3 added 47 characters in body

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:

=========================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$./indefCycle Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2 -1 5 1 0 form -1 5 1 1 0 0 1 To Return 1 0 0 1 0 form -1 5 1 delta 5 1 form 1 5 -1 delta -5 2 form -1 5 1 minimum was 1rep 1 0 disc 29 dSqrt 5.3851648071 M_Ratio 29 Automorph, written on right of Gram matrix: -1 5 5 -26 Trace: -27 gcd(a21, a22 - a11, a12) : 5 ========================================= jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./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
=========================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$./indefCycle Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2 -1 5 5 0 form -1 5 5 1 0 0 1 To Return 1 0 0 1 0 form -1 5 5 delta 1 1 form 5 5 -1 delta -5 2 form -1 5 5 minimum was 1rep 1 0 disc 45 dSqrt 6.7082039325 M_Ratio 45 Automorph, written on right of Gram matrix: -1 5 1 -6 Trace: -7 gcd(a21, a22 - a11, a12) : 1 ========================================= jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./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
=========================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$./indefCycle Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2 -1 7 3 0 form -1 7 3 1 0 0 1 To Return 1 0 0 1 0 form -1 7 3 delta 2 1 form 3 5 -3 delta -2 2 form -3 7 1 delta 7 3 form 1 7 -3 delta -2 4 form -3 5 3 delta 2 5 form 3 7 -1 delta -7 6 form -1 7 3 minimum was 1rep 1 0 disc 61 dSqrt 7.8102496759 M_Ratio 61 Automorph, written on right of Gram matrix: -79 585 195 -1444 Trace: -1523 gcd(a21, a22 - a11, a12) : 195 ========================================= jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./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
=========================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$./indefCycle Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2 -1 7 7 0 form -1 7 7 1 0 0 1 To Return 1 0 0 1 0 form -1 7 7 delta 1 1 form 7 7 -1 delta -7 2 form -1 7 7 minimum was 1rep 1 0 disc 77 dSqrt 8.7749643874 M_Ratio 77 Automorph, written on right of Gram matrix: -1 7 1 -8 Trace: -9 gcd(a21, a22 - a11, a12) : 1 ========================================= jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$

2 added 256 characters in body

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:

=========================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$./indefCycle Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2 -1 5 1 0 form -1 5 1 1 0 0 1 To Return 1 0 0 1 0 form -1 5 1 delta 5 1 form 1 5 -1 delta -5 2 form -1 5 1 minimum was 1rep 1 0 disc 29 dSqrt 5.3851648071 M_Ratio 29 Automorph, written on right of Gram matrix: -1 5 5 -26 Trace: -27 gcd(a21, a22 - a11, a12) : 5 ========================================= jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./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
=========================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$./indefCycle Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2 -1 5 5 0 form -1 5 5 1 0 0 1 To Return 1 0 0 1 0 form -1 5 5 delta 1 1 form 5 5 -1 delta -5 2 form -1 5 5 minimum was 1rep 1 0 disc 45 dSqrt 6.7082039325 M_Ratio 45 Automorph, written on right of Gram matrix: -1 5 1 -6 Trace: -7 gcd(a21, a22 - a11, a12) : 1 ========================================= jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./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
=========================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$./indefCycle Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2 -1 7 3 0 form -1 7 3 1 0 0 1 To Return 1 0 0 1 0 form -1 7 3 delta 2 1 form 3 5 -3 delta -2 2 form -3 7 1 delta 7 3 form 1 7 -3 delta -2 4 form -3 5 3 delta 2 5 form 3 7 -1 delta -7 6 form -1 7 3 minimum was 1rep 1 0 disc 61 dSqrt 7.8102496759 M_Ratio 61 Automorph, written on right of Gram matrix: -79 585 195 -1444 Trace: -1523 gcd(a21, a22 - a11, a12) : 195 ========================================= jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./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
=========================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$./indefCycle Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2 -1 7 7 0 form -1 7 7 1 0 0 1 To Return 1 0 0 1 0 form -1 7 7 delta 1 1 form 7 7 -1 delta -7 2 form -1 7 7 minimum was 1rep 1 0 disc 77 dSqrt 8.7749643874 M_Ratio 77 Automorph, written on right of Gram matrix: -1 7 1 -8 Trace: -9 gcd(a21, a22 - a11, a12) : 1 ========================================= jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$

1

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 $\langle -1, n, m \rangle .$

Examples:

=========================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$./indefCycle Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2 -1 5 1 0 form -1 5 1 1 0 0 1 To Return 1 0 0 1 0 form -1 5 1 delta 5 1 form 1 5 -1 delta -5 2 form -1 5 1 minimum was 1rep 1 0 disc 29 dSqrt 5.3851648071 M_Ratio 29 Automorph, written on right of Gram matrix: -1 5 5 -26 Trace: -27 gcd(a21, a22 - a11, a12) : 5 ========================================= jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./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
=========================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$./indefCycle Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2 -1 5 5 0 form -1 5 5 1 0 0 1 To Return 1 0 0 1 0 form -1 5 5 delta 1 1 form 5 5 -1 delta -5 2 form -1 5 5 minimum was 1rep 1 0 disc 45 dSqrt 6.7082039325 M_Ratio 45 Automorph, written on right of Gram matrix: -1 5 1 -6 Trace: -7 gcd(a21, a22 - a11, a12) : 1 ========================================= jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./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
=========================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$./indefCycle Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2 -1 7 3 0 form -1 7 3 1 0 0 1 To Return 1 0 0 1 0 form -1 7 3 delta 2 1 form 3 5 -3 delta -2 2 form -3 7 1 delta 7 3 form 1 7 -3 delta -2 4 form -3 5 3 delta 2 5 form 3 7 -1 delta -7 6 form -1 7 3 minimum was 1rep 1 0 disc 61 dSqrt 7.8102496759 M_Ratio 61 Automorph, written on right of Gram matrix: -79 585 195 -1444 Trace: -1523 gcd(a21, a22 - a11, a12) : 195 ========================================= jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./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
=========================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$./indefCycle Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2 -1 7 7 0 form -1 7 7 1 0 0 1 To Return 1 0 0 1 0 form -1 7 7 delta 1 1 form 7 7 -1 delta -7 2 form -1 7 7 minimum was 1rep 1 0 disc 77 dSqrt 8.7749643874 M_Ratio 77 Automorph, written on right of Gram matrix: -1 7 1 -8 Trace: -9 gcd(a21, a22 - a11, a12) : 1 ========================================= jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$