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Integer solutions to the equations Sum_{i} (((-1)^i)*binomial(m, i)*binomial(x - m, t - i)) = 0.
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%I #17 Jun 30 2019 20:22:58

%S 9,16,17,22,25,33,34,36,41,49,57,64,65,66,67,73,81,86,89,97,98,100,

%T 105,113,121,129,132,134,137,144,145,153,161,162,169,177,185,193,196,

%U 201,209,214,217,225,226,233,241,249,256,257,262,265,273,281,289,297,305

%N Integer solutions to the equations Sum_{i} (((-1)^i)*binomial(m, i)*binomial(x - m, t - i)) = 0.

%C 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.

%C 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.

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

%C 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.

%C 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.

%C 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.

%H Konrad Tschernig, Roberto de J. León-Montiel, Omar S. Magaña-Loaiza, Alexander Szameit, Kurt Busch, Armando Perez-Leija, <a href="https://doi.org/10.1364/JOSAB.35.001985">Multiphoton Discrete Fractional Fourier Dynamics in Waveguide Beam Splitters</a>, Journal of the Optical Society of America B (2018) Vol. 35, Issue 8, 1985-1989. <a href="https://arxiv.org/abs/1807.07463">arXiv:1807.07463</a> [physics.optics], 2018.

%F There are some patterns in the sequence (quite easy to see with elementary algebra):

%F For t(resp. m)=2*k, s=8*k+1, m(resp. t)=4*k-1 is an infinite family of solutions (k>=1). This arithmetic progression (from 9), belongs to the sequence.

%F For t(resp. m)=2, s=(k+2)^2, m(resp. t)=((k + 2)*(k + 1))/2 is another infinite family of solutions (k>=1). All the squares (from 9) belongs to the sequence.

%F 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 solutions (k>=1).

%F 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 solutions (k>=0). These polynomial progressions belong to the sequence.

%t f[n_, m_, t_] := Sum[(-1)^i*Binomial[m, i]*Binomial[n - m, t - i], {i, 0, t}]; lim = 200; list = {};

%t Do[ Do[Do[If[f[n, m, t] == 0, AppendTo[list, n]], {t, 0, m}], {m, 0, n/2 - 1}], {n, 0, lim}]; Print[Union[list]]

%o (PARI) isok(s) = {for (m=0, s\2-1, for (t=0, m, if (sum(i=0, t, (-1)^i*binomial(m, i)*binomial(s-m, t-i)) == 0, return (1));););} \\ _Michel Marcus_, Mar 01 2016

%Y Cf. A269499.

%K nonn

%O 1,1

%A _René Gy_, Feb 29 2016

%E More terms from _Michel Marcus_, Mar 01 2016