OFFSET
2,2
COMMENTS
This sequence is based on Gary Detlefs's conjecture, which he posted as a comment to A005809. His conjecture is equivalent to the conjecture that the Diophantine equation ((n+1)*(3*n)!/((2*n-1)!*(n+1)!*2*n)-3)/n^3 = m has integer solutions m for all odd primes n.
Additionally I conjecture that all m are divisible by 3, therefore terms of this sequence a(n) = m/3.
It is also notable that for quite a few values of n (2, 3, 4, 5, 6, 7, 17, 19, 21, 22, 23, 24, 25, 26, 35, 39, 43, ...) a(n+1) = a(n) mod 7.
The values of this sequence's terms are replicated by conjectured general formula, given in A223886 (and also added to the formula section here) for k=3, j=1 and n>=2. - Alexander R. Povolotsky, Apr 18 2013
For n>=3 and k>=2 ((binomial(k*p,p)-k)/p^3)/k is an integer. For k=2 this is the Wolstenholme quotient (A034602) and for k=3 the current sequence. - Peter Luschny, Feb 09 2016
FORMULA
a(n) = (binomial(j*k*prime(n), j*prime(n)) - binomial(k*j, j))/(k*prime(n)^3) for k=3, j=1 and n>=2 (conjectured). - Alexander R. Povolotsky, Apr 18 2013
MAPLE
WQ := proc(n, k) local p; p := ithprime(n); ((binomial(k*p, p)-k)/p^3)/k end:
seq(WQ(n, 3), n=2..14); # Peter Luschny, Feb 09 2016
PROG
(PARI) a(n)=my(p=prime(n)); (binomial(3*p, p+1)*(p+1)/(2*p)-3)/(3*p^3) \\ Charles R Greathouse IV, Mar 26 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
Alexander R. Povolotsky, Mar 24 2013
STATUS
approved