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A269499 Nontrivial integer solutions s to the equations Sum_{i} ((-1)^i)*binomial(m,i)*binomial(s-m,t-i) = 0 listed in increasing order. 2
36, 66, 67, 98, 132, 177, 214, 289, 345, 465, 514, 576, 774, 932, 1029, 1219, 1252 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

A nontrivial integer solution s 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 3 < m,t < s/2 such that S(m,s,t)=0 and m,s,t are not such that s=8*k+1 and t=2*k or m=2*k for some integer k.

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.

For t(resp. m)=2, s=(k+2)^2, m(resp. t)=((k + 2)*(k + 1))/2 is an infinite family of trivial solutions.

For t(resp. m)=3, s=3*k^2 + 8*k + 6, m(resp. t)=((k + 1)*(3*k + 2))/2 is another infinite family of trivial solutions.

For t(resp. m)=3, s=3*k^2 + 10*k + 9, m(resp. t)=((k + 1)*(3*k + 4))/2 is another infinite family of trivial solutions.

For t(resp. m)=2*k, s=8*k+1, m(resp. t)=4*k-1 is another infinite family of trivial solutions.

1521, 3193, 3362, 10882, 15043, 19600 also belong to the sequence, but the list has been checked to be complete only up to 1252.

A082639(k) (from k=4) is included in the sequence, because  Sum_{i} (((-1)^i)*binomial(m(k), i)*binomial(s(k) - m, t(k) - i)) = 0, with s(k)= A082639(k) and m(k)=(g^k + g^(-k) - 10)/4 with g=3+2*sqrt(2) and t(k)=(h*g^k + 2 h^(-1)*g^(-k) - 4)/8 with h=2-2*sqrt(2). In other words they are positive integers s of the form s=2*m+4 where (m,t) m>6 is any couple of positive integer solutions to the Diophantine equation m^2 - 4*m*t + 2*t^2 + 3*m - 8*t + 2 = 0 (there are infinitely many).

Nine such infinite subsequences for the present sequence are known.

The eight other similar subsequences are:

- positive integers s of the form s=2*m+5  where (m,t) is solution to the Diophantine equation 5*m^2 - 10*m*t + 4*t^2 + 25*m - 26*t + 32 = 0  (producing 2 subsequences).

- positive integers s of the form s=2*m+5  where (m,t) m>10 is solution to the Diophantine equation m^2 - 6*m*t + 4*t^2 + 3*m - 14*t + 2 = 0     (producing 2 subsequences).

- positive integers s of the form s=2*m+6  where (m,t) is solution to the Diophantine equation m^2 - 8*m*t + 4*t^2 + 3*m - 24*t + 2 = 0  (producing 2 subsequences).

- positive integers s of the form s=2*m+8  where (m,t) m>7 is solution to the Diophantine equation m^2 - 4*m*t + 2*t^2 + 7*m - 16*t + 16 = 0     (producing 2 subsequences).

LINKS

Table of n, a(n) for n=1..17.

EXAMPLE

36=14+22 belongs to the sequence because Sum_{i=0..5} (((-1)^i)*binomial(14, i)*binomial(22,5-i)) = 0, both 5 and 14 are less than 18 and (14,36,5) is not in one of the above trivial families.

MATHEMATICA

f[n_, m_, t_]:= Sum[(-1)^i*Binomial[m, i]*Binomial[n-m, t-i], {i, 0, t}]; lim=200; list={}; Do[ Do[Do [If[ Mod[n, 8]==1&& t==2*Quotient[n, 8], Continue, If[f[n, m, t]==0  , AppendTo[list, n]]], {t, 4, m}] , {m, 4, n/2-1}], {n, 10, lim}]; Print [Union [list]]

PROG

(PARI) isok(s) = {for (m=4, s\2-1, for (t=4, m, if (!(((s % 8) == 1) && (t == 2*(s\8))), if (sum(i=0, t, (-1)^i*binomial(m, i)*binomial(s-m, t-i)) == 0, return (1)); ); ); ); } \\ Michel Marcus, Mar 01 2016

CROSSREFS

Cf. A269563, A082639.

Sequence in context: A326666 A272190 A060671 * A074315 A240520 A126789

Adjacent sequences:  A269496 A269497 A269498 * A269500 A269501 A269502

KEYWORD

nonn,more

AUTHOR

René Gy, Feb 28 2016

EXTENSIONS

a(16)-a(17) from Michel Marcus, Apr 04 2016

STATUS

approved

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Last modified September 24 17:17 EDT 2021. Contains 347651 sequences. (Running on oeis4.)