This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A320625 #13 Oct 26 2018 00:59:21
%S A320625 1,1,11,520783,1777232132705889910073,
%T A320625 1989655738014873996462170980393276816167557169374094238588991602837393
%N A320625 a(n) = A006134((3^n-1)/2)/3^n.
%C A320625 a(n) is always an integer.
%C A320625 a(6) has 215 digits and a(7) has 654 digits.
%C A320625 For primes p we have A006134((p-1)/2) == Legendre(p, 3) (mod p). For composite n equal to a power of 3, n is also divisible by A006134((n-1)/2). Odd composite n not a power of 3 such that n divides A006134((n-1)/2) are n = 99, 1539, ... What's the next? Are there similar values of n not divisible by 3?
%C A320625 Conjecture: for n > 0, a(n) == 1 (mod 3) for odd n, a(n) == 2 (mod 3) for even n.
%e A320625 a(1) = (binomial(0, 0) + binomial(2, 1))/3 = 3/3 = 1.
%e A320625 a(2) = (binomial(0, 0) + binomial(2, 1) + binomial(4, 2) + binomial(6, 3) + binomial(8, 4))/9 = 99/9 = 11.
%p A320625 a:=n->add(binomial(2*k,k),k=0..(3^n-1)/2)/3^n: seq(a(n),n=0..5); # _Muniru A Asiru_, Oct 23 2018
%t A320625 Array[Sum[Binomial[2 k, k], {k, 0, #}] &[(3^# - 1)/2]/3^# &, 5] (* _Michael De Vlieger_, Oct 22 2018 *)
%o A320625 (PARI) A006134(n) = sum(k=0,n,binomial(2*k,k))
%o A320625 a(n) = A006134((3^n-1)/2)/3^n
%o A320625 (GAP) List([0..5],n->Sum([0..(3^n-1)/2],k->Binomial(2*k,k))/3^n); # _Muniru A Asiru_, Oct 23 2018
%Y A320625 Cf. A000244, A006134, A320627.
%K A320625 nonn
%O A320625 0,3
%A A320625 _Jianing Song_, Oct 18 2018