OFFSET
0,3
COMMENTS
Conjecture: For any prime p > 5 and positive integer n, the number (a(p*n)-a(n))/(p*n)^3 is always a p-adic integer.
We have proved that for any prime p > 5 and positive integer n the number (a(p*n)-a(n))/(p^3*n^2) is always a p-adic integer.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 0..200
Zhi-Wei Sun, Supercongruences involving Lucas sequences, arXiv:1610.03384 [math.NT], 2016.
EXAMPLE
a(3) = 19 since a(3) = C(3,2*0)^2*C(3-0,0) + C(3,2*1)^2*C(3-1,1) = 1 + 3^2*2 = 19.
G.f. = 1 + x + 2*x^2 + 19*x^3 + 110*x^4 + 476*x^5 + 2477*x^6 + 15093*x^7 + ...
MATHEMATICA
a[n_]:=a[n]=Sum[Binomial[n, 2k]^2*Binomial[n-k, k], {k, 0, n/2}]
Table[a[n], {n, 0, 27}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Nov 20 2016
STATUS
approved