# exact beta_1..beta_281 (23s)

== Observation 1, Table 1 (caption), Table 4 (upper part) ==
PASS  beta_1..beta_8 = 4, 2, 4/3, 7/6, 11/10, 414/385, 16868/15939, 102259729/97766928
PASS  all beta_k > 1 and decreasing for 2 <= k <= 281
   k =  10: beta_k = 1.0304820 (table 1.030482),  k(beta_k - 1) = 0.30482 (table 0.3048)  ok
   k =  20: beta_k = 1.0101770 (table 1.010177),  k(beta_k - 1) = 0.20354 (table 0.2035)  ok
   k =  50: beta_k = 1.0031056 (table 1.003106),  k(beta_k - 1) = 0.15528 (table 0.1553)  ok
   k = 100: beta_k = 1.0014666 (table 1.001467),  k(beta_k - 1) = 0.14666 (table 0.1467)  ok
   k = 150: beta_k = 1.0009468 (table 1.000947),  k(beta_k - 1) = 0.14201 (table 0.1420)  ok
   k = 200: beta_k = 1.0007018 (table 1.000702),  k(beta_k - 1) = 0.14035 (table 0.1404)  ok
   k = 240: beta_k = 1.0005790 (table 1.000579),  k(beta_k - 1) = 0.13895 (table 0.1389)  ok
   k = 280: beta_k = 1.0004931 (table 1.000493),  k(beta_k - 1) = 0.13808 (table 0.1381)  ok
PASS  Table 4, upper part: all 16 entries agree to the digits printed
PASS  k(beta_k - 1) is decreasing for 150 <= k <= 281 (still decreasing at the end of the data)   values at k = 150, 280, 281: 0.14201, 0.13808, 0.13806

== Observations 2 and 3, Table 4 (lower part): Gauss nodes of mu ==
   K = 100: nodes 2..5 = 2.0114856926, 2.001463, 1.99739, 1.99095; weight in [-2,2] = 0.49045; largest node 2.5858266060482, weight 0.2513316546384   ok
   K = 153: nodes 2..5 = 2.0114856947, 2.001744, 1.99998, 1.99749; weight in [-2,2] = 0.49066; largest node 2.5858266060482, weight 0.2513316546384   ok
   K = 200: nodes 2..5 = 2.0114856947, 2.001759, 2.00048, 1.99914; weight in [-2,2] = 0.49047; largest node 2.5858266060482, weight 0.2513316546384   ok
   K = 280: nodes 2..5 = 2.0114856947, 2.001760, 2.00065, 2.00002; weight in [-2,2] = 0.49041; largest node 2.5858266060482, weight 0.2513316546384   ok
            weights of nodes 2..5 at K = 280: 0.0031285, 0.000202, 0.000058, 0.000073   ok
PASS  Table 4, lower part: all entries agree to the digits printed
   K = 100: largest node = 2.5858266060482044486315152361168, weight = 0.251331654638419314530505674329;  node - c = 2.11e-50, weight - w = 1.77e-45;  inside the intervals of Theorem 1.2(d): True
   K = 153: largest node = 2.5858266060482044486315152361168, weight = 0.251331654638419314530505674329;  node - c = 2.11e-50, weight - w = 1.77e-45;  inside the intervals of Theorem 1.2(d): True
   K = 200: largest node = 2.5858266060482044486315152361168, weight = 0.251331654638419314530505674329;  node - c = 2.11e-50, weight - w = 1.77e-45;  inside the intervals of Theorem 1.2(d): True
   K = 280: largest node = 2.5858266060482044486315152361168, weight = 0.251331654638419314530505674329;  node - c = 2.11e-50, weight - w = 1.77e-45;  inside the intervals of Theorem 1.2(d): True
PASS  caption of Table 4: for K = 100, 153, 200, 280 the largest node and its weight lie in the intervals of Theorem 1.2(d) (so they agree with c and w to the digits shown there)
   second node: K = 280: 2.011485694658159583435, weight 0.0031285045344519;  K = 200: 2.011485694658159580793, weight 0.0031285045344519
PASS  Observation 2: c_2 = 2.011485694658159583, w_2 = 0.0031285045345 (the values at K = 280)
      second node at K = 153: 2.01148569465811608550203   weight 0.003128504534644972
      second node at K = 200: 2.01148569465815958079342   weight 0.003128504534451935
      second node at K = 240: 2.01148569465815958343431   weight 0.00312850453445192
      second node at K = 260: 2.01148569465815958343494   weight 0.00312850453445192
      second node at K = 270: 2.01148569465815958343494   weight 0.00312850453445192
      second node at K = 280: 2.01148569465815958343495   weight 0.00312850453445192
      second node at K = 281: 2.01148569465815958343495   weight 0.00312850453445192
   third node (K = 280): 2.00175971727, weight 0.000201584;  fourth: 2.000652;  fifth: 2.000020
PASS  Observation 2: third node near 2.0017597 with weight about 2.016e-4
PASS  Observation 3: total weight of the nodes in [-2,2] between 0.4904 and 0.4907 for the four K of Table 4   min 0.49041, max 0.49066
      over all K = 100..281: min 0.49028 (K = 154), max 0.49066 (K = 153)
PASS  ... and between 0.4902 and 0.4907 for every K from 100 to 281 (the note's range 0.4904..0.4907 holds for the four K of Table 4, not for all K)
      K in 100..281 with weight in [-2,2] below 0.4904: [154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176]
PASS  Question (2): the sum of the weights 2 w_k of the nodes above 2 (K = 280) is about 0.5096   0.50959

== Observation 3: rough density (continued fraction cut after K levels, tail beta = 1) ==
   x = 0: density between 0.1320 and 0.1321 for K = 100, 120, .., 280 (note: 0.132)  ok
   x = 0.5: density between 0.1297 and 0.1298 for K = 100, 120, .., 280 (note: 0.13)  ok
   x = 1: density between 0.1253 and 0.1255 for K = 100, 120, .., 280 (note: 0.125)  ok
   x = 1.5: density between 0.1191 and 0.1193 for K = 100, 120, .., 280 (note: 0.119)  ok
   x = 1.75: density between 0.1099 and 0.1104 for K = 100, 120, .., 280 (note: 0.11)  ok
   x = 1.9: density between 0.1013 and 0.1016 for K = 100, 120, .., 280 (note: 0.101)  ok
PASS  Observation 3: the six density values, for K between 100 and 280

== Observation 4: the largest Gauss node of nu ==
   K = 100: 2.10199076046602191987523  weight 0.09980958798055
   K = 153: 2.10199076046602191987523  weight 0.09980958798055
   K = 200: 2.10199076046602191987523  weight 0.09980958798055
   K = 280: 2.10199076046602191987523  weight 0.09980958798055
PASS  Observation 4: 2.101990760466021919875 with weight 0.0998095879805 for K = 100, 153, 200, 280
PASS  it is below sqrt 6 and above the proved lower bound 2.1019907

== Question (3): the heuristic fit ==
   alpha = 0.0689, k_0 = 1.357, predicted c_4 = 2.00068
PASS  alpha = 0.069, k_0 = 1.36, c_4 = 2.0007

== Section 1.3 and Remark 8.3: measures with Jacobi parameters (theta, 1, 1, ...) ==
PASS  exact: zero of z - theta G_sc(z) at theta/sqrt(theta-1) and residue (theta-2)/(2(theta-1)) (five rational cases)
   theta = 2.5: largest eigenvalue 2.0412414523 (formula 2.0412414523), weight 0.1666666667 (formula 0.1666666667); second eigenvalue 1.999999
   theta = 3.0: largest eigenvalue 2.1213203436 (formula 2.1213203436), weight 0.2500000000 (formula 0.2500000000); second eigenvalue 1.999999
   theta = 4.0: largest eigenvalue 2.3094010768 (formula 2.3094010768), weight 0.3333333333 (formula 0.3333333333); second eigenvalue 1.999999
   theta = 7.0: largest eigenvalue 2.8577380332 (formula 2.8577380332), weight 0.4166666667 (formula 0.4166666667); second eigenvalue 1.999999
PASS  numerical: Jacobi matrices (theta, 1, 1, ...) of size 4000: one eigenvalue above 2, at theta/sqrt(theta-1), with weight (theta-2)/(2(theta-1))
PASS  theta = 4: atoms at 4/sqrt 3 = 2.3094 with mass 1/3
PASS  Remark 8.3: <(x1+x3)^(2n)> is the n-th Catalan number for n <= 30 (exact)
PASS  Remark 8.3: <x1^a x3^b> equals the moment of the uniform distribution on the disc of radius sqrt 2 for a, b <= 20 (exact)
TOTAL failures: 0   (42s)
