A270615 Sporadic solutions s to the equations Sum_i (-1)^i * binomial(m,i) * binomial(s-m,t-i) = 0 listed in increasing order.

67, 289, 345, 1029

Offset: 1

Author

René Gy, Mar 20 2016

Keywords

• nonn
• more
• hard

Comments

"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.

Examples

			67 is in the sequence because Sum_i (-1)^i * binomial(m,i) * binomial(67-m,t-i) = 0, when m=22 and t=5. And m=22 and t=5 do not belong to any of the above progressions.
		

Crossrefs

Cf. A269563, A269499.