cp's OEIS Frontend

An integer solution to the equations S(m,s,t) = Sum_{i} (((-1)^i)*binomial(m, i)*binomial(s - m, t - i)) = 0 is an integer s such that there exist integers m, t and 0 < m,t < s/2 such that S(m,s,t)=0.

S(m,s,t)=0 iff S(t,s,m)=0 iff S(s-m,s,t)=0 iff S(s-t,s,m)=0.

If m or t > s, the equation is trivially true, if m or t = s, it is never true.

There are m,t such that 0 < m,t < s/2 and S(m,s,t)=0 iff there are m',t' such that s/2 < m',t' < s and S(m',s,t')=0.

When s is even S(s/2,s,t)=0 (resp. S(m,s,s/2)=0) whenever t (resp. m) is odd. These kinds of super-trivial solutions are not considered.

Therefore the sequence only contains the s for which there exist integers m, t such that 0 < m,t < s/2 and S(m,s,t)=0.

"Sporadic" solutions s: these are the solutions that remain from A269563, when we remove the four known infinite families of solutions in polynomial progression (see the comments in A269563) and also remove all the nine known infinite families of solutions in exponential progression (see the comments in A269499). These nine families are the s = 2*m + p, where p=4,5,6 or 8 and (m,t) are positive integer solutions to some Diophantine bivariate polynomial equation of degree 2:

p=4 m^2 - 4*m*t + 2*t^2 + 3*m - 8*t + 2 = 0

p=5 5*m^2 - 10*m*t + 4*t^2 + 25*m - 26*t + 32 = 0

p=5 m^2 - 6*m*t + 4*t^2 + 3*m - 14*t + 2 = 0

p=6 m^2 - 8*m*t + 4*t^2 + 3*m - 24*t + 2 = 0

p=8 m^2 - 4*m*t + 2*t^2 + 7*m - 16*t + 16 = 0

1521, 10882, 15043 and 48324 are also "sporadic" solutions, but the list has been checked to be complete up to 1029 only.