Part A: Theorem 1 (A_2, n = 2, C = 2)
  A  DFS over tridiagonal matrices with entries in {-2..2} (5120000 transitions examined): 16383 recognized matrices of size <= 28, exactly the M_s, s in {U,D}^N; windows as in Lemma 3.3; dim ker = [#U = #D] with the kernel vector of Lemma 3.5; c_d = coefficients of x^2(1-4x^4)^(-1/2) for d <= 28; F(1/2) = sqrt(3)/6, minimal polynomial 12t^2 - 1  (21.7 s)
     c_d (d = 2, 6, 10, ...): [1, 2, 6, 20, 70, 252, 924]
Part B: Remark 3.6 (A^(n) for n = 3 and n = 4)
  B  n = 3: all 1953125 first letters of Sigma_{3,2}; 3928125 transitions examined; recognized matrices of size <= 16: 254, exactly the M_s with 3 <= 2N+2; coefficients [d >= n] c_d; F^(3)(1/2) = sqrt(3)/6 - 1/4  (2.1 s)
  B  n = 4: all 9765625 first letters without far non-zero entries (the other 5^16 - 5^10 letters are dead by the first rule); 21484375 transitions examined; recognized matrices of size <= 12: 62, exactly the M_s with 4 <= 2N+2; coefficients [d >= n] c_d; F^(4)(1/2) = sqrt(3)/6 - 1/4  (27.6 s)
Part C: Remark 3.7
  C  x^-1 F(x) at 1/2 = sqrt(3)/3; F(1/3) = sqrt(77)/77 = 1/sqrt(77): ok
Part D: Proposition 4.1
  D  p/s = 1/1: C = 2, alphabet of 625 letters, 1875 transitions: the recognized matrices of size 2..7 are the 2 matrices diag(0,lam,...,lam) per size, kernel 1; F(x) = 2x^2/(1-x), F(1/2) = 1/1  (0.0 s)
  D  p/s = 1/2: C = 3, alphabet of 2401 letters, 7546 transitions: the recognized matrices of size 2..7 are the 3 matrices diag(0,lam,...,lam) per size, kernel 1; F(x) = 3x^2/(1-x), F(1/3) = 1/2  (0.0 s)
  D  p/s = 2/3: C = 8, alphabet of 83521 letters, 280041 transitions: the recognized matrices of size 2..7 are the 8 matrices diag(0,lam,...,lam) per size, kernel 1; F(x) = 8x^2/(1-x), F(1/4) = 2/3  (0.0 s)
  D  p/s = 3/2: C = 9, alphabet of 130321 letters, 438976 transitions: the recognized matrices of size 2..7 are the 9 matrices diag(0,lam,...,lam) per size, kernel 1; F(x) = 9x^2/(1-x), F(1/3) = 3/2  (0.1 s)
Part E: Proposition 4.2 (L = no two consecutive a's, c = (2,-1,0), odd labels 1, 3, 5)
  E  alphabet Sigma_{2,6} (28561 letters), 1419262 transitions: 258 recognized matrices of size <= 12, exactly the M^c_s with s in L; kernels [c(s) = 0]; coefficients of x^2 G(x^2): G = [1, 1, 1, 4, 13, 31]  (0.2 s)
Part F: Example 4.4 and Corollary 4.3
  F  (1-2y-3y^2) G(y)^2 = 1 to y^60; Example 4.4 value 1/sqrt(5); diagonal of H(a,b) = G(y^K) for Example 4.4 (K = 1) and for the language of Part E (K = 2); for L = {a,b}^*, c = (2,-1): g(z) = sum binom(3k,k) z^k satisfies 27*w**3*z - 4*w**3 + 3*w + 1 = 0 (to order z^220), with G(y) = g(y^3)
TOTAL checks: 32, failures: 0; running time 52.1 s
ALL CHECKS PASSED
