A332114 a(n) = (10^(2*n+1) - 1)/9 + 3*10^n.

4, 141, 11411, 1114111, 111141111, 11111411111, 1111114111111, 111111141111111, 11111111411111111, 1111111114111111111, 111111111141111111111, 11111111111411111111111, 1111111111114111111111111, 111111111111141111111111111, 11111111111111411111111111111, 1111111111111114111111111111111

Offset

0,1

Comments

See A107124 = {2, 3, 32, 45, 1544, ...} for the indices of primes.

Links

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

Patrick De Geest, Palindromic Wing Primes: (1)4(1), updated: June 25, 2017.

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

Makoto Kamada, Factorization of 11...11411...11, updated Dec 11 2018.

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

Formula

a(n) = A138148(n) + 4*10^n = A002275(2*n+1) + 3*10^n.

G.f.: (4 - 303*x + 200*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) + 27*exp(9*x) - 1)/9. - Elmo R. Oliveira, Dec 15 2025

Maple

A332114 := n -> (10^(2*n+1)-1)/9+3*10^n;

Mathematica

Array[(10^(2 # + 1)-1)/9 + 3*10^# &, 15, 0]

Prog

(PARI) apply( {A332114(n)=10^(n*2+1)\9+3*10^n}, [0..15])
(Python) def A332114(n): return 10**(n*2+1)//9+3*10**n

Crossrefs

Cf. (A077780-1)/2 = A107124: indices of primes; A331866 and A331867 (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. A332124 .. A332194 (variants with different repeated digit 2, ..., 9).

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

Sequence in context: A215606 A119038 A048432 * A262653 A299061 A299722

Adjacent sequences: A332111 A332112 A332113 * A332115 A332116 A332117

Keyword

nonn,base,easy

Author

M. F. Hasler, Feb 09 2020

Status

approved