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 10 results.

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

Original entry on oeis.org

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

Views

Author

M. F. Hasler, Feb 08 2020

Keywords

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), A002113 (palindromes).
Cf. A332120 .. A332180 (variants with different repeated digit 2, ..., 8).
Cf. A332191 .. A332197, A181965 (variants with different middle digit 1, ..., 8).

Programs

  • Maple
    A332190 := n -> 10^(2*n+1)-1-9*10^n;
  • Mathematica
    Array[10^(2 # + 1)-1-9*10^# &, 15, 0]
LinearRecurrence[{111,-1110,1000},{0,909,99099},20] (* Harvey P. Dale, May 28 2021 *)
  • PARI
    apply( {A332190(n)=10^(n*2+1)-1-9*10^n}, [0..15])
  • Python
    def A332190(n): return 10**(n*2+1)-1-9*10^n

Formula

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.

A332189 a(n) = 8*(10^(2n+1)-1)/9 + 10^n.

Original entry on oeis.org

9, 898, 88988, 8889888, 888898888, 88888988888, 8888889888888, 888888898888888, 88888888988888888, 8888888889888888888, 888888888898888888888, 88888888888988888888888, 8888888888889888888888888, 888888888888898888888888888, 88888888888888988888888888888, 8888888888888889888888888888888
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), A002113 (palindromes).
Cf. A332119 .. A332189 (variants with different "wing" digit 1, ..., 8).
Cf. A332180 .. A332187 (variants with different middle digit 0, ..., 7).

Programs

  • Maple
    A332189 := n -> 8*(10^(2*n+1)-1)/9+10^n;
  • Mathematica
    Array[8 (10^(2 # + 1)-1)/9 + 10^# &, 15, 0]
  • PARI
    apply( {A332189(n)=10^(n*2+1)\9*8+10^n}, [0..15])
  • Python
    def A332189(n): return 10**(n*2+1)//9*8+10**n

Formula

a(n) = 8*A138148(n) + 9*10^n = A002282(2n+1) + 10^n.
G.f.: (9 - 101*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.

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.

A332183 a(n) = 8*(10^(2n+1)-1)/9 - 5*10^n.

Original entry on oeis.org

3, 838, 88388, 8883888, 888838888, 88888388888, 8888883888888, 888888838888888, 88888888388888888, 8888888883888888888, 888888888838888888888, 88888888888388888888888, 8888888888883888888888888, 888888888888838888888888888, 88888888888888388888888888888, 8888888888888883888888888888888
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. A332113 .. A332193 (variants with different repeated digit 1, ..., 9).
Cf. A332180 .. A332189 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332183 := n -> 8*(10^(2*n+1)-1)/9-5*10^n;
  • Mathematica
    Array[8 (10^(2 # + 1)-1)/9 - 5*10^# &, 15, 0]
  • PARI
    apply( {A332183(n)=10^(n*2+1)\9*8-5*10^n}, [0..15])
  • Python
    def A332183(n): return 10**(n*2+1)//9*8-5*10**n

Formula

a(n) = 8*A138148(n) + 3*10^n = A002282(2n+1) - 5*10^n.
G.f.: (3 + 505*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.

A332184 a(n) = 8*(10^(2n+1)-1)/9 - 4*10^n.

Original entry on oeis.org

4, 848, 88488, 8884888, 888848888, 88888488888, 8888884888888, 888888848888888, 88888888488888888, 8888888884888888888, 888888888848888888888, 88888888888488888888888, 8888888888884888888888888, 888888888888848888888888888, 88888888888888488888888888888, 8888888888888884888888888888888
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. A332180 .. A332189 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332184 := n -> 8*(10^(2*n+1)-1)/9-4*10^n;
  • Mathematica
    Array[8 (10^(2 # + 1)-1)/9- 4*10^# &, 15, 0]
  • PARI
    apply( {A332184(n)=10^(n*2+1)\9*8-4*10^n}, [0..15])
  • Python
    def A332184(n): return 10**(n*2+1)//9*8-4*10**n

Formula

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.

A332185 a(n) = 8*(10^(2n+1)-1)/9 - 3*10^n.

Original entry on oeis.org

5, 858, 88588, 8885888, 888858888, 88888588888, 8888885888888, 888888858888888, 88888888588888888, 8888888885888888888, 888888888858888888888, 88888888888588888888888, 8888888888885888888888888, 888888888888858888888888888, 88888888888888588888888888888, 8888888888888885888888888888888
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. A332180 .. A332189 (variants with different middle digit 0, ..., 9).
Cf. A332115 .. A332195 (variants with different "wing" digit 1, ..., 9).

Programs

  • Maple
    A332185 := n -> 8*(10^(2*n+1)-1)/9-3*10^n;
  • Mathematica
    Array[8 (10^(2 # + 1)-1)/9 - 3*10^# &, 15, 0]
  • PARI
    apply( {A332185(n)=10^(n*2+1)\9*8-3*10^n}, [0..15])
  • Python
    def A332185(n): return 10**(n*2+1)//9*8-3*10**n

Formula

a(n) = 8*A138148(n) + 5*10^n = A002282(2n+1) - 3*10^n.
G.f.: (5 + 303*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.

A332186 a(n) = 8*(10^(2n+1)-1)/9 - 2*10^n.

Original entry on oeis.org

6, 868, 88688, 8886888, 888868888, 88888688888, 8888886888888, 888888868888888, 88888888688888888, 8888888886888888888, 888888888868888888888, 88888888888688888888888, 8888888888886888888888888, 888888888888868888888888888, 88888888888888688888888888888, 8888888888888886888888888888888
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), A002113 (palindromes).
Cf. A332180 .. A332189 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332186 := n -> 8*(10^(2*n+1)-1)/9-2*10^n;
  • Mathematica
    Array[8 (10^(2 # + 1)-1)/9 - 2*10^# &, 15, 0]
LinearRecurrence[{111,-1110,1000},{6,868,88688},20] (*or *) Table[FromDigits[Join[PadRight[ {},n,8],PadRight[ {6},n+1,8]]],{n,0,20}] (* Harvey P. Dale, May 30 2023 *)
  • PARI
    apply( {A332186(n)=10^(n*2+1)\9*8-2*10^n}, [0..15])
  • Python
    def A332186(n): return 10**(n*2+1)//9*8-2*10**n

Formula

a(n) = 8*A138148(n) + 6*10^n = A002282(2n+1) - 2*10^n = 2*A332143(n).
G.f.: (6 + 202*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.
E.g.f.: 2*exp(x)*(40*exp(99*x) - 9*exp(9*x) - 4)/9. - Stefano Spezia, Jul 13 2024

A337842 The smallest palindrome with exactly n circular loops (or holes) in its decimal representation.

Original entry on oeis.org

1, 0, 8, 606, 88, 808, 888, 68086, 8888, 88088, 88888, 6880886, 888888, 8880888, 8888888, 688808886, 88888888, 888808888, 888888888, 68888088886, 8888888888, 88888088888, 88888888888, 6888880888886, 888888888888, 8888880888888, 8888888888888, 688888808888886
Offset: 0

Views

Author

Bernard Schott, Sep 25 2020

Keywords

Comments

The decimal digits 1, 2, 3, 5, 7 have no hole, and 4 is represented without a hole; otherwise, 0, 6, 9 have one hole each and 8 has two holes.
Least palindrome q such that A064532(q) = n.
Except for a(0) = 1, each term has only digits 0, 6 or 8 in its decimal expansion.

Examples

			a(3) = 606 because 6 and 0 have each one circular loop for a total of 3.

Links

Crossrefs

Cf. A000533, A002113, A002282, A064532, A332180.
Cf. A331898 (similar for primes).

Programs

Formula

a(2m) = A002282(m) for m >= 1.
a(4m+1) = A332180(m) for m >= 1.
a(4m+3) = 6 * A000533(2m+2) + 10 * A332180(m) for m >= 0.

Extensions

More terms from Amiram Eldar, Sep 25 2020
Showing 1-10 of 10 results.

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