== alternating 3/10, 7/10 (liminf 3/10)
   phases (k, M_k, N_k+1) within range: [(0, 2, 4), (1, 6, 18), (2, 40, 160), (3, 374, 1870), (4, 4364, 26184), (5, 61096, 427672)]
   increments of g in [0,1]: True ; |delta_A(n) - rho(n)| <= 1/n for all n: True
   all m            min over all = 0.0000 ; over tail m >= 374: min = 0.2997941916304596 max = 0.7
   m = 2^i          min over all = 0.0000 ; over tail m >= 374: min = 0.2999267578125 max = 0.6999969482421875
   m = 3^i          min over all = 0.0000 ; over tail m >= 374: min = 0.29995427526291724 max = 0.6995884773662552
   m = 5^i          min over all = 0.0000 ; over tail m >= 374: min = 0.299968 max = 0.6999936
   m = 2^i - 1      min over all = 0.0000 ; over tail m >= 374: min = 0.2999450650064091 max = 0.6999923704890516
   m = 3^i - 1      min over all = 0.3000 ; over tail m >= 374: min = 0.29996951529316124 max = 0.6991758241758241
   m = (3^i-1)/2    min over all = 0.0000 ; over tail m >= 374: min = 0.29996951529316124 max = 0.6999988709877728
   m = (5^i-1)/4    min over all = 0.0000 ; over tail m >= 374: min = 0.29998463980338946 max = 0.6999979519947571
== q_k = 1/2 + (-1)^k/(k+2) (liminf 1/2)
   phases (k, M_k, N_k+1) within range: [(0, 2, 4), (1, 6, 18), (2, 48, 192), (3, 480, 2400), (4, 5040, 30240), (5, 56448, 395136)]
   increments of g in [0,1]: True ; |delta_A(n) - rho(n)| <= 1/n for all n: True
   all m            min over all = 0.0000 ; over tail m >= 480: min = 0.2981366459627329 max = 0.6666666666666666
   m = 2^i          min over all = 0.0000 ; over tail m >= 480: min = 0.298828125 max = 0.6666259765625
   m = 3^i          min over all = 0.0000 ; over tail m >= 480: min = 0.299039780521262 max = 0.6666666666666666
   m = 5^i          min over all = 0.0000 ; over tail m >= 480: min = 0.2992 max = 0.666624
   m = 2^i - 1      min over all = 0.0000 ; over tail m >= 480: min = 0.2991202346041056 max = 0.6666666666666666
   m = 3^i - 1      min over all = 0.2500 ; over tail m >= 480: min = 0.29945054945054944 max = 0.6666497307184229
   m = (3^i-1)/2    min over all = 0.0000 ; over tail m >= 480: min = 0.2991765782250686 max = 0.6666440861671861
   m = (5^i-1)/4    min over all = 0.0000 ; over tail m >= 480: min = 0.29961587708066584 max = 0.6666325328964211
== q_k = 0 (liminf 0)
   phases (k, M_k, N_k+1) within range: [(0, 2, 4), (1, 6, 18), (2, 24, 96), (3, 120, 600), (4, 720, 4320), (5, 5040, 35280), (6, 40320, 322560)]
   increments of g in [0,1]: True ; |delta_A(n) - rho(n)| <= 1/n for all n: True
   all m            min over all = 0.0000 ; over tail m >= 720: min = 0.12497830237805936 max = 0.16666666666666666
   m = 2^i          min over all = 0.0000 ; over tail m >= 720: min = 0.125 max = 0.16650390625
   m = 3^i          min over all = 0.0000 ; over tail m >= 720: min = 0.12499788311402395 max = 0.16643804298125287
   m = 5^i          min over all = 0.0000 ; over tail m >= 720: min = 0.124992 max = 0.1664
   m = 2^i - 1      min over all = 0.0000 ; over tail m >= 720: min = 0.12498664835584039 max = 0.16658524670249145
   m = 3^i - 1      min over all = 0.1250 ; over tail m >= 720: min = 0.125 max = 0.16651418115279049
   m = (3^i-1)/2    min over all = 0.0000 ; over tail m >= 720: min = 0.12499294367357998 max = 0.16651418115279049
   m = (5^i-1)/4    min over all = 0.0000 ; over tail m >= 720: min = 0.125 max = 0.16666666666666666
== q_k = 1 (liminf 1)
   phases (k, M_k, N_k+1) within range: [(0, 2, 4), (1, 6, 18), (2, 24, 96), (3, 120, 600), (4, 720, 4320), (5, 5040, 35280), (6, 40320, 322560)]
   increments of g in [0,1]: True ; |delta_A(n) - rho(n)| <= 1/n for all n: True
   all m            min over all = 0.0000 ; over tail m >= 720: min = 0.8321775312066574 max = 0.875
   m = 2^i          min over all = 0.0000 ; over tail m >= 720: min = 0.8330078125 max = 0.875
   m = 3^i          min over all = 0.0000 ; over tail m >= 720: min = 0.8326474622770919 max = 0.8749851817981676
   m = 5^i          min over all = 0.0000 ; over tail m >= 720: min = 0.83328 max = 0.8749952
   m = 2^i - 1      min over all = 0.0000 ; over tail m >= 720: min = 0.8328445747800587 max = 0.8749980926222629
   m = 3^i - 1      min over all = 0.5000 ; over tail m >= 720: min = 0.8324175824175825 max = 0.875
   m = (3^i-1)/2    min over all = 0.0000 ; over tail m >= 720: min = 0.8325709057639524 max = 0.874995766204148
   m = (5^i-1)/4    min over all = 0.0000 ; over tail m >= 720: min = 0.8322663252240717 max = 0.875
== q_k = 1 - 1/(k+1) if k even else 1/(k+1) (liminf 0)
   phases (k, M_k, N_k+1) within range: [(0, 2, 4), (1, 5, 15), (2, 23, 92), (3, 246, 1230), (4, 4613, 27678)]
   increments of g in [0,1]: True ; |delta_A(n) - rho(n)| <= 1/n for all n: True
   all m            min over all = 0.0000 ; over tail m >= 23: min = 0.22142 max = 0.8
   m = 2^i          min over all = 0.0000 ; over tail m >= 23: min = 0.25 max = 0.79998779296875
   m = 3^i          min over all = 0.0000 ; over tail m >= 23: min = 0.2496570644718793 max = 0.7999796778946299
   m = 5^i          min over all = 0.0000 ; over tail m >= 23: min = 0.2496 max = 0.8
   m = 2^i - 1      min over all = 0.0000 ; over tail m >= 23: min = 0.24705882352941178 max = 0.7999755844472929
   m = 3^i - 1      min over all = 0.2500 ; over tail m >= 23: min = 0.25 max = 0.8
   m = (3^i-1)/2    min over all = 0.0000 ; over tail m >= 23: min = 0.24977127172918573 max = 0.7999187074484301
   m = (5^i-1)/4    min over all = 0.0000 ; over tail m >= 23: min = 0.22673466044073073 max = 0.7999590394757052
