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 A357958 #9 Nov 06 2022 07:50:09
%S A357958 39,407,7491,167063,4112539,107461667,2923006251,81853622423,
%T A357958 2343591359499,68288538877907,2018394003648391,60366962358086243,
%U A357958 1823569260750104179,55557874330437332267,1705172670555862322491,52672612525369663916183
%N A357958 a(n) = 5*A005259(n) + 14*A005258(n-1).
%C A357958 Conjectures:
%C A357958 1) a(p) == a(1) (mod p^5) for all primes p >= 5 (checked up to p = 271).
%C A357958 2) a(p^r) == a(p^(r-1)) ( mod p^(3*r+3) ) for r >= 2 and for all primes p >= 3.
%C A357958 These are stronger supercongruences than those satisfied separately by the two types of Apéry numbers A005258 and A005259. Cf. A357959.
%C A357958 There is also a product version of these conjectures:
%C A357958 3) the sequence {u(n): n>= 1} defined by u(n) = A005259(n)^25 * A005258(n-1)^14 conjecturally satisfies the congruences in 1) and 2) above.
%F A357958 a(n) = 5*Sum_{k = 0..n} binomial(n,k)^2*binomial(n+k,k)^2 + 14*Sum_{k = 0..n-1} binomial(n-1,k)^2*binomial(n+k-1,k).
%F A357958 a(p^r) == a(p^(r-1)) ( mod p^(3*r) ) for positive integer r and for all primes p >= 5.
%e A357958 Examples of supercongruences:
%e A357958 a(13) - a(1) = 1823569260750104179 - 39 = (2^2)*5*7*(13^5)*35081444357 == 0 (mod 13^5).
%e A357958 a(7^2) - a(7) = (2^3)*(7^9)* 10412078726049425470554760052126170543547100055154203726400782433 == 0 (mod 7^9).
%p A357958 seq( add( 5*binomial(n,k)^2*binomial(n+k,k)^2 + 14*binomial(n-1,k)^2* binomial(n+k-1,k), k = 0..n ), n = 1..20);
%Y A357958 Cf. A005258, A005259, A212334, A352655, A357567, A357956, A357957, A357959, A357960.
%K A357958 nonn,easy
%O A357958 1,1
%A A357958 _Peter Bala_, Oct 25 2022