A332192 a(n) = 10^(2n+1) - 1 - 7*10^n.

2, 929, 99299, 9992999, 999929999, 99999299999, 9999992999999, 999999929999999, 99999999299999999, 9999999992999999999, 999999999929999999999, 99999999999299999999999, 9999999999992999999999999, 999999999999929999999999999, 99999999999999299999999999999, 9999999999999992999999999999999

Offset: 0

Author

M. F. Hasler, Feb 08 2020

Keywords

• nonn
• base
• easy

Comments

See A115073 = {1, 8, 9, 352, 530, 697, ...} for the indices of primes.

Links

• Patrick De Geest, Palindromic Wing Primes: (9)2(9), updated: June 25, 2017.
• Makoto Kamada, Factorization of 99...99299...99, updated Dec 11 2018.
• Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Crossrefs

Cf. (A077778-1)/2 = A115073: 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. A332190 .. A332197, A181965 (variants with different middle digit 0, ..., 8).

Cf. A332112 .. A332182 (variants with different repeated digit 1, ..., 8).

Programs

Maple

A332192 := n -> 10^(n*2+1)-1-7*10^n;

Mathematica

Array[ 10^(2 # +1) -1 -7*10^# &, 15, 0]
LinearRecurrence[{111,-1110,1000},{2,929,99299},20] (* Harvey P. Dale, Nov 07 2022 *)

PARI

apply( {A332192(n)=10^(n*2+1)-1-7*10^n}, [0..15])

Python

def A332192(n): return 10**(n*2+1)-1-7*10^n

Formula

a(n) = 9*A138148(n) + 2*10^n = A002283(2n+1) - 7*10^n.

G.f.: (2 + 707*x - 1600*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.