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

9, 16, 17, 22, 25, 33, 34, 36, 41, 49, 57, 64, 65, 66, 67, 73, 81, 86, 89, 97, 98, 100, 105, 113, 121, 129, 132, 134, 137, 144, 145, 153, 161, 162, 169, 177, 185, 193, 196, 201, 209, 214, 217, 225, 226, 233, 241, 249, 256, 257, 262, 265, 273, 281, 289, 297, 305

Offset: 1

Author

René Gy, Feb 29 2016

Keywords

• nonn

Comments

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.

Links

• Konrad Tschernig, Roberto de J. León-Montiel, Omar S. Magaña-Loaiza, Alexander Szameit, Kurt Busch, Armando Perez-Leija, Multiphoton Discrete Fractional Fourier Dynamics in Waveguide Beam Splitters, Journal of the Optical Society of America B (2018) Vol. 35, Issue 8, 1985-1989. arXiv:1807.07463 [physics.optics], 2018.

Crossrefs

Cf. A269499.

Programs

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

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

Formula

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

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.

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.

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).

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.

Extensions

More terms from Michel Marcus, Mar 01 2016