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

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

Offset: 0

Author

M. F. Hasler, Feb 09 2020

Keywords

• nonn
• base
• easy

Comments

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

Links

• Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).
• Patrick De Geest, Palindromic Wing Primes: (1)9(1), updated: June 25, 2017.
• Makoto Kamada, Factorization of 11...11911...11, updated Dec 11 2018.
• Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

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.