A332116 a(n) = (10^(2n+1)-1)/9 + 5*10^n.

6, 161, 11611, 1116111, 111161111, 11111611111, 1111116111111, 111111161111111, 11111111611111111, 1111111116111111111, 111111111161111111111, 11111111111611111111111, 1111111111116111111111111, 111111111111161111111111111, 11111111111111611111111111111, 1111111111111116111111111111111

Offset: 0

Author

M. F. Hasler, Feb 09 2020

Keywords

• nonn
• base
• easy

Comments

See A107126 = {10, 14, 40, 59, 160, 412, ...} for the indices of primes.

Links

• Patrick De Geest, Palindromic Wing Primes: (1)6(1), updated: June 25, 2017.
• Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).
• Makoto Kamada, Factorization of 11...11611...11, updated Dec 11 2018.
• Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Crossrefs

Cf. (A077706-1)/2 = A107126: 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. A332126 .. A332196 (variants with different repeated digit 2, ..., 9).

Cf. A332112 .. A332119 (variants with different middle digit 2, ..., 9).

Programs

Maple

A332116 := n -> (10^(2*n+1)-1)/9+5*10^n;

Mathematica

Array[(10^(2 # + 1)-1)/9 + 5*10^# &, 15, 0]

PARI

apply( {A332116(n)=10^(n*2+1)\9+5*10^n}, [0..15])

Python

def A332116(n): return 10**(n*2+1)//9+5*10**n

Formula

a(n) = A138148(n) + 6*10^n = A002275(2n+1) + 5*10^n.

G.f.: (6 - 505*x + 400*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) + 45*exp(9*x) - 1)/9. - Stefano Spezia, Jul 13 2024