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A269563 Integer solutions to the equations Sum_{i} (((-1)^i)*binomial(m, i)*binomial(x - m, t - i)) = 0.

%I A269563 #17 Jun 30 2019 20:22:58
%S A269563 9,16,17,22,25,33,34,36,41,49,57,64,65,66,67,73,81,86,89,97,98,100,
%T A269563 105,113,121,129,132,134,137,144,145,153,161,162,169,177,185,193,196,
%U A269563 201,209,214,217,225,226,233,241,249,256,257,262,265,273,281,289,297,305
%N A269563 Integer solutions to the equations Sum_{i} (((-1)^i)*binomial(m, i)*binomial(x - m, t - i)) = 0.
%C A269563 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 A269563 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 A269563 If m or t > s, the equation is trivially true, if m or t = s, it is never true.
%C A269563 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 A269563 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 A269563 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 A269563 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 A269563 There are some patterns in the sequence (quite easy to see with elementary algebra):
%F A269563 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 A269563 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 A269563 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 A269563 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 A269563 f[n_, m_, t_] := Sum[(-1)^i*Binomial[m, i]*Binomial[n - m, t - i], {i, 0, t}]; lim = 200; list = {};
%t A269563 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 A269563 (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 A269563 Cf. A269499.
%K A269563 nonn
%O A269563 1,1
%A A269563 _René Gy_, Feb 29 2016
%E A269563 More terms from _Michel Marcus_, Mar 01 2016