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 A357567 #14 Nov 06 2022 12:24:16
%S A357567 -9,-17,99,5167,147491,3937483,105834699,2907476527,81702447651,
%T A357567 2342097382483,68273597307599,2018243113678027,60365426282638091,
%U A357567 1823553517258576723,55557712038989195099,1705170989220937925167,52672595030914982754851,1636296525812843554700323
%N A357567 a(n) = 5*A005259(n) - 14*A005258(n).
%C A357567 Conjectures:
%C A357567 1) a(p) == a(1) (mod p^5) for all primes p >= 5.
%C A357567 2) For r >= 2, a(p^r) == a(p^(r-1)) ( mod p^(3*r+3) ) for all primes p >= 5.
%C A357567 These are stronger supercongruences than those satisfied separately by the two types of Apéry numbers A005258 and A005259.
%C A357567 From _Peter Bala_, Oct 25 2022: (Start)
%C A357567 Additional conjectures:
%C A357567 3) the sequence {u(n): n>= 1} defined by u(n) = (3^42)*A005259(n)^25 - (5^25)* A005258(n)^42 also satisfies the congruences in 1) and 2) above.
%C A357567 4) u(n) == 0 (mod n^5) for integer n of the form 3^i*5^j (see A003593). (End)
%F A357567 a(n) = 5*Sum_{k = 0..n} binomial(n,k)^2*binomial(n+k,k)^2 - 14*Sum_{k = 0..n} binomial(n,k)^2*binomial(n+k,k).
%F A357567 For positive integers n and r, a(n*p^r) == a(n*p^(r-1)) ( mod p^(3*r) ) for all primes p >= 5.
%e A357567 a(11) - a(1) = 2018243113678027 + 17 = (2^2)*(3^2)*(11^5)*17*20476637 == 0 (mod 11^5).
%p A357567 seq(add(5*binomial(n,k)^2*binomial(n+k,k)^2 - 14*binomial(n,k)^2*binomial(n+k,k), k = 0..n), n = 0..20);
%Y A357567 Cf. A005258, A005259, A212334, A352655, A357506, A357507, A357508, A357509, A357568, A357569, A357956, A357957, A357958, A357959, A357960.
%K A357567 sign,easy
%O A357567 0,1
%A A357567 _Peter Bala_, Oct 19 2022