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A332118 a(n) = (10^(2n+1) - 1)/9 + 7*10^n.

%I A332118 #11 Dec 21 2022 03:28:35
%S A332118 8,181,11811,1118111,111181111,11111811111,1111118111111,
%T A332118 111111181111111,11111111811111111,1111111118111111111,
%U A332118 111111111181111111111,11111111111811111111111,1111111111118111111111111,111111111111181111111111111,11111111111111811111111111111,1111111111111118111111111111111
%N A332118 a(n) = (10^(2n+1) - 1)/9 + 7*10^n.
%C A332118 See A107648 = {1, 4, 6, 7, 384, 666, ...} for the indices of primes.
%H A332118 Brady Haran and Simon Pampena, <a href="https://youtu.be/HPfAnX5blO0">Glitch Primes and Cyclops Numbers</a>, Numberphile video (2015).
%H A332118 Patrick De Geest, <a href="http://www.worldofnumbers.com/wing.htm#pwp181">Palindromic Wing Primes: (1)8(1)</a>, updated: June 25, 2017.
%H A332118 Makoto Kamada, <a href="https://stdkmd.net/nrr/1/11811.htm">Factorization of 11...11811...11</a>, updated Dec 11 2018.
%H A332118 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (111,-1110,1000).
%F A332118 a(n) = A138148(n) + 8*10^n = A002275(2n+1) + 7*10^n.
%F A332118 G.f.: (8 - 707*x + 600*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
%F A332118 a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.
%p A332118 A332118 := n -> (10^(2*n+1)-1)/9+7*10^n;
%t A332118 Array[(10^(2 # + 1)-1)/9 + 7*10^# &, 15, 0]
%o A332118 (PARI) apply( {A332118(n)=10^(n*2+1)\9+7*10^n}, [0..15])
%o A332118 (Python) def A332118(n): return 10**(n*2+1)//9+7*10**n
%Y A332118 Cf. (A077791-1)/2 = A107648: indices of primes.
%Y A332118 Cf. A002275 (repunits R_n = (10^n-1)/9), A011557 (10^n).
%Y A332118 Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes), A077798 (palindromic wing primes), A088281 (primes 1..1x1..1), A068160 (smallest of given length), A053701 (vertically symmetric numbers).
%Y A332118 Cf. A332128 .. A332178, A181965 (variants with different repeated digit 2, ..., 9).
%Y A332118 Cf. A332112 .. A332119 (variants with different middle digit 2, ..., 9).
%K A332118 nonn,base,easy
%O A332118 0,1
%A A332118 _M. F. Hasler_, Feb 09 2020