OEIS Triangle A114176 - Revised

[Paul D. Hanna]

Dear Seqfans, 
     Copied below is a very clear statement of the problem presented by 
Jocelyn Veilleux in a prior e-mail. 
A nice example (of N=4) was also given as a .tif file that I will provide
upon request.  
I will be on vacation for a week starting Saturday, so please copy 
Jocelyn on all related correspondence. 
 
Most likely Jocelyn's triangle is NOT the same as A114176, since 
it seems that the coefficients should remain symmetric for all rows. 
 
A very neat problem - I am glad that the OEIS brought Jocelyn's triangle 
to our attention.  
    Paul 
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On Wed, 21 Jun 2006 10:09:52 -0400 Jocelyn Veilleux
<Jocelyn.Veilleux at USherbrooke.ca> writes:
Dear Paul,
 
In fact, I think that it would be a good idea to restate the problem.
 
Note that I have corrected the formula for A(N) to obtain a simpler
expression.
Since I did not reach manually the rows for which the triangle A114176
coefficient symmetry is lost, I don't know which order is the good one to
generate the coefficients... so either this formula for A(N) or the
previous one
might be correct. My triangle might also differ from yours.
 
 
Consider the asymptotic formula for the maximum amplitude reflected from
the
N-th glass plate for which we must also add the contribution of all
equivalent
optical paths within the N-1 glass plates located above the N-th glass
plate.
 
In this case, we have :
 
A(N) = t1*t2 * Sum[T(N,i) * teq^(2(N-i)) * req^(i) * r2^(i-1), i=1..N]
 
A(N): amplitude of the signal detected
t1  : transmission coefficient in amplitude at an air-glass interface
t2  : transmission coefficient in amplitude at a glass-air interface
teq : equivalent transmission coefficient in amplitude at a plate-plate
interface
req : equivalent reflection coefficient in amplitude at a plate-plate
interface
 
T(N,i): the coefficients T(N,i) equal the number of distinct optical
paths that
have undergone 2i-1 reflections and 2(N-i) transmissions in the N-1 glass
plates located above the N-th plate and that have an optical path length
equivalent to the single optical path that has undergone 2(N-1)
transmissions
and 1 reflection from the N-th glass plate.
 
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Here are some examples for N=1,2,3,4 and 5 (exact contributions from all
optical
paths), followed by the asymptotic approximation used to derive the
expression
for A(N).
 
Exact contributions:
 
N=1:        A(1) = t1*t2*req
 
N=2:        A(2) = t1*t2*[teq^2*req + req^2*r2]
 
N=3:        A(3) = t1*t2*[teq^4*req + teq^2*(2*req^2*r2 + req^3) +
req^3*r2^2]
 
N=4:        A(4) = t1*t2*[teq^6*req + teq^4*(3*req^2*r2 + 3*req^3) +
teq^2*(3*req^3*r2^2 + 2*req^4*r2 + req^5) + req^4*r2^3]
 
N=5:        A(5) = t1*t2*[teq^8*req + teq^6*(4*req^2*r2 + 6*req^3) +
teq^4*(6*req^3*r2^2 + 6*req^4*r2 + 6*req^5) + teq^2*(4*req^4*r2^3 +
3*req^5*r2^2 + 2*req^6*r2 + req^7) + req^5*r2^4]
 
 
Asymptotic approximation for maximum amplitude, based on the fact that r2
> req
 
N=1:        A(1) = t1*t2*req
 
N=2:        A(2) = t1*t2*[teq^2*req + req^2*r2]
 
N=3:        A(3) < t1*t2*[teq^4*req + 3*teq^2*req^2*r2 + req^3*r2^2]
 
N=4:        A(4) < t1*t2*[teq^6*req + 6*teq^4*req^2*r2 +
6*teq^2*req^3*r2^2 +
req^4*r2^3]
 
N=5:        A(5) < t1*t2*[teq^8*req + 10*teq^6*req^2*r2 +
18*teq^4*req^3*r2^2 +
10*teq^2*req^4*r2^3 + req^5*r2^4]
 
 
 From which we obtain A(N) = t1*t2 * Sum[T(N,i) * teq^(2(N-i)) * req^(i) *
r2^(i-1), i=1..N].
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I also add in attach file a schematic representation for the case N = 4.
 
Hope this helps. And thank you very much for your help.
 
Best wishes,
 
Jocelyn