A332119 - OEIS

A332119

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

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

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OFFSET

0,1

COMMENTS

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

LINKS

Table of n, a(n) for n=0..15.

Patrick De Geest, Palindromic Wing Primes: (1)9(1), updated: June 25, 2017.

Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).

Makoto Kamada, Factorization of 11...11911...11, updated Dec 11 2018.

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

FORMULA

a(n) = A138148(n) + 9*10^n = A002275(2*n+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.

E.g.f.: exp(x)*(10*exp(99*x) + 72*exp(9*x) - 1)/9. - Elmo R. Oliveira, Dec 16 2025

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 *)

PROG

(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

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).

Sequence in context: A113269 A376041 A307387 * A261826 A184688 A189178

Adjacent sequences: A332116 A332117 A332118 * A332120 A332121 A332122

KEYWORD

nonn,base,easy

AUTHOR

M. F. Hasler, Feb 09 2020

STATUS

approved