A332124 -id:A332124 - cp's OEIS frontend

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

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

Offset: 0

M. F. Hasler, Feb 09 2020

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

Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).
Patrick De Geest, Palindromic Wing Primes: (1)4(1), updated: June 25, 2017.
Makoto Kamada, Factorization of 11...11411...11, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. (A077780-1)/2 = A107124: indices of primes; A331866 & 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).

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

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

apply( {A332114(n)=10^(n*2+1)\9+3*10^n}, [0..15])

def A332114(n): return 10**(n*2+1)//9+3*10**n

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

A332134 a(n) = (10^(2n+1)-1)/3 + 10^n.

Original entry on oeis.org

4, 343, 33433, 3334333, 333343333, 33333433333, 3333334333333, 333333343333333, 33333333433333333, 3333333334333333333, 333333333343333333333, 33333333333433333333333, 3333333333334333333333333, 333333333333343333333333333, 33333333333333433333333333333, 3333333333333334333333333333333

Offset: 0

M. F. Hasler, Feb 09 2020

There are no primes in this sequence because a(n) = round(n*2/3)*(5*10^n-1).

Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).
Patrick De Geest, Palindromic Wing Primes: (3)4(3), updated: June 25, 2017.
Makoto Kamada, Factorization of 33...33433...33, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002275 (repunits R_n = (10^n-1)/9), A002277 (3*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332124 .. A332194 (variants with different repeated digit 2, ..., 9).
Cf. A332130 .. A332139 (variants with different middle digit 0, ..., 9).

Maple
```
A332134 := n -> (10^(2*n+1)-1)/3+10^n;
```

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

apply( {A332134(n)=10^(n*2+1)\3+10^n}, [0..15])

def A332134(n): return 10**(n*2+1)//3+10**n

a(n) = 3*A138148(n) + 4*10^n = A002277(2n+1) + 10^n.
G.f.: (4 - 101*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.

cp's OEIS Frontend

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

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