A332191 a(n) = 10^(2n+1) - 1 - 8*10^n.

1, 919, 99199, 9991999, 999919999, 99999199999, 9999991999999, 999999919999999, 99999999199999999, 9999999991999999999, 999999999919999999999, 99999999999199999999999, 9999999999991999999999999, 999999999999919999999999999, 99999999999999199999999999999, 9999999999999991999999999999999

Offset: 0

Author

M. F. Hasler, Feb 08 2020

Keywords

• nonn
• base
• easy

Comments

See A183184 = {1, 5, 13, 43, 169, 181, ...} for the indices of primes.

Links

• Patrick De Geest, Palindromic Wing Primes: (9)1(9), updated: June 25, 2017.
• Makoto Kamada, Factorization of 9999199...99, updated Dec 11 2018.
• Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Crossrefs

Cf. (A077776-1)/2 = A183184: indices of primes.

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. A332121 .. A332181 (variants with different repeated digit 2, ..., 8).

Cf. A332190 .. A332197, A181965 (variants with different middle digit 0, ..., 8).

Programs

Maple

A332191 := n -> 10^(n*2+1)-1-8*10^n;

Mathematica

Array[ 10^(2 # + 1)-1-8*10^# &, 15, 0]

PARI

apply( {A332191(n)=10^(n*2+1)-1-8*10^n}, [0..15])

Python

def A332191(n): return 10**(n*2+1)-1-8*10^n

Formula

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

G.f.: (1 + 808*x - 1700*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.