.

Original entry on oeis.org

9, 191, 11911, 1119111, 111191111, 11111911111, 1111119111111, 111111191111111, 11111111911111111, 1111111119111111111, 111111111191111111111, 11111111111911111111111, 1111111111119111111111111, 111111111111191111111111111, 11111111111111911111111111111, 1111111111111119111111111111111
Offset: 0

Views

Author

M. F. Hasler, Feb 09 2020

Keywords

Comments

See A107649 = {1, 4, 26, 187, 226, 874, ...} for the indices of primes.

Links

Crossrefs

Cf. (A077795-1)/2 = A107649: 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. A332129 .. A332189 (variants with different repeated digit 2, ..., 8).
Cf. A332112 .. A332118 (variants with different middle digit 2, ..., 8).

Programs

  • Maple
    A332119 := n -> (10^(2*n+1)-1)/9+8*10^n;
  • Mathematica
    Array[(10^(2 # + 1)-1)/9 + 8*10^# &, 15, 0]
Table[FromDigits[Join[PadRight[{},n,1],{9},PadRight[{},n,1]]],{n,0,20}] (* or *) LinearRecurrence[ {111,-1110,1000},{9,191,11911},20] (* Harvey P. Dale, Mar 30 2024 *)
  • PARI
    apply( {A332119(n)=10^(n*2+1)\9+8*10^n}, [0..15])
  • Python
    def A332119(n): return 10**(n*2+1)//9+8*10**n

Formula

a(n) = A138148(n) + 9*10^n = A002275(2n+1) + 8*10^n.
G.f.: (9 - 808*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.

A332129 a(n) = 2*(10^(2n+1)-1)/9 + 7*10^n.

Original entry on oeis.org

9, 292, 22922, 2229222, 222292222, 22222922222, 2222229222222, 222222292222222, 22222222922222222, 2222222229222222222, 222222222292222222222, 22222222222922222222222, 2222222222229222222222222, 222222222222292222222222222, 22222222222222922222222222222, 2222222222222229222222222222222
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. A332119 .. A332189 (variants with different repeated digit 1, ..., 8).
Cf. A332120 .. A332128 (variants with different middle digit 0, ..., 8).

Programs

  • Maple
    A332129 := n -> 2*(10^(2*n+1)-1)/9+7*10^n;
  • Mathematica
    Array[2 (10^(2 # + 1)-1)/9 + 7*10^# &, 15, 0]
LinearRecurrence[{111,-1110,1000},{9,292,22922},20] (* Harvey P. Dale, Jun 25 2020 *)
  • PARI
    apply( {A332129(n)=10^(n*2+1)\9*2+7*10^n}, [0..15])
  • Python
    def A332129(n): return 10**(n*2+1)//9*2+7*10**n

Formula

a(n) = 2*A138148(n) + 9*10^n = A002276(2n+1) + 7*10^n.
G.f.: (9 - 707*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.

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

A332139 a(n) = (10^(2*n+1)-1)/3 + 6*10^n.

Original entry on oeis.org

9, 393, 33933, 3339333, 333393333, 33333933333, 3333339333333, 333333393333333, 33333333933333333, 3333333339333333333, 333333333393333333333, 33333333333933333333333, 3333333333339333333333333, 333333333333393333333333333, 33333333333333933333333333333, 3333333333333339333333333333333
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. A332129 .. A332189 (variants with different repeated digit 2, ..., 8).
Cf. A332130 .. A332138 (variants with different middle digit 0, ..., 8).

Programs

  • Maple
    A332139 := n -> (10^(2*n+1)-1)/3+6*10^n;
  • Mathematica
    Array[ (10^(2 # + 1)-1)/3 + 6*10^# &, 15, 0]
LinearRecurrence[{111,-1110,1000},{9,393,33933},20] (* Harvey P. Dale, Sep 17 2020 *)
  • PARI
    apply( {A332139(n)=10^(n*2+1)\3+6*10^n}, [0..15])
  • Python
    def A332139(n): return 10**(n*2+1)//3+6*10**n

Formula

a(n) = 3*A138148(n) + 9*10^n = A002277(2n+1) + 6*10^n = 3*A332113(n).
G.f.: (9 - 606*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.

A332149 a(n) = 4*(10^(2*n+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

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

Programs

  • Maple
    A332149 := n -> 4*(10^(2*n+1)-1)/9+5*10^n;
  • Mathematica
    Array[4 (10^(2 # + 1)-1)/9 + 5*10^# &, 15, 0]
  • PARI
    apply( {A332149(n)=10^(n*2+1)\9*4+5*10^n}, [0..15])
  • Python
    def A332149(n): return 10**(n*2+1)//9*4+5*10**n

Formula

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.

A332159 a(n) = 5*(10^(2*n+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

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. A332119 .. A332189 (variants with different repeated digit 1, ..., 8).
Cf. A332150 .. A332159 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332159 := n -> 5*(10^(2*n+1)-1)/9+4*10^n;
  • Mathematica
    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 *)
  • PARI
    apply( {A332159(n)=10^(n*2+1)\9*5+4*10^n}, [0..15])
  • Python
    def A332159(n): return 10**(n*2+1)//9*5+4*10**n

Formula

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.

A332169 a(n) = 6*(10^(2*n+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

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

Programs

  • Maple
    A332169 := n -> 6*(10^(2*n+1)-1)/9+3*10^n;
  • Mathematica
    Array[6 (10^(2 # + 1)-1)/9 + 3*10^# &, 15, 0]
  • PARI
    apply( {A332169(n)=10^(n*2+1)\9*6+3*10^n}, [0..15])
  • Python
    def A332169(n): return 10**(n*2+1)//9*6+3*10**n

Formula

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.

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

Original entry on oeis.org

1, 818, 88188, 8881888, 888818888, 88888188888, 8888881888888, 888888818888888, 88888888188888888, 8888888881888888888, 888888888818888888888, 88888888888188888888888, 8888888888881888888888888, 888888888888818888888888888, 88888888888888188888888888888, 8888888888888881888888888888888
Offset: 0

Views

Author

M. F. Hasler, Feb 08 2020

Keywords

Links

Crossrefs

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).

Programs

  • Maple
    A332181 := n -> 8*(10^(2*n+1)-1)/9-7*10^n;
  • Mathematica
    Array[8 (10^(2 # + 1)-1)/9 - 7*10^# &, 15, 0]
  • PARI
    apply( {A332181(n)=10^(n*2+1)\9*8-7*10^n}, [0..15])
  • Python
    def A332181(n): return 10**(n*2+1)//9*8-7*10**n

Formula

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.

A332187 a(n) = 8*(10^(2n+1)-1)/9 - 10^n.

Original entry on oeis.org

7, 878, 88788, 8887888, 888878888, 88888788888, 8888887888888, 888888878888888, 88888888788888888, 8888888887888888888, 888888888878888888888, 88888888888788888888888, 8888888888887888888888888, 888888888888878888888888888, 88888888888888788888888888888, 8888888888888887888888888888888
Offset: 0

Views

Author

M. F. Hasler, Feb 08 2020

Keywords

Links

Crossrefs

Cf. (A077776-1)/2 = A183190: 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), A002113 (palindromes).
Cf. A332117 .. A332197 (variants with different "wing" digit 1, ..., 9).
Cf. A332180 .. A332189 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332187 := n -> 8*(10^(2*n+1)-1)/9-10^n;
  • Mathematica
    Array[8 (10^(2 # + 1)-1)/9 - 10^# &, 15, 0]
LinearRecurrence[{111,-1110,1000},{7,878,88788},20] (* Harvey P. Dale, Jul 21 2024 *)
  • PARI
    apply( {A332187(n)=10^(n*2+1)\9*8-10^n}, [0..15])
  • Python
    def A332187(n): return 10**(n*2+1)//9*8-10**n

Formula

a(n) = 8*A138148(n) + 7*10^n = A002282(2n+1) - 10^n.
G.f.: (7 + 101*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.

A332182 a(n) = 8*(10^(2n+1)-1)/9 - 6*10^n.

Original entry on oeis.org

2, 828, 88288, 8882888, 888828888, 88888288888, 8888882888888, 888888828888888, 88888888288888888, 8888888882888888888, 888888888828888888888, 88888888888288888888888, 8888888888882888888888888, 888888888888828888888888888, 88888888888888288888888888888, 8888888888888882888888888888888
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).
Cf. A332112 .. A332192 (variants with different repeated digit 1, ..., 9).
Cf. A332180 .. A332189 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332182 := n -> 8*(10^(2*n+1)-1)/9-6*10^n;
  • Mathematica
    Array[8 (10^(2 # + 1)-1)/9 - 6*10^# &, 15, 0]
  • PARI
    apply( {A332182(n)=10^(n*2+1)\9*8-6*10^n}, [0..15])
  • Python
    def A332182(n): return 10**(n*2+1)//9*8-6*10**n

Formula

a(n) = 8*A138148(n) + 2*10^n = A002282(2n+1)- 6*10^n.
G.f.: (2 + 606*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.
Showing 1-10 of 14 results. Next

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