A332113 a(n) = (10^(2*n+1) - 1)/9 + 2*10^n.

3, 131, 11311, 1113111, 111131111, 11111311111, 1111113111111, 111111131111111, 11111111311111111, 1111111113111111111, 111111111131111111111, 11111111111311111111111, 1111111111113111111111111, 111111111111131111111111111, 11111111111111311111111111111, 1111111111111113111111111111111

Offset

0,1

Comments

See A107123 = {0, 1, 2, 19, 97, 9818, ...} for the indices of primes.

Links

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

Patrick De Geest, Palindromic Wing Primes: (1)3(1), updated: June 25, 2017.

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

Makoto Kamada, Factorization of 11...11311...11, updated Dec 11 2018.

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

Formula

a(n) = A138148(n) + 3*10^n = A002275(2*n+1) + 2*10^n.

G.f.: (3 - 202*x + 100*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) + 18*exp(9*x) - 1)/9. - Elmo R. Oliveira, Dec 15 2025

Maple

A332113 := n -> (10^(2*n+1)-1)/9+2*10^n;

Mathematica

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

Prog

(PARI) apply( {A332113(n)=10^(n*2+1)\9+2*10^n}, [0..15])
(Python) def A332113(n): return 10**(n*2+1)//9+2*10**n

Crossrefs

Cf. (A077779-1)/2 = A107123: indices of primes; A331864 and A331865 (non-palindromic variants).

Cf. A002275 (repunits R_n = (10^n-1)/9), A011557 (10^n).

Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).

Cf. A332123 .. A332193 (variants with different repeated digit 2, ..., 9).

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

Sequence in context: A347985 A082439 A082622 * A075597 A262639 A082563

Adjacent sequences: A332110 A332111 A332112 * A332114 A332115 A332116

Keyword

nonn,base,easy

Author

M. F. Hasler, Feb 09 2020

Status

approved