A332115 - OEIS

%I #8 Feb 11 2020 07:57:15

%S 5,151,11511,1115111,111151111,11111511111,1111115111111,

%T 111111151111111,11111111511111111,1111111115111111111,

%U 111111111151111111111,11111111111511111111111,1111111111115111111111111,111111111111151111111111111,11111111111111511111111111111,1111111111111115111111111111111

%N a(n) = (10^(2n+1)-1)/9 + 4*10^n.

%C See A107125 = {0, 1, 7, 45, 115, 681, 1248, ...} for the indices of primes.

%H Brady Haran and Simon Pampena, <a href="https://youtu.be/HPfAnX5blO0">Glitch Primes and Cyclops Numbers</a>, Numberphile video (2015).

%H Patrick De Geest, <a href="http://www.worldofnumbers.com/wing.htm#pwp151">Palindromic Wing Primes: (1)5(1)</a>, updated: June 25, 2017.

%H Makoto Kamada, <a href="https://stdkmd.net/nrr/1/11511.htm">Factorization of 11...11511...11</a>, updated Dec 11 2018.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (111,-1110,1000).

%F a(n) = A138148(n) + 5*10^n = A002275(2n+1) + 4*10^n.

%F G.f.: (5 - 404*x + 300*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).

%F a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

%p A332115 := n -> (10^(2*n+1)-1)/9+4*10^n;

%t Array[(10^(2 # + 1)-1)/9 + 4*10^# &, 15, 0]

%o (PARI) apply( {A332115(n)=10^(n*2+1)\9+4*10^n}, [0..15])

%o (Python) def A332115(n): return 10**(n*2+1)//9+4*10**n

%Y Cf. (A077783-1)/2 = A107125: indices of primes; A331868 & A331869 (non-palindromic variants).

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

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

%Y Cf. A332125 .. A332195 (variants with different repeated digit 2, ..., 9).

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

%K nonn,base,easy

%O 0,1

%A _M. F. Hasler_, Feb 09 2020