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 A277640 #21 Dec 12 2023 18:35:25
%S A277640 -2,1,-3,-1,7,-1,6,4,-15,-15,-13,1,-23,1,8,-15,-22,13,-33,27,25,11,
%T A277640 -17,24,-32,-53,31,42,-19,18,-35,55,-5,38,-29,76,34,44,-71,-21,-13,16,
%U A277640 46,70,92,70,-39,88,-84,-118,-120,64,107,111,-56,124,-13,-23
%N A277640 a(n) is the integer r with |r| < prime(n)/2 such that (T(prime(n)^2)-T(prime(n)))/prime(n)^4 == r (mod prime(n)), where T(k) denotes the central trinomial coefficient A002426(k).
%C A277640 Conjecture: (i) For any prime p > 3 and positive integer n, the number (T(p*n)-T(n))/(p*n)^2 is always a p-adic integer.
%C A277640 (ii) For any prime p > 3 and positive integer k, we have (T(p^k)-T(p^(k-1)))/p^(2k) == 1/6*(p^k/3)*B_{p-2}(1/3) (mod p), where (p^k/3) denotes the Legendre symbol and B_{p-2}(x) is the Bernoulli polynomial of degree p-2.
%C A277640 For any prime p > 3, the author has proved that (T(p*n)-T(n))/(p^2*n) is a p-adic integer for each positive integer n, and that T(p) == 1 + p^2/6*(p/3)*B_{p-2}(1/3) (mod p^3).
%H A277640 Hao Pan and Zhi-Wei Sun, <a href="https://arxiv.org/abs/2012.05121">Supercongruences for central trinomial coefficients</a>, arXiv:2012.05121 [math.NT], 2020.
%H A277640 Zhi-Wei Sun, <a href="http://dx.doi.org/10.1007/s11425-014-4809-z">Congruences involving generalized central trinomial coefficients</a>, Sci. China Math. 57(2014), no.7, 1375-1400.
%H A277640 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1610.03384">Supercongruences involving Lucas sequences</a>, arXiv:1610.03384 [math.NT], 2016.
%e A277640 a(3) = -2 since (T(prime(3)^2)-T(prime(3)))/prime(3)^4 = (T(25)-T(5))/5^4 = (82176836301-51)/5^4 = 131482938 is congruent to -2 modulo prime(3) = 5 with |-2| < 5/2.
%t A277640 T[n_]:=T[n]=Sum[Binomial[n,2k]Binomial[2k,k],{k,0,n/2}]
%t A277640 rMod[m_,n_]:=Mod[Numerator[m]*PowerMod[Denominator[m],-1,n],n,-n/2]
%t A277640 Do[Print[n," ",rMod[(T[Prime[n]^2]-T[Prime[n]])/Prime[n]^4,Prime[n]]],{n,3,60}]
%Y A277640 Cf. A000040, A002426, A245089, A277860.
%K A277640 sign
%O A277640 3,1
%A A277640 _Zhi-Wei Sun_, Oct 25 2016