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 1-2 of 2 results.

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).

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.

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

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.
Showing 1-2 of 2 results.