on some "floorantin" equations

[benoit]

Hi again,

I considered the following equation for r real > 0 :

(i) x^2=ceiling(r*x*floor(x/r))

Sequence (b(n)) n>=1 of smallest solution to :  
x^2=ceiling(sqrt(n)*x*floor(x/sqrt(n))) begins for n>=1 :

1,3,2,2,9,5,8,3,3,19,10,7,649,15,4,4,33,17,170,9,55,197,24,5,5,51,26,127 
,9801,11,
1520,17,23,35,6,6,73,37,25,19,2049,13,3482,199,161,24335,48,7,7,99,50,64 
9,66249,485,89,

not in OEIS.

It turns out the set of solutions for (i) when r=sqrt(m), m integer>0,  
consists in a sequence (a(n)) n>=1 satisfying the reccurence :

a(n)= 2*A002350(m)*a(n-1)-a(n-2)

ex :

r=sqrt(2) gives solutions = A001541 satisfying : a(n)=6a(n-1)-a(n-2)
r=sqrt(3) gives solutions = A001075 satisfying : a(n)=4a(n-1)-a(n-2)
r=sqrt(5) gives solutions = A023039  satisfying : a(n)=9a(n-1)-a(n-2)
r=sqrt(5) gives solutions = A001079  satisfying : a(n)=10a(n-1)-a(n-2)

Hence, when r=sqrt(m), from the above sequence (b(k))_k>=1, we get all  
the solutions for (i) by evaluating Chebyshev's polynomials T(n,x)  at  
x=b(m).

Any evidence for that?

thanks

B. Cloitre



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