A364119 a(n) = 7*A364115(n) - 17*A364115(n-1).

46, 1870, 95950, 6111054, 445850046, 35606390254, 3031075759870, 270542736416590, 25045919145436366, 2386963634176587870, 232926731552238831054, 23180020599857593886190, 2345286553765877009107710, 240670553547813070050900126, 25001383450621552178261089950

Offset

1,1

Comments

It is conjectured that the sequence {A364115(n)} and the shifted sequence {A364115(n-1)} both satisfy the supercongruences A364115(p^r) == A364115(p^(r-1)) (mod p^(3*r)) for all primes p >= 5 and positive integers r. Stronger supercongruences may hold for the present sequence, a linear combination of A364115(n) and A364115(n-1).

Conjectures: 1) the supercongruences a(p) == a(1) (mod p^5) hold for all primes p >= 7 (checked up to p = 101).

2) for r >= 2, the supercongruences a(p^r) == a(p^(r-1)) (mod p^(3*r+3)) hold for all primes p >= 7. Cf. A212334.

There is also a multiplicative version of this sequence. Define a sequence of rational numbers {b(n) : n >= 1} by b(n) = A364115(n)^63 / A364115(n-1)^17. Then we conjecture that the above pair of supercongruences also hold for the sequence {b(n)}.

Links

Table of n, a(n) for n=1..15.

Peter Bala, Some conjectural supercongruences for the Apéry numbers.

Maple

A364115 := n -> coeff(series( 1/(1-x)* LegendreP(n, (1+x)/(1-x))^4, x, 21), x, n):
seq(7*A364115(n) - 17*A364115(n-1), n = 1..20);

Crossrefs

Cf. A212334, A357506, A357507, A357568, A364115, A364118.

Sequence in context: A115608 A318044 A263190 * A331707 A223813 A324451

Adjacent sequences: A364116 A364117 A364118 * A364120 A364121 A364122

Keyword

nonn,easy

Author

Peter Bala, Jul 12 2023

Status

approved