|
%I #11 Mar 30 2024 16:10:00
%S 9,191,11911,1119111,111191111,11111911111,1111119111111,
%T 111111191111111,11111111911111111,1111111119111111111,
%U 111111111191111111111,11111111111911111111111,1111111111119111111111111,111111111111191111111111111,11111111111111911111111111111,1111111111111119111111111111111
%N a(n) = (10^(2n+1)-1)/9 + 8*10^n.
%C See A107649 = {1, 4, 26, 187, 226, 874, ...} 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#pwp191">Palindromic Wing Primes: (1)9(1)</a>, updated: June 25, 2017.
%H Makoto Kamada, <a href="https://stdkmd.net/nrr/1/11911.htm">Factorization of 11...11911...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) + 9*10^n = A002275(2n+1) + 8*10^n.
%F G.f.: (9 - 808*x + 700*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 A332119 := n -> (10^(2*n+1)-1)/9+8*10^n;
%t Array[(10^(2 # + 1)-1)/9 + 8*10^# &, 15, 0]
%t Table[FromDigits[Join[PadRight[{},n,1],{9},PadRight[{},n,1]]],{n,0,20}] (* or *) LinearRecurrence[ {111,-1110,1000},{9,191,11911},20] (* _Harvey P. Dale_, Mar 30 2024 *)
%o (PARI) apply( {A332119(n)=10^(n*2+1)\9+8*10^n}, [0..15])
%o (Python) def A332119(n): return 10**(n*2+1)//9+8*10**n
%Y Cf. (A077795-1)/2 = A107649: indices of primes.
%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. A332129 .. A332189 (variants with different repeated digit 2, ..., 8).
%Y Cf. A332112 .. A332118 (variants with different middle digit 2, ..., 8).
%K nonn,base,easy
%O 0,1
%A _M. F. Hasler_, Feb 09 2020
|
