Thread: Heptagon Flat
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Old 2018-04-22, 15:35   #2
Dr Sardonicus
 
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The numbers may be formulated exactly in terms of

f = x^3 - x^2 - 2*x + 1

by taking

a = Mod(x,f) and b = Mod(x^2 - 1, f).

The quantity b has characteristic polynomial x^3 - 2*x^2 - x + 1, the reciprocal polynomial to f.

In order to express

a^n * b^m = C1 + C2*a + C3*b,

it suffices to find the dual basis, with respect to the trace, of

[1, Mod(x, f), Mod(x^2 - 1, f)].

The dual basis [t1, t2, t3] with respect to [Mod(1, f), Mod(x, f), Mod(x^2, f)] is easily found (Newton's identities). The required basis is obviously

[w1, w2, w3] = [t1 + t3, t2, t3].

Then we obtain

C1 = trace(w1 * a^n * b^m), C2 = trace(w2 * a^n * b^m), C3 = trace(w3 * a^n * b^m).

We have

[w1, w2, w3] = [Mod(-1/7*x^2 + 4/7, x^3 - x^2 - 2*x + 1), Mod(-1/7*x^2 + 2/7*x + 1/7, x^3 - x^2 - 2*x + 1), Mod(2/7*x^2 - 1/7*x - 3/7, x^3 - x^2 - 2*x + 1)].

In particular, the row of the table in the web page for b^m is given by

trace(w1 * b^m * a^n), n = -2 to 6.

The column for a^n is given by

trace(w[1]* a^n * b^m), m = -1 to 7.

The numbers in any given row satisfy the recurrence

Cn+3 = Cn+2 + 2*Cn+1 - Cn

while those in a given column satisfy

Cn+3 = 2*Cn+2 + Cn+1 - Cn.
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