EDIT: I got your fact right here.
ORIGINAL: It seems Henry got it. Meanwhile, let me point out how things appear from the Lagrange viewpoint of right-adjacent reduced forms: given odd numbers $n$ and $1 \leq m \leq n,$ the cycle for the form $\langle -1, n, m \rangle $ has penultimate form $\langle m, n, -1 \rangle, $ then "digit" $\delta = -n,$ then the end of the cycle is again $\langle -1, n, m \rangle .$ Well, see the method in my answer at http://mathoverflow.net/questions/22811/upper-bound-of-period-length-of-continued-fraction-representation-of-very-composi where the fact you need, the final $\delta = -n,$ follows from the definition of the $\delta$'s.
Examples:
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jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle
Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2
-1 5 1
0 form -1 5 1
1 0
0 1
To Return
1 0
0 1
0 form -1 5 1 delta 5
1 form 1 5 -1 delta -5
2 form -1 5 1
minimum was 1rep 1 0 disc 29 dSqrt 5.3851648071 M_Ratio 29
Automorph, written on right of Gram matrix:
-1 5
5 -26
Trace: -27 gcd(a21, a22 - a11, a12) : 5
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jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle
Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2
-1 5 3
0 form -1 5 3
1 0
0 1
To Return
1 0
0 1
0 form -1 5 3 delta 1
1 form 3 1 -3 delta -1
2 form -3 5 1 delta 5
3 form 1 5 -3 delta -1
4 form -3 1 3 delta 1
5 form 3 5 -1 delta -5
6 form -1 5 3
minimum was 1rep 1 0 disc 37 dSqrt 6.0827625303 M_Ratio 37
Automorph, written on right of Gram matrix:
-13 72
24 -133
Trace: -146 gcd(a21, a22 - a11, a12) : 24
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jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle
Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2
-1 5 5
0 form -1 5 5
1 0
0 1
To Return
1 0
0 1
0 form -1 5 5 delta 1
1 form 5 5 -1 delta -5
2 form -1 5 5
minimum was 1rep 1 0 disc 45 dSqrt 6.7082039325 M_Ratio 45
Automorph, written on right of Gram matrix:
-1 5
1 -6
Trace: -7 gcd(a21, a22 - a11, a12) : 1
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jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle
Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2
-1 7 1
0 form -1 7 1
1 0
0 1
To Return
1 0
0 1
0 form -1 7 1 delta 7
1 form 1 7 -1 delta -7
2 form -1 7 1
minimum was 1rep 1 0 disc 53 dSqrt 7.2801098893 M_Ratio 53
Automorph, written on right of Gram matrix:
-1 7
7 -50
Trace: -51 gcd(a21, a22 - a11, a12) : 7
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jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle
Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2
-1 7 3
0 form -1 7 3
1 0
0 1
To Return
1 0
0 1
0 form -1 7 3 delta 2
1 form 3 5 -3 delta -2
2 form -3 7 1 delta 7
3 form 1 7 -3 delta -2
4 form -3 5 3 delta 2
5 form 3 7 -1 delta -7
6 form -1 7 3
minimum was 1rep 1 0 disc 61 dSqrt 7.8102496759 M_Ratio 61
Automorph, written on right of Gram matrix:
-79 585
195 -1444
Trace: -1523 gcd(a21, a22 - a11, a12) : 195
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jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle
Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2
-1 7 5
0 form -1 7 5
1 0
0 1
To Return
1 0
0 1
0 form -1 7 5 delta 1
1 form 5 3 -3 delta -1
2 form -3 3 5 delta 1
3 form 5 7 -1 delta -7
4 form -1 7 5
minimum was 1rep 1 0 disc 69 dSqrt 8.3066238629 M_Ratio 69
Automorph, written on right of Gram matrix:
2 -15
-3 23
Trace: 25 gcd(a21, a22 - a11, a12) : 3
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jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle
Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2
-1 7 7
0 form -1 7 7
1 0
0 1
To Return
1 0
0 1
0 form -1 7 7 delta 1
1 form 7 7 -1 delta -7
2 form -1 7 7
minimum was 1rep 1 0 disc 77 dSqrt 8.7749643874 M_Ratio 77
Automorph, written on right of Gram matrix:
-1 7
1 -8
Trace: -9 gcd(a21, a22 - a11, a12) : 1
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jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$
