Generating functions and Growth rates for the Enumerating Sequences for Como\ sitions Avoiding Each of the compositions of, 1 By Shalosh B. Ekhad The compositions of, 1, that yield polynomial growth (hence growth ratio 1, that is the lowest) are , [1] Theorem Number, 1, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = 1 / ----- n = 0 and in Maple format 1 The first, 31, terms of a(n), starting at n=0, are [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ---------------------------------------------------------------------------- This ends this article, that took, 0.028, seconds to generate. ---------------------------------------------------------- ---------------------------------------- Generating functions and Growth rates for the Enumerating Sequences for Como\ sitions Avoiding Each of the compositions of, 2 By Shalosh B. Ekhad The compositions of, 2, that yield polynomial growth (hence growth ratio 1, that is the lowest) are , [2], [1, 1] Theorem Number, 1, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 1 ) a(n) x = - ------ / -1 + x ----- n = 0 and in Maple format -1/(-1+x) The first, 31, terms of a(n), starting at n=0, are [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] The limit of a(n+1)/a(n) as n goes to infinity is 1. a(n) is asymptotic to 1.*1.^n ---------------------------------------------------------------------------- This ends this article, that took, 0.019, seconds to generate. ---------------------------------------------------------- ---------------------------------------- Generating functions and Growth rates for the Enumerating Sequences for Como\ sitions Avoiding Each of the compositions of, 3 By Shalosh B. Ekhad The compositions of, 3, that yield polynomial growth (hence growth ratio 1, that is the lowest) are , [1, 2], [2, 1], [1, 1, 1] Theorem Number, 1, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 2 \ n x - x + 1 ) a(n) x = ---------- / 2 ----- (-1 + x) n = 0 and in Maple format (x^2-x+1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30] ---------------------------------------------------------------------------- The compositions of, 3, that yield the, 2, -th largest growth, that is, 1.6180339887498948482, are , [3] Theorem Number, 2, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 1 ) a(n) x = - ---------- / 2 ----- x + x - 1 n = 0 and in Maple format -1/(x^2+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269] The limit of a(n+1)/a(n) as n goes to infinity is 1.61803398875 a(n) is asymptotic to .723606797750*1.61803398875^n ---------------------------------------------------------------------------- This ends this article, that took, 0.028, seconds to generate. ---------------------------------------------------------- ---------------------------------------- Generating functions and Growth rates for the Enumerating Sequences for Como\ sitions Avoiding Each of the compositions of, 4 By Shalosh B. Ekhad The compositions of, 4, that yield polynomial growth (hence growth ratio 1, that is the lowest) are , [1, 1, 2], [1, 2, 1], [2, 1, 1], [1, 1, 1, 1] Theorem Number, 1, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 2 \ n 2 x - 2 x + 1 ) a(n) x = - -------------- / 3 ----- (-1 + x) n = 0 and in Maple format -(2*x^2-2*x+1)/(-1+x)^3 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, 232, 254, 277, 301, 326, 352, 379, 407, 436] ---------------------------------------------------------------------------- The compositions of, 4, that yield the, 2, -th largest growth, that is, 1.6180339887498948482, are , [1, 3], [3, 1] Theorem Number, 2, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 3 \ n x - x + 1 ) a(n) x = --------------------- / 2 ----- (-1 + x) (x + x - 1) n = 0 and in Maple format (x^3-x+1)/(-1+x)/(x^2+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 7, 12, 20, 33, 54, 88, 143, 232, 376, 609, 986, 1596, 2583, 4180, 6764, 10945, 17710, 28656, 46367, 75024, 121392, 196417, 317810, 514228, 832039, 1346268, 2178308] The limit of a(n+1)/a(n) as n goes to infinity is 1.61803398875 a(n) is asymptotic to 1.17082039325*1.61803398875^n ---------------------------------------------------------------------------- The compositions of, 4, that yield the, 3, -th largest growth, that is, 1.7548776662466927601, are , [2, 2] Theorem Number, 3, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 2 \ n x - x + 1 ) a(n) x = - ----------------- / 3 2 ----- x - x + 2 x - 1 n = 0 and in Maple format -(x^2-x+1)/(x^3-x^2+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 7, 12, 21, 37, 65, 114, 200, 351, 616, 1081, 1897, 3329, 5842, 10252, 17991, 31572, 55405, 97229, 170625, 299426, 525456, 922111, 1618192, 2839729, 4983377, 8745217, 15346786] The limit of a(n+1)/a(n) as n goes to infinity is 1.75487766625 a(n) is asymptotic to .722124418303*1.75487766625^n ---------------------------------------------------------------------------- The compositions of, 4, that yield the, 4, -th largest growth, that is, 1.8392867552141611326, are , [4] Theorem Number, 4, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 1 ) a(n) x = - --------------- / 3 2 ----- x + x + x - 1 n = 0 and in Maple format -1/(x^3+x^2+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, 5768, 10609, 19513, 35890, 66012, 121415, 223317, 410744, 755476, 1389537, 2555757, 4700770, 8646064, 15902591, 29249425, 53798080] The limit of a(n+1)/a(n) as n goes to infinity is 1.83928675521 a(n) is asymptotic to .618419922319*1.83928675521^n ---------------------------------------------------------------------------- This ends this article, that took, 0.056, seconds to generate. ---------------------------------------------------------- ---------------------------------------- Generating functions and Growth rates for the Enumerating Sequences for Como\ sitions Avoiding Each of the compositions of, 5 By Shalosh B. Ekhad The compositions of, 5, that yield polynomial growth (hence growth ratio 1, that is the lowest) are , [1, 1, 1, 2], [1, 1, 2, 1], [1, 2, 1, 1], [2, 1, 1, 1], [1, 1, 1, 1, 1] Theorem Number, 1, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 4 3 2 \ n x - 2 x + 4 x - 3 x + 1 ) a(n) x = -------------------------- / 4 ----- (-1 + x) n = 0 and in Maple format (x^4-2*x^3+4*x^2-3*x+1)/(-1+x)^4 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 15, 26, 42, 64, 93, 130, 176, 232, 299, 378, 470, 576, 697, 834, 988, 1160, 1351, 1562, 1794, 2048, 2325, 2626, 2952, 3304, 3683, 4090] ---------------------------------------------------------------------------- The compositions of, 5, that yield the, 2, -th largest growth, that is, 1.6180339887498948482, are , [1, 1, 3], [1, 3, 1], [3, 1, 1] Theorem Number, 2, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 3 2 \ n x + x - 2 x + 1 ) a(n) x = - ---------------------- / 2 2 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format -(x^3+x^2-2*x+1)/(x^2+x-1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 15, 27, 47, 80, 134, 222, 365, 597, 973, 1582, 2568, 4164, 6747, 10927, 17691, 28636, 46346, 75002, 121369, 196393, 317785, 514202, 832012, 1346240, 2178279, 3524547] The limit of a(n+1)/a(n) as n goes to infinity is 1.61803398875 a(n) is asymptotic to 1.89442719100*1.61803398875^n ---------------------------------------------------------------------------- The compositions of, 5, that yield the, 3, -th largest growth, that is, 1.7548776662466927601, are , [2, 1, 2] Theorem Number, 3, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 5 2 \ n x + 2 x - 2 x + 1 ) a(n) x = - -------------------------------- / 3 3 2 ----- (x - x + 1) (x - x + 2 x - 1) n = 0 and in Maple format -(x^5+2*x^2-2*x+1)/(x^3-x+1)/(x^3-x^2+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 15, 27, 47, 81, 140, 244, 428, 753, 1325, 2329, 4089, 7174, 12584, 22076, 38735, 67975, 119295, 209361, 367416, 644776, 1131496, 1985617, 3484489, 6114833, 10730785, 18831242] The limit of a(n+1)/a(n) as n goes to infinity is 1.75487766625 a(n) is asymptotic to .886082866099*1.75487766625^n ---------------------------------------------------------------------------- The compositions of, 5, that yield the, 4, -th largest growth, that is, 1.8392867552141611326, are , [1, 4], [4, 1] Theorem Number, 4, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 4 \ n x - x + 1 ) a(n) x = -------------------------- / 3 2 ----- (-1 + x) (x + x + x - 1) n = 0 and in Maple format (x^4-x+1)/(-1+x)/(x^3+x^2+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 15, 28, 52, 96, 177, 326, 600, 1104, 2031, 3736, 6872, 12640, 23249, 42762, 78652, 144664, 266079, 489396, 900140, 1655616, 3045153, 5600910, 10301680, 18947744, 34850335, 64099760] The limit of a(n+1)/a(n) as n goes to infinity is 1.83928675521 a(n) is asymptotic to .736839844639*1.83928675521^n ---------------------------------------------------------------------------- The compositions of, 5, that yield the, 5, -th largest growth, that is, 1.8667603991738620930, are , [2, 3], [3, 2] Theorem Number, 5, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 3 \ n x - x + 1 ) a(n) x = - ----------------- / 4 3 ----- x - x + 2 x - 1 n = 0 and in Maple format -(x^3-x+1)/(x^4-x^3+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 15, 28, 52, 97, 181, 338, 631, 1178, 2199, 4105, 7663, 14305, 26704, 49850, 93058, 173717, 324288, 605368, 1130077, 2109583, 3938086, 7351463, 13723420, 25618337, 47823297, 89274637] The limit of a(n+1)/a(n) as n goes to infinity is 1.86676039917 a(n) is asymptotic to .657764927635*1.86676039917^n ---------------------------------------------------------------------------- The compositions of, 5, that yield the, 6, -th largest growth, that is, 1.9275619754829253043, are , [5] Theorem Number, 6, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 1 ) a(n) x = - -------------------- / 4 3 2 ----- x + x + x + x - 1 n = 0 and in Maple format -1/(x^4+x^3+x^2+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, 2872, 5536, 10671, 20569, 39648, 76424, 147312, 283953, 547337, 1055026, 2033628, 3919944, 7555935, 14564533, 28074040, 54114452, 104308960, 201061985] The limit of a(n+1)/a(n) as n goes to infinity is 1.92756197548 a(n) is asymptotic to .566342887703*1.92756197548^n ---------------------------------------------------------------------------- This ends this article, that took, 0.075, seconds to generate. ---------------------------------------------------------- ---------------------------------------- Generating functions and Growth rates for the Enumerating Sequences for Como\ sitions Avoiding Each of the compositions of, 6 By Shalosh B. Ekhad The compositions of, 6, that yield polynomial growth (hence growth ratio 1, that is the lowest) are , [1, 1, 1, 1, 2], [1, 1, 1, 2, 1], [1, 1, 2, 1, 1], [1, 2, 1, 1, 1], [2, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1] Theorem Number, 1, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 2 2 \ n (x - x + 1) (3 x - 3 x + 1) ) a(n) x = - ----------------------------- / 5 ----- (-1 + x) n = 0 and in Maple format -(x^2-x+1)*(3*x^2-3*x+1)/(-1+x)^5 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 31, 57, 99, 163, 256, 386, 562, 794, 1093, 1471, 1941, 2517, 3214, 4048, 5036, 6196, 7547, 9109, 10903, 12951, 15276, 17902, 20854, 24158, 27841] ---------------------------------------------------------------------------- The compositions of, 6, that yield the, 2, -th largest growth, that is, 1.6180339887498948482, are , [1, 1, 1, 3], [1, 1, 3, 1], [1, 3, 1, 1], [3, 1, 1, 1] Theorem Number, 2, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 5 4 2 \ n x - x + 3 x - 3 x + 1 ) a(n) x = ------------------------ / 2 3 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format (x^5-x^4+3*x^2-3*x+1)/(x^2+x-1)/(-1+x)^3 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 31, 58, 105, 185, 319, 541, 906, 1503, 2476, 4058, 6626, 10790, 17537, 28464, 46155, 74791, 121137, 196139, 317508, 513901, 831686, 1345888, 2177900, 3524140, 5702419] The limit of a(n+1)/a(n) as n goes to infinity is 1.61803398875 a(n) is asymptotic to 3.06524758425*1.61803398875^n ---------------------------------------------------------------------------- The compositions of, 6, that yield the, 3, -th largest growth, that is, 1.7548776662466927601, are , [2, 1, 1, 2] Theorem Number, 3, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 11 9 8 7 6 5 4 3 2 x + 2 x - x + 3 x - 4 x + 5 x - 6 x + 7 x - 7 x + 4 x - 1 - ------------------------------------------------------------------- 3 2 9 6 5 4 3 2 (x - x + 2 x - 1) (x - x + 2 x - x + x - 3 x + 3 x - 1) and in Maple format -(x^11+2*x^9-x^8+3*x^7-4*x^6+5*x^5-6*x^4+7*x^3-7*x^2+4*x-1)/(x^3-x^2+2*x-1)/(x^ 9-x^6+2*x^5-x^4+x^3-3*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 31, 58, 105, 185, 320, 549, 942, 1625, 2824, 4941, 8686, 15306, 26983, 47529, 83604, 146856, 257686, 451873, 792225, 1389052, 2436112, 4273686, 7499219, 13161377, 23100431] The limit of a(n+1)/a(n) as n goes to infinity is 1.75487766625 a(n) is asymptotic to 1.08726810186*1.75487766625^n ---------------------------------------------------------------------------- The compositions of, 6, that yield the, 4, -th largest growth, that is, 1.8392867552141611326, are , [1, 1, 4], [1, 4, 1], [4, 1, 1] Theorem Number, 4, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 4 2 \ n x + x - 2 x + 1 ) a(n) x = - --------------------------- / 3 2 2 ----- (x + x + x - 1) (-1 + x) n = 0 and in Maple format -(x^4+x^2-2*x+1)/(x^3+x^2+x-1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 31, 59, 111, 207, 384, 710, 1310, 2414, 4445, 8181, 15053, 27693, 50942, 93704, 172356, 317020, 583099, 1072495, 1972635, 3628251, 6673404, 12274314, 22575994, 41523738, 76374073] The limit of a(n+1)/a(n) as n goes to infinity is 1.83928675521 a(n) is asymptotic to .877935747301*1.83928675521^n ---------------------------------------------------------------------------- The compositions of, 6, that yield the, 5, -th largest growth, that is, 1.8667603991738620930, are , [2, 1, 3], [3, 1, 2] Theorem Number, 5, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 7 3 2 \ n x + x + x - 2 x + 1 ) a(n) x = - -------------------------------- / 4 4 3 ----- (x - x + 1) (x - x + 2 x - 1) n = 0 and in Maple format -(x^7+x^3+x^2-2*x+1)/(x^4-x+1)/(x^4-x^3+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 31, 59, 111, 207, 385, 716, 1333, 2485, 4637, 8657, 16165, 30184, 56356, 105212, 196410, 366647, 684429, 1277643, 2385027, 4452251, 8311276, 15515173, 28963149, 54067321, 100930805] The limit of a(n+1)/a(n) as n goes to infinity is 1.86676039917 a(n) is asymptotic to .743646261889*1.86676039917^n ---------------------------------------------------------------------------- The compositions of, 6, that yield the, 6, -th largest growth, that is, 1.9087907387871591034, are , [2, 2, 2] Theorem Number, 6, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 4 3 2 \ n x - x + 2 x - 2 x + 1 ) a(n) x = - ------------------------------- / 5 4 3 2 ----- x - x + 2 x - 3 x + 3 x - 1 n = 0 and in Maple format -(x^4-x^3+2*x^2-2*x+1)/(x^5-x^4+2*x^3-3*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 31, 59, 112, 213, 406, 775, 1480, 2826, 5395, 10298, 19656, 37518, 71613, 136694, 260921, 498045, 950665, 1814621, 3463731, 6611536, 12620037, 24089009, 45980878, 87767876, 167530511] The limit of a(n+1)/a(n) as n goes to infinity is 1.90879073879 a(n) is asymptotic to .632913467972*1.90879073879^n ---------------------------------------------------------------------------- The compositions of, 6, that yield the, 7, -th largest growth, that is, 1.9275619754829253043, are , [1, 5], [5, 1] Theorem Number, 7, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 5 \ n x - x + 1 ) a(n) x = ------------------------------- / 4 3 2 ----- (-1 + x) (x + x + x + x - 1) n = 0 and in Maple format (x^5-x+1)/(-1+x)/(x^4+x^3+x^2+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 31, 60, 116, 224, 432, 833, 1606, 3096, 5968, 11504, 22175, 42744, 82392, 158816, 306128, 590081, 1137418, 2192444, 4226072, 8146016, 15701951, 30266484, 58340524, 112454976, 216763936] The limit of a(n+1)/a(n) as n goes to infinity is 1.92756197548 a(n) is asymptotic to .610571479504*1.92756197548^n ---------------------------------------------------------------------------- The compositions of, 6, that yield the, 8, -th largest growth, that is, 1.9331849818995204468, are , [2, 4], [4, 2] Theorem Number, 8, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 4 \ n x - x + 1 ) a(n) x = - ----------------- / 5 4 ----- x - x + 2 x - 1 n = 0 and in Maple format -(x^4-x+1)/(x^5-x^4+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 31, 60, 116, 224, 433, 837, 1618, 3128, 6047, 11690, 22599, 43688, 84457, 163271, 315633, 610177, 1179585, 2280356, 4408350, 8522156, 16474904, 31849037, 61570080, 119026354, 230099960] The limit of a(n+1)/a(n) as n goes to infinity is 1.93318498190 a(n) is asymptotic to .593901147371*1.93318498190^n ---------------------------------------------------------------------------- The compositions of, 6, that yield the, 9, -th largest growth, that is, 1.9417130342786384772, are , [3, 3] Theorem Number, 9, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 3 \ n x - x + 1 ) a(n) x = - ---------------------- / 5 4 3 ----- x + x - x + 2 x - 1 n = 0 and in Maple format -(x^3-x+1)/(x^5+x^4-x^3+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 31, 60, 116, 225, 437, 849, 1649, 3202, 6217, 12071, 23438, 45510, 88368, 171586, 333171, 646922, 1256136, 2439055, 4735945, 9195847, 17855697, 34670640, 67320433, 130716961, 253814826] The limit of a(n+1)/a(n) as n goes to infinity is 1.94171303428 a(n) is asymptotic to .574071001256*1.94171303428^n ---------------------------------------------------------------------------- The compositions of, 6, that yield the, 10, -th largest growth, that is, 1.9659482366454853372, are , [6] Theorem Number, 10, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [6] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 1 ) a(n) x = - ------------------------- / 5 4 3 2 ----- x + x + x + x + x - 1 n = 0 and in Maple format -1/(x^5+x^4+x^3+x^2+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525, 6930, 13624, 26784, 52656, 103519, 203513, 400096, 786568, 1546352, 3040048, 5976577, 11749641, 23099186, 45411804, 89277256, 175514464, 345052351] The limit of a(n+1)/a(n) as n goes to infinity is 1.96594823665 a(n) is asymptotic to .537926116819*1.96594823665^n ---------------------------------------------------------------------------- This ends this article, that took, 0.213, seconds to generate. ---------------------------------------------------------- ---------------------------------------- Generating functions and Growth rates for the Enumerating Sequences for Como\ sitions Avoiding Each of the compositions of, 7 By Shalosh B. Ekhad The compositions of, 7, that yield polynomial growth (hence growth ratio 1, that is the lowest) are , [1, 1, 1, 1, 1, 2], [1, 1, 1, 1, 2, 1], [1, 1, 1, 2, 1, 1], [1, 1, 2, 1, 1, 1], [1, 2, 1, 1, 1, 1], [2, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1] Theorem Number, 1, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 1, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 6 5 4 3 2 \ n x - 3 x + 9 x - 13 x + 11 x - 5 x + 1 ) a(n) x = ------------------------------------------ / 6 ----- (-1 + x) n = 0 and in Maple format (x^6-3*x^5+9*x^4-13*x^3+11*x^2-5*x+1)/(-1+x)^6 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 63, 120, 219, 382, 638, 1024, 1586, 2380, 3473, 4944, 6885, 9402, 12616, 16664, 21700, 27896, 35443, 44552, 55455, 68406, 83682, 101584, 122438, 146596] ---------------------------------------------------------------------------- The compositions of, 7, that yield the, 2, -th largest growth, that is, 1.6180339887498948482, are , [1, 1, 1, 1, 3], [1, 1, 1, 3, 1], [1, 1, 3, 1, 1], [1, 3, 1, 1, 1], [3, 1, 1, 1, 1] Theorem Number, 2, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 5 4 3 2 \ n 2 x - x - 3 x + 6 x - 4 x + 1 ) a(n) x = - --------------------------------- / 2 4 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format -(2*x^5-x^4-3*x^3+6*x^2-4*x+1)/(x^2+x-1)/(-1+x)^4 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 63, 121, 226, 411, 730, 1271, 2177, 3680, 6156, 10214, 16840, 27630, 45167, 73631, 119786, 194577, 315714, 511853, 829361, 1343262, 2174948, 3520836, 5698736, 9222876] The limit of a(n+1)/a(n) as n goes to infinity is 1.61803398875 a(n) is asymptotic to 4.95967477525*1.61803398875^n ---------------------------------------------------------------------------- The compositions of, 7, that yield the, 3, -th largest growth, that is, 1.7548776662466927601, are , [2, 1, 1, 1, 2] Theorem Number, 3, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 20 18 17 15 14 13 12 11 ) a(n) x = - (x + 2 x + x + 2 x - 3 x + 3 x - 3 x - x / ----- n = 0 10 9 8 7 6 5 4 3 2 + 6 x - 10 x + 9 x + 3 x - 24 x + 44 x - 51 x + 41 x - 22 x / 3 3 2 + 7 x - 1) / ((x - x + 1) (x - x + 2 x - 1) / 12 8 7 6 5 4 3 2 3 (x + x - x - x + x + x - 4 x + 6 x - 4 x + 1) (x + x - 1)) and in Maple format -(x^20+2*x^18+x^17+2*x^15-3*x^14+3*x^13-3*x^12-x^11+6*x^10-10*x^9+9*x^8+3*x^7-\ 24*x^6+44*x^5-51*x^4+41*x^3-22*x^2+7*x-1)/(x^3-x+1)/(x^3-x^2+2*x-1)/(x^12+x^8-x ^7-x^6+x^5+x^4-4*x^3+6*x^2-4*x+1)/(x^3+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 63, 121, 226, 411, 730, 1272, 2187, 3734, 6368, 10897, 18764, 32549, 56855, 99863, 176050, 310927, 549289, 969586, 1708992, 3007219, 5283242, 9269885, 16250201, 28472951] The limit of a(n+1)/a(n) as n goes to infinity is 1.75487766625 a(n) is asymptotic to 1.33413247287*1.75487766625^n ---------------------------------------------------------------------------- The compositions of, 7, that yield the, 4, -th largest growth, that is, 1.8392867552141611326, are , [1, 1, 1, 4], [1, 1, 4, 1], [1, 4, 1, 1], [4, 1, 1, 1] Theorem Number, 4, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 6 5 4 3 2 \ n x - x + x - x + 3 x - 3 x + 1 ) a(n) x = ---------------------------------- / 3 2 3 ----- (x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^6-x^5+x^4-x^3+3*x^2-3*x+1)/(x^3+x^2+x-1)/(-1+x)^3 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 63, 122, 233, 440, 824, 1534, 2844, 5258, 9703, 17884, 32937, 60630, 111572, 205276, 377632, 694652, 1277751, 2350246, 4322881, 7951132, 14624536, 26898850, 49474844, 90998582] The limit of a(n+1)/a(n) as n goes to infinity is 1.83928675521 a(n) is asymptotic to 1.04604980580*1.83928675521^n ---------------------------------------------------------------------------- The compositions of, 7, that yield the, 5, -th largest growth, that is, 1.8667603991738620930, are , [2, 1, 1, 3], [3, 1, 1, 2] Theorem Number, 5, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 3 12 10 9 7 6 5 3 2 (x - x + 1) (x + x + x + x + x - 2 x + x - 3 x + 3 x - 1) - -------------------------------------------------------------------- 4 3 12 8 7 6 5 3 2 (x - x + 2 x - 1) (x - x + x + x - x + x - 3 x + 3 x - 1) and in Maple format -(x^3-x+1)*(x^12+x^10+x^9+x^7+x^6-2*x^5+x^3-3*x^2+3*x-1)/(x^4-x^3+2*x-1)/(x^12- x^8+x^7+x^6-x^5+x^3-3*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 63, 122, 233, 440, 824, 1535, 2852, 5296, 9841, 18309, 34108, 63610, 118722, 221683, 414015, 773226, 1443982, 2696289, 5034099, 9398052, 17543959, 32749370, 61132622, 114115265] The limit of a(n+1)/a(n) as n goes to infinity is 1.86676039917 a(n) is asymptotic to .840740726039*1.86676039917^n ---------------------------------------------------------------------------- The compositions of, 7, that yield the, 6, -th largest growth, that is, 1.8908049048980778372, are , [2, 1, 2, 2], [2, 2, 1, 2] Theorem Number, 6, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 2 5 4 2 \ n (x - x + 1) (x + x + x - 2 x + 1) ) a(n) x = - ------------------------------------------------ / 8 7 6 5 4 3 2 ----- x - x + x + x - 3 x + 5 x - 6 x + 4 x - 1 n = 0 and in Maple format -(x^2-x+1)*(x^5+x^4+x^2-2*x+1)/(x^8-x^7+x^6+x^5-3*x^4+5*x^3-6*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 63, 122, 233, 441, 831, 1564, 2945, 5552, 10480, 19802, 37440, 70811, 133938, 253331, 479106, 906009, 1713170, 3239274, 6124701, 11580293, 21895548, 41399521, 78277699, 148007345] The limit of a(n+1)/a(n) as n goes to infinity is 1.89080490490 a(n) is asymptotic to .742822192761*1.89080490490^n ---------------------------------------------------------------------------- The compositions of, 7, that yield the, 7, -th largest growth, that is, 1.9087907387871591034, are , [1, 2, 2, 2], [2, 2, 2, 1] Theorem Number, 7, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 2 4 2 \ n (x - x + 1) (x + x - 2 x + 1) ) a(n) x = ------------------------------------------ / 5 4 3 2 ----- (-1 + x) (x - x + 2 x - 3 x + 3 x - 1) n = 0 and in Maple format (x^2-x+1)*(x^4+x^2-2*x+1)/(-1+x)/(x^5-x^4+2*x^3-3*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 63, 122, 234, 447, 853, 1628, 3108, 5934, 11329, 21627, 41283, 78801, 150414, 287108, 548029, 1046074, 1996739, 3811360, 7275091, 13886627, 26506664, 50595673, 96576551, 184344427] The limit of a(n+1)/a(n) as n goes to infinity is 1.90879073879 a(n) is asymptotic to .696434768709*1.90879073879^n ---------------------------------------------------------------------------- The compositions of, 7, that yield the, 8, -th largest growth, that is, 1.9275619754829253043, are , [1, 1, 5], [1, 5, 1], [5, 1, 1] Theorem Number, 8, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 5 2 \ n x + x - 2 x + 1 ) a(n) x = - -------------------------------- / 4 3 2 2 ----- (x + x + x + x - 1) (-1 + x) n = 0 and in Maple format -(x^5+x^2-2*x+1)/(x^4+x^3+x^2+x-1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 63, 123, 239, 463, 895, 1728, 3334, 6430, 12398, 23902, 46077, 88821, 171213, 330029, 636157, 1226238, 2363656, 4556100, 8782172, 16928188, 32630139, 62896623, 121237147, 233692123] The limit of a(n+1)/a(n) as n goes to infinity is 1.92756197548 a(n) is asymptotic to .658254106618*1.92756197548^n ---------------------------------------------------------------------------- The compositions of, 7, that yield the, 9, -th largest growth, that is, 1.9331849818995204468, are , [2, 1, 4], [4, 1, 2] Theorem Number, 9, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 9 4 2 \ n x + x + x - 2 x + 1 ) a(n) x = - -------------------------------- / 5 5 4 ----- (x - x + 1) (x - x + 2 x - 1) n = 0 and in Maple format -(x^9+x^4+x^2-2*x+1)/(x^5-x+1)/(x^5-x^4+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 63, 123, 239, 463, 895, 1729, 3340, 6453, 12470, 24102, 46590, 90066, 174117, 336608, 650738, 1258013, 2431989, 4701499, 9088874, 17570469, 33966945, 65664349, 126941280, 245400916] The limit of a(n+1)/a(n) as n goes to infinity is 1.93318498190 a(n) is asymptotic to .633393735410*1.93318498190^n ---------------------------------------------------------------------------- The compositions of, 7, that yield the, 10, -th largest growth, that is, 1.9417130342786384772, are , [3, 1, 3] Theorem Number, 10, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 8 7 3 2 \ n x + x + x + x - 2 x + 1 ) a(n) x = - ------------------------------------------ / 5 4 5 4 3 ----- (x + x - x + 1) (x + x - x + 2 x - 1) n = 0 and in Maple format -(x^8+x^7+x^3+x^2-2*x+1)/(x^5+x^4-x+1)/(x^5+x^4-x^3+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 63, 123, 239, 463, 896, 1735, 3363, 6525, 12669, 24607, 47798, 92839, 180301, 350120, 679834, 1319997, 2562947, 4976338, 9662422, 18761496, 36429440, 70735795, 137349260, 266693847] The limit of a(n+1)/a(n) as n goes to infinity is 1.94171303428 a(n) is asymptotic to .603201328844*1.94171303428^n ---------------------------------------------------------------------------- The compositions of, 7, that yield the, 11, -th largest growth, that is, 1.9454365275632690792, are , [2, 2, 3], [2, 3, 2], [3, 2, 2] Theorem Number, 11, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 5 4 3 2 \ n x - x + x + x - 2 x + 1 ) a(n) x = - ------------------------------------ / 6 5 4 3 2 ----- x - x + 2 x - x - 2 x + 3 x - 1 n = 0 and in Maple format -(x^5-x^4+x^3+x^2-2*x+1)/(x^6-x^5+2*x^4-x^3-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 63, 123, 239, 464, 901, 1751, 3405, 6624, 12888, 25076, 48788, 94918, 184659, 359241, 698875, 1359608, 2645021, 5145713, 10010657, 19475106, 37887600, 73707944, 143394148, 278964224] The limit of a(n+1)/a(n) as n goes to infinity is 1.94543652756 a(n) is asymptotic to .595712202984*1.94543652756^n ---------------------------------------------------------------------------- The compositions of, 7, that yield the, 12, -th largest growth, that is, 1.9659482366454853372, are , [1, 6], [6, 1] Theorem Number, 12, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 6] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 6 \ n x - x + 1 ) a(n) x = ------------------------------------ / 5 4 3 2 ----- (-1 + x) (x + x + x + x + x - 1) n = 0 and in Maple format (x^6-x+1)/(-1+x)/(x^5+x^4+x^3+x^2+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 63, 124, 244, 480, 944, 1856, 3649, 7174, 14104, 27728, 54512, 107168, 210687, 414200, 814296, 1600864, 3147216, 6187264, 12163841, 23913482, 47012668, 92424472, 181701728, 357216192] The limit of a(n+1)/a(n) as n goes to infinity is 1.96594823665 a(n) is asymptotic to .556889175229*1.96594823665^n ---------------------------------------------------------------------------- The compositions of, 7, that yield the, 13, -th largest growth, that is, 1.9671682128139660358, are , [2, 5], [5, 2] Theorem Number, 13, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 5 \ n x - x + 1 ) a(n) x = - ----------------- / 6 5 ----- x - x + 2 x - 1 n = 0 and in Maple format -(x^5-x+1)/(x^6-x^5+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 63, 124, 244, 480, 944, 1857, 3653, 7186, 14136, 27808, 54703, 107610, 211687, 416424, 819176, 1611457, 3170007, 6235937, 12267137, 24131522, 47470763, 93382976, 183700022, 361368844] The limit of a(n+1)/a(n) as n goes to infinity is 1.96716821281 a(n) is asymptotic to .552975339403*1.96716821281^n ---------------------------------------------------------------------------- The compositions of, 7, that yield the, 14, -th largest growth, that is, 1.9693144732632464526, are , [3, 4], [4, 3] Theorem Number, 14, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 4 \ n x - x + 1 ) a(n) x = - ---------------------- / 6 5 4 ----- x + x - x + 2 x - 1 n = 0 and in Maple format -(x^4-x+1)/(x^6+x^5-x^4+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 63, 124, 244, 480, 945, 1861, 3665, 7218, 14215, 27994, 55129, 108566, 213800, 421039, 829158, 1632873, 3215641, 6332609, 12470899, 24559122, 48364634, 95245173, 187567697, 369379780] The limit of a(n+1)/a(n) as n goes to infinity is 1.96931447326 a(n) is asymptotic to .547042335160*1.96931447326^n ---------------------------------------------------------------------------- The compositions of, 7, that yield the, 15, -th largest growth, that is, 1.9835828434243263304, are , [7] Theorem Number, 15, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [7] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 1 ) a(n) x = - ------------------------------ / 6 5 4 3 2 ----- x + x + x + x + x + x - 1 n = 0 and in Maple format -1/(x^6+x^5+x^4+x^3+x^2+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492, 976, 1936, 3840, 7617, 15109, 29970, 59448, 117920, 233904, 463968, 920319, 1825529, 3621088, 7182728, 14247536, 28261168, 56058368, 111196417, 220567305, 437513522] The limit of a(n+1)/a(n) as n goes to infinity is 1.98358284342 a(n) is asymptotic to .521772494287*1.98358284342^n ---------------------------------------------------------------------------- This ends this article, that took, 0.590, seconds to generate. ---------------------------------------------------------- ---------------------------------------- Generating functions and Growth rates for the Enumerating Sequences for Como\ sitions Avoiding Each of the compositions of, 8 By Shalosh B. Ekhad The compositions of, 8, that yield polynomial growth (hence growth ratio 1, that is the lowest) are , [2, 1, 1, 1, 1, 2] Theorem Number, 1, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 1, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 32 30 29 28 27 26 25 24 23 ) a(n) x = - (x + 2 x + x + x + x - x + x - x - 2 x / ----- n = 0 22 21 20 19 18 17 16 15 14 + 2 x + 2 x - 6 x + 8 x - 6 x + 5 x - 12 x + 32 x - 57 x 13 12 11 10 9 8 7 6 + 60 x - 22 x - 54 x + 161 x - 306 x + 488 x - 652 x + 702 x 5 4 3 2 / 3 2 - 590 x + 376 x - 175 x + 56 x - 11 x + 1) / ((x - x + 2 x - 1) / 15 10 9 8 7 5 4 3 2 (x - x + 3 x - 3 x + x + x - 5 x + 10 x - 10 x + 5 x - 1) 15 10 9 8 7 6 5 4 3 2 (x + 2 x - 5 x + 3 x + x - x + x - 5 x + 10 x - 10 x + 5 x - 1) ) and in Maple format -(x^32+2*x^30+x^29+x^28+x^27-x^26+x^25-x^24-2*x^23+2*x^22+2*x^21-6*x^20+8*x^19-\ 6*x^18+5*x^17-12*x^16+32*x^15-57*x^14+60*x^13-22*x^12-54*x^11+161*x^10-306*x^9+ 488*x^8-652*x^7+702*x^6-590*x^5+376*x^4-175*x^3+56*x^2-11*x+1)/(x^3-x^2+2*x-1)/ (x^15-x^10+3*x^9-3*x^8+x^7+x^5-5*x^4+10*x^3-10*x^2+5*x-1)/(x^15+2*x^10-5*x^9+3* x^8+x^7-x^6+x^5-5*x^4+10*x^3-10*x^2+5*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 248, 474, 885, 1615, 2886, 5064, 8755, 14975, 25455, 43190, 73428, 125451, 215775, 373875, 652374, 1145083, 2018792, 3569071, 6317906, 11184155, 19780765, 34932849] ---------------------------------------------------------------------------- The compositions of, 8, that yield the, 2, -th largest growth, that is, 1.6180339887498948482, are , [1, 1, 1, 1, 1, 3], [1, 1, 1, 1, 3, 1], [1, 1, 1, 3, 1, 1], [1, 1, 3, 1, 1, 1], [1, 3, 1, 1, 1, 1], [3, 1, 1, 1, 1, 1] Theorem Number, 2, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 1, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 7 6 5 4 3 2 \ n x - 2 x + 3 x + 2 x - 9 x + 10 x - 5 x + 1 ) a(n) x = ------------------------------------------------ / 2 5 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format (x^7-2*x^6+3*x^5+2*x^4-9*x^3+10*x^2-5*x+1)/(x^2+x-1)/(-1+x)^5 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 248, 474, 885, 1615, 2886, 5063, 8743, 14899, 25113, 41953, 69583, 114750, 188381, 308167, 502744, 818458, 1330311, 2159672, 3502934, 5677882, 9198718, 14897454] The limit of a(n+1)/a(n) as n goes to infinity is 1.61803398875 a(n) is asymptotic to 8.02492235950*1.61803398875^n ---------------------------------------------------------------------------- The compositions of, 8, that yield the, 3, -th largest growth, that is, 1.7548776662466927601, are , [1, 1, 2, 1, 1, 2], [1, 2, 1, 1, 2, 1], [2, 1, 1, 2, 1, 1] Theorem Number, 3, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 2, 1, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 2 ) a(n) x = - (x - x + 1) / ----- n = 0 11 9 8 7 6 5 4 3 2 (2 x + 3 x - 2 x + 3 x - 7 x + 7 x - 7 x + 10 x - 10 x + 5 x - 1) / 3 2 9 6 5 4 3 2 / ((x - x + 2 x - 1) (x - x + 2 x - x + x - 3 x + 3 x - 1) / 2 (-1 + x) ) and in Maple format -(x^2-x+1)*(2*x^11+3*x^9-2*x^8+3*x^7-7*x^6+7*x^5-7*x^4+10*x^3-10*x^2+5*x-1)/(x^ 3-x^2+2*x-1)/(x^9-x^6+2*x^5-x^4+x^3-3*x^2+3*x-1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 248, 474, 885, 1616, 2896, 5118, 8965, 15636, 27248, 47546, 83150, 145737, 255853, 449573, 790149, 1388411, 2438546, 4280906, 7512318, 13179842, 23121052, 40561481] The limit of a(n+1)/a(n) as n goes to infinity is 1.75487766625 a(n) is asymptotic to 1.90802250917*1.75487766625^n ---------------------------------------------------------------------------- The compositions of, 8, that yield the, 4, -th largest growth, that is, 1.8392867552141611326, are , [1, 1, 1, 1, 4], [1, 1, 1, 4, 1], [1, 1, 4, 1, 1], [1, 4, 1, 1, 1], [4, 1, 1, 1, 1] Theorem Number, 4, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 1, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 6 5 4 3 2 \ n 2 x - 2 x + 2 x - 4 x + 6 x - 4 x + 1 ) a(n) x = - ------------------------------------------ / 3 2 4 ----- (x + x + x - 1) (-1 + x) n = 0 and in Maple format -(2*x^6-2*x^5+2*x^4-4*x^3+6*x^2-4*x+1)/(x^3+x^2+x-1)/(-1+x)^4 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 249, 482, 922, 1746, 3280, 6124, 11382, 21085, 38969, 71906, 132536, 244108, 449384, 827016, 1521668, 2799419, 5149665, 9472546, 17423678, 32048214, 58947064, 108421908] The limit of a(n+1)/a(n) as n goes to infinity is 1.83928675521 a(n) is asymptotic to 1.24635566962*1.83928675521^n ---------------------------------------------------------------------------- The compositions of, 8, that yield the, 5, -th largest growth, that is, 1.8667603991738620930, are , [2, 1, 1, 1, 3], [3, 1, 1, 1, 2] Theorem Number, 5, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 27 24 23 22 21 20 19 18 17 ) a(n) x = - (x + 2 x + x - x + 2 x + x - 3 x + x + x / ----- n = 0 16 15 14 13 12 11 10 9 8 + 2 x - 8 x + 5 x + 5 x - 10 x + 4 x + 4 x - 5 x + 8 x 7 6 5 4 3 2 / - 17 x + 16 x + 5 x - 29 x + 34 x - 21 x + 7 x - 1) / ( / 4 4 3 4 (x - x + 1) (x - x + 2 x - 1) (x + x - 1) 16 11 10 9 7 6 4 3 2 (x - x + 3 x - 2 x - x + x + x - 4 x + 6 x - 4 x + 1)) and in Maple format -(x^27+2*x^24+x^23-x^22+2*x^21+x^20-3*x^19+x^18+x^17+2*x^16-8*x^15+5*x^14+5*x^ 13-10*x^12+4*x^11+4*x^10-5*x^9+8*x^8-17*x^7+16*x^6+5*x^5-29*x^4+34*x^3-21*x^2+7 *x-1)/(x^4-x+1)/(x^4-x^3+2*x-1)/(x^4+x-1)/(x^16-x^11+3*x^10-2*x^9-x^7+x^6+x^4-4 *x^3+6*x^2-4*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 249, 482, 922, 1746, 3280, 6125, 11392, 21142, 39210, 72748, 135113, 251271, 467904, 872301, 1627646, 3038906, 5675820, 10602440, 19805504, 36993860, 69090383, 129016404] The limit of a(n+1)/a(n) as n goes to infinity is 1.86676039917 a(n) is asymptotic to .950512366759*1.86676039917^n ---------------------------------------------------------------------------- The compositions of, 8, that yield the, 6, -th largest growth, that is, 1.8908049048980778372, are , [1, 2, 1, 2, 2], [1, 2, 2, 1, 2], [2, 1, 2, 2, 1], [2, 2, 1, 2, 1] Theorem Number, 6, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 1, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 9 8 7 5 4 3 2 \ n x - x + 2 x - 2 x + 5 x - 7 x + 7 x - 4 x + 1 ) a(n) x = ----------------------------------------------------------- / 8 7 6 5 4 3 2 ----- (-1 + x) (x - x + x + x - 3 x + 5 x - 6 x + 4 x - 1) n = 0 and in Maple format (x^9-x^8+2*x^7-2*x^5+5*x^4-7*x^3+7*x^2-4*x+1)/(-1+x)/(x^8-x^7+x^6+x^5-3*x^4+5*x ^3-6*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 249, 482, 923, 1754, 3318, 6263, 11815, 22295, 42097, 79537, 150348, 284286, 537617, 1016723, 1922732, 3635902, 6875176, 12999877, 24580170, 46475718, 87875239, 166152938] The limit of a(n+1)/a(n) as n goes to infinity is 1.89080490490 a(n) is asymptotic to .833877528824*1.89080490490^n ---------------------------------------------------------------------------- The compositions of, 8, that yield the, 7, -th largest growth, that is, 1.8923110706522823122, are , [2, 1, 1, 2, 2], [2, 2, 1, 1, 2] Theorem Number, 7, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 10 8 6 5 4 3 2 x + x + x - 2 x + 5 x - 7 x + 7 x - 4 x + 1 - --------------------------------------------------------------------- 11 10 9 8 7 6 5 4 3 2 x - x + x - x + x - x + 4 x - 8 x + 11 x - 10 x + 5 x - 1 and in Maple format -(x^10+x^8+x^6-2*x^5+5*x^4-7*x^3+7*x^2-4*x+1)/(x^11-x^10+x^9-x^8+x^7-x^6+4*x^5-\ 8*x^4+11*x^3-10*x^2+5*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 249, 482, 923, 1753, 3311, 6235, 11730, 22077, 41599, 78493, 148301, 280481, 530840, 1005052, 1903140, 3603611, 6822606, 12914983, 24443975, 46259097, 87535889, 165635897] The limit of a(n+1)/a(n) as n goes to infinity is 1.89231107065 a(n) is asymptotic to .811466235213*1.89231107065^n ---------------------------------------------------------------------------- The compositions of, 8, that yield the, 8, -th largest growth, that is, 1.9087907387871591034, are , [2, 1, 2, 1, 2] Theorem Number, 8, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 2, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = - ( / ----- n = 0 14 11 10 9 8 7 5 4 3 2 x + x - x + 2 x - x - x + 9 x - 21 x + 24 x - 16 x + 6 x - 1) / 5 4 3 2 / ((x - x + 2 x - 3 x + 3 x - 1) / 10 8 7 5 4 3 2 (x - x + 2 x - 3 x + 2 x + 3 x - 6 x + 4 x - 1)) and in Maple format -(x^14+x^11-x^10+2*x^9-x^8-x^7+9*x^5-21*x^4+24*x^3-16*x^2+6*x-1)/(x^5-x^4+2*x^3 -3*x^2+3*x-1)/(x^10-x^8+2*x^7-3*x^5+2*x^4+3*x^3-6*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 249, 482, 923, 1755, 3326, 6301, 11953, 22720, 43270, 82535, 157592, 301064, 575232, 1098948, 2098976, 4007962, 7651424, 14604699, 27874367, 53199129, 101533210, 193787063] The limit of a(n+1)/a(n) as n goes to infinity is 1.90879073879 a(n) is asymptotic to .732089369699*1.90879073879^n ---------------------------------------------------------------------------- The compositions of, 8, that yield the, 9, -th largest growth, that is, 1.9275619754829253043, are , [1, 1, 1, 5], [1, 1, 5, 1], [1, 5, 1, 1], [5, 1, 1, 1] Theorem Number, 9, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 7 6 5 3 2 \ n x - x + x - x + 3 x - 3 x + 1 ) a(n) x = ---------------------------------- / 4 3 2 3 ----- (x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^7-x^6+x^5-x^3+3*x^2-3*x+1)/(x^4+x^3+x^2+x-1)/(-1+x)^3 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 250, 489, 952, 1847, 3575, 6909, 13339, 25737, 49639, 95716, 184537, 355750, 685779, 1321936, 2548174, 4911830, 9467930, 18250102, 35178290, 67808429, 130705052, 251942199] The limit of a(n+1)/a(n) as n goes to infinity is 1.92756197548 a(n) is asymptotic to .709660512200*1.92756197548^n ---------------------------------------------------------------------------- The compositions of, 8, that yield the, 10, -th largest growth, that is, 1.9331849818995204468, are , [2, 1, 1, 4], [4, 1, 1, 2] Theorem Number, 10, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 19 15 14 13 11 10 9 8 7 ) a(n) x = - (x + 2 x - x + x + 2 x - 3 x + x - x + 4 x / ----- n = 0 6 5 4 3 2 / - 5 x + 3 x - 2 x + 4 x - 6 x + 4 x - 1) / ( / 15 10 9 7 6 3 2 5 4 (x - x + x + x - x + x - 3 x + 3 x - 1) (x - x + 2 x - 1)) and in Maple format -(x^19+2*x^15-x^14+x^13+2*x^11-3*x^10+x^9-x^8+4*x^7-5*x^6+3*x^5-2*x^4+4*x^3-6*x ^2+4*x-1)/(x^15-x^10+x^9+x^7-x^6+x^3-3*x^2+3*x-1)/(x^5-x^4+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 250, 489, 952, 1847, 3575, 6910, 13347, 25775, 49779, 96157, 185787, 359037, 693957, 1341448, 2593249, 5013366, 9692144, 18737428, 36224057, 70029397, 135381750, 261720071] The limit of a(n+1)/a(n) as n goes to infinity is 1.93318498190 a(n) is asymptotic to .675512458314*1.93318498190^n ---------------------------------------------------------------------------- The compositions of, 8, that yield the, 11, -th largest growth, that is, 1.9407101328380924652, are , [2, 1, 2, 3], [2, 1, 3, 2], [2, 2, 1, 3], [2, 3, 1, 2], [3, 1, 2, 2], [3, 2, 1, 2] Theorem Number, 11, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 9 5 4 2 \ n x + 2 x - 2 x + 3 x - 3 x + 1 ) a(n) x = - ------------------------------------------------------ / 10 9 7 6 5 4 3 2 ----- x - x + x + x - 3 x + 3 x + x - 5 x + 4 x - 1 n = 0 and in Maple format -(x^9+2*x^5-2*x^4+3*x^2-3*x+1)/(x^10-x^9+x^7+x^6-3*x^5+3*x^4+x^3-5*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 250, 489, 952, 1848, 3582, 6940, 13448, 26069, 50556, 98078, 190316, 369352, 716859, 1391341, 2700402, 5240996, 10171609, 19740485, 38310763, 74349980, 144291136, 280025926] The limit of a(n+1)/a(n) as n goes to infinity is 1.94071013284 a(n) is asymptotic to .643241925447*1.94071013284^n ---------------------------------------------------------------------------- The compositions of, 8, that yield the, 12, -th largest growth, that is, 1.9417130342786384772, are , [3, 1, 1, 3] Theorem Number, 12, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 1, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 18 17 16 15 14 13 10 9 ) a(n) x = - (x + 3 x + 3 x + x + 2 x + 3 x + 3 x - x / ----- n = 0 8 7 6 5 4 3 2 / 2 - 3 x + 3 x + x - 4 x + 2 x + 3 x - 6 x + 4 x - 1) / ((x + 1) / 5 4 3 13 12 11 10 9 8 7 (x + x - x + 2 x - 1) (x + 3 x + 2 x - 2 x - 2 x + 2 x + x 6 5 4 3 2 - 2 x + x + 2 x - 2 x - 2 x + 3 x - 1)) and in Maple format -(x^18+3*x^17+3*x^16+x^15+2*x^14+3*x^13+3*x^10-x^9-3*x^8+3*x^7+x^6-4*x^5+2*x^4+ 3*x^3-6*x^2+4*x-1)/(x^2+1)/(x^5+x^4-x^3+2*x-1)/(x^13+3*x^12+2*x^11-2*x^10-2*x^9 +2*x^8+x^7-2*x^6+x^5+2*x^4-2*x^3-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 250, 489, 952, 1847, 3576, 6918, 13385, 25915, 50220, 97406, 189063, 367149, 713178, 1385465, 2691421, 5227890, 10153586, 19718017, 38288440, 74343896, 144347147, 280263162] The limit of a(n+1)/a(n) as n goes to infinity is 1.94171303428 a(n) is asymptotic to .633809828964*1.94171303428^n ---------------------------------------------------------------------------- The compositions of, 8, that yield the, 13, -th largest growth, that is, 1.9454365275632690792, are , [1, 2, 2, 3], [1, 2, 3, 2], [1, 3, 2, 2], [2, 2, 3, 1], [2, 3, 2, 1], [3, 2, 2, 1] Theorem Number, 13, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 7 6 5 4 2 \ n x - x + 2 x - 2 x + 3 x - 3 x + 1 ) a(n) x = ----------------------------------------------- / 6 5 4 3 2 ----- (-1 + x) (x - x + 2 x - x - 2 x + 3 x - 1) n = 0 and in Maple format (x^7-x^6+2*x^5-2*x^4+3*x^2-3*x+1)/(-1+x)/(x^6-x^5+2*x^4-x^3-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 250, 489, 953, 1854, 3605, 7010, 13634, 26522, 51598, 100386, 195304, 379963, 739204, 1438079, 2797687, 5442708, 10588421, 20599078, 40074184, 77961784, 151669728, 295063876] The limit of a(n+1)/a(n) as n goes to infinity is 1.94543652756 a(n) is asymptotic to .630092222605*1.94543652756^n ---------------------------------------------------------------------------- The compositions of, 8, that yield the, 14, -th largest growth, that is, 1.9611865309023902347, are , [2, 2, 2, 2] Theorem Number, 14, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 2, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 2 4 2 \ n (x - x + 1) (x + x - 2 x + 1) ) a(n) x = - --------------------------------------------- / 7 6 5 4 3 2 ----- x - x + 2 x - 3 x + 5 x - 6 x + 4 x - 1 n = 0 and in Maple format -(x^2-x+1)*(x^4+x^2-2*x+1)/(x^7-x^6+2*x^5-3*x^4+5*x^3-6*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 250, 490, 959, 1877, 3676, 7204, 14125, 27703, 54339, 106585, 209055, 410018, 804135, 1577052, 3092857, 6065601, 11895683, 23329562, 45753571, 89731321, 175980016, 345129929] The limit of a(n+1)/a(n) as n goes to infinity is 1.96118653090 a(n) is asymptotic to .578649889688*1.96118653090^n ---------------------------------------------------------------------------- The compositions of, 8, that yield the, 15, -th largest growth, that is, 1.9659482366454853372, are , [1, 1, 6], [1, 6, 1], [6, 1, 1] Theorem Number, 15, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 6] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 6 2 \ n x + x - 2 x + 1 ) a(n) x = - ------------------------------------- / 5 4 3 2 2 ----- (x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format -(x^6+x^2-2*x+1)/(x^5+x^4+x^3+x^2+x-1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 251, 495, 975, 1919, 3775, 7424, 14598, 28702, 56430, 110942, 218110, 428797, 842997, 1657293, 3258157, 6405373, 12592637, 24756478, 48669960, 95682628, 188107100, 369808828] The limit of a(n+1)/a(n) as n goes to infinity is 1.96594823665 a(n) is asymptotic to .576520722438*1.96594823665^n ---------------------------------------------------------------------------- The compositions of, 8, that yield the, 16, -th largest growth, that is, 1.9671682128139660358, are , [2, 1, 5], [5, 1, 2] Theorem Number, 16, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 11 5 2 \ n x + x + x - 2 x + 1 ) a(n) x = - -------------------------------- / 6 5 6 ----- (x - x + 2 x - 1) (x - x + 1) n = 0 and in Maple format -(x^11+x^5+x^2-2*x+1)/(x^6-x^5+2*x-1)/(x^6-x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 251, 495, 975, 1919, 3775, 7425, 14604, 28725, 56502, 111143, 218631, 430079, 846035, 1664294, 3273952, 6440426, 12669419, 24922901, 49027564, 96445490, 189724521, 373220054] The limit of a(n+1)/a(n) as n goes to infinity is 1.96716821281 a(n) is asymptotic to .571110296331*1.96716821281^n ---------------------------------------------------------------------------- The compositions of, 8, that yield the, 17, -th largest growth, that is, 1.9693144732632464526, are , [3, 1, 4], [4, 1, 3] Theorem Number, 17, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 1, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 10 9 4 2 \ n x + x + x + x - 2 x + 1 ) a(n) x = - -------------------------------------------------------- / 2 4 3 2 6 5 4 ----- (x + 1) (x + x - x - x + 1) (x + x - x + 2 x - 1) n = 0 and in Maple format -(x^10+x^9+x^4+x^2-2*x+1)/(x^2+1)/(x^4+x^3-x^2-x+1)/(x^6+x^5-x^4+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 251, 495, 975, 1919, 3776, 7431, 14627, 28797, 56703, 111663, 219905, 433080, 852904, 1679685, 3307887, 6514326, 12828789, 25263901, 49752465, 97978041, 192949251, 379977341] The limit of a(n+1)/a(n) as n goes to infinity is 1.96931447326 a(n) is asymptotic to .562736849592*1.96931447326^n ---------------------------------------------------------------------------- The compositions of, 8, that yield the, 18, -th largest growth, that is, 1.9703230372932668084, are , [2, 2, 4], [2, 4, 2], [4, 2, 2] Theorem Number, 18, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 2, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 6 5 4 2 \ n x - x + x + x - 2 x + 1 ) a(n) x = - ------------------------------------ / 7 6 5 4 2 ----- x - x + 2 x - x - 2 x + 3 x - 1 n = 0 and in Maple format -(x^6-x^5+x^4+x^2-2*x+1)/(x^7-x^6+2*x^5-x^4-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 251, 495, 975, 1920, 3781, 7447, 14670, 28902, 56945, 112201, 221076, 435598, 858278, 1691095, 3332012, 6565144, 12935451, 25487004, 50217608, 98944879, 194953341, 384121031] The limit of a(n+1)/a(n) as n goes to infinity is 1.97032303729 a(n) is asymptotic to .560202503508*1.97032303729^n ---------------------------------------------------------------------------- The compositions of, 8, that yield the, 19, -th largest growth, that is, 1.9717270001741243154, are , [3, 2, 3] Theorem Number, 19, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 8 5 4 3 2 \ n x + x - x + x + x - 2 x + 1 ) a(n) x = - --------------------------------------------------------- / 10 9 8 7 6 5 4 3 2 ----- x + x - x + x + x - x + 2 x - x - 2 x + 3 x - 1 n = 0 and in Maple format -(x^8+x^5-x^4+x^3+x^2-2*x+1)/(x^10+x^9-x^8+x^7+x^6-x^5+2*x^4-x^3-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 251, 495, 975, 1920, 3782, 7453, 14692, 28968, 57121, 112637, 222105, 437949, 863532, 1702655, 3357157, 6619357, 13051498, 25733907, 50740161, 100045708, 197262868, 388948683] The limit of a(n+1)/a(n) as n goes to infinity is 1.97172700017 a(n) is asymptotic to .555250679255*1.97172700017^n ---------------------------------------------------------------------------- The compositions of, 8, that yield the, 20, -th largest growth, that is, 1.9735704833094816886, are , [2, 3, 3], [3, 3, 2] Theorem Number, 20, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 3, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 2 4 3 2 \ n (x - x + 1) (x + x - x - x + 1) ) a(n) x = - ------------------------------------ / 7 6 4 3 2 ----- x - x + 2 x - x - 2 x + 3 x - 1 n = 0 and in Maple format -(x^2-x+1)*(x^4+x^3-x^2-x+1)/(x^7-x^6+2*x^4-x^3-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 251, 495, 976, 1925, 3798, 7495, 14792, 29194, 57618, 113715, 224426, 442921, 874135, 1725165, 3404732, 6719476, 13261358, 26172225, 51652733, 101940313, 201186397, 397055538] The limit of a(n+1)/a(n) as n goes to infinity is 1.97357048331 a(n) is asymptotic to .551152801880*1.97357048331^n ---------------------------------------------------------------------------- The compositions of, 8, that yield the, 21, -th largest growth, that is, 1.9835828434243263304, are , [1, 7], [7, 1] Theorem Number, 21, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 7] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 7 \ n x - x + 1 ) a(n) x = ----------------------------------------- / 6 5 4 3 2 ----- (-1 + x) (x + x + x + x + x + x - 1) n = 0 and in Maple format (x^7-x+1)/(-1+x)/(x^6+x^5+x^4+x^3+x^2+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 252, 500, 992, 1968, 3904, 7744, 15361, 30470, 60440, 119888, 237808, 471712, 935680, 1855999, 3681528, 7302616, 14485344, 28732880, 56994048, 113052416, 224248833, 444816138] The limit of a(n+1)/a(n) as n goes to infinity is 1.98358284342 a(n) is asymptotic to .530481492001*1.98358284342^n ---------------------------------------------------------------------------- The compositions of, 8, that yield the, 22, -th largest growth, that is, 1.9838613961621262283, are , [2, 6], [6, 2] Theorem Number, 22, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 6] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 6 \ n x - x + 1 ) a(n) x = - ----------------- / 7 6 ----- x - x + 2 x - 1 n = 0 and in Maple format -(x^6-x+1)/(x^7-x^6+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 252, 500, 992, 1968, 3904, 7745, 15365, 30482, 60472, 119968, 238000, 472159, 936698, 1858279, 3686568, 7313640, 14509248, 28784337, 57104135, 113286689, 224745089, 445863106] The limit of a(n+1)/a(n) as n goes to infinity is 1.98386139616 a(n) is asymptotic to .529494848582*1.98386139616^n ---------------------------------------------------------------------------- The compositions of, 8, that yield the, 23, -th largest growth, that is, 1.9843858253440954550, are , [3, 5], [5, 3] Theorem Number, 23, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 5 \ n x - x + 1 ) a(n) x = - ---------------------- / 7 6 5 ----- x + x - x + 2 x - 1 n = 0 and in Maple format -(x^5-x+1)/(x^7+x^6-x^5+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 252, 500, 992, 1968, 3905, 7749, 15377, 30514, 60552, 120159, 238442, 473161, 938934, 1863207, 3697321, 7336911, 14559262, 28891193, 57331274, 113767368, 225758353, 447991676] The limit of a(n+1)/a(n) as n goes to infinity is 1.98438582534 a(n) is asymptotic to .527820750127*1.98438582534^n ---------------------------------------------------------------------------- The compositions of, 8, that yield the, 24, -th largest growth, that is, 1.9853288885629234253, are , [4, 4] Theorem Number, 24, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [4, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 4 \ n x - x + 1 ) a(n) x = - --------------------------- / 7 6 5 4 ----- x + x + x - x + 2 x - 1 n = 0 and in Maple format -(x^4-x+1)/(x^7+x^6+x^5-x^4+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 252, 500, 992, 1969, 3909, 7761, 15409, 30593, 60738, 120585, 239400, 475286, 943597, 1873349, 3719214, 7383865, 14659404, 29103742, 57780502, 114713299, 227743622, 452145985] The limit of a(n+1)/a(n) as n goes to infinity is 1.98532888856 a(n) is asymptotic to .525175931290*1.98532888856^n ---------------------------------------------------------------------------- The compositions of, 8, that yield the, 25, -th largest growth, that is, 1.9919641966050350211, are , [8] Theorem Number, 25, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [8] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 1 ) a(n) x = - ----------------------------------- / 7 6 5 4 3 2 ----- x + x + x + x + x + x + x - 1 n = 0 and in Maple format -1/(x^7+x^6+x^5+x^4+x^3+x^2+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000, 3984, 7936, 15808, 31489, 62725, 124946, 248888, 495776, 987568, 1967200, 3918592, 7805695, 15548665, 30972384, 61695880, 122895984, 244804400, 487641600] The limit of a(n+1)/a(n) as n goes to infinity is 1.99196419661 a(n) is asymptotic to .512454017228*1.99196419661^n ---------------------------------------------------------------------------- This ends this article, that took, 1.744, seconds to generate. ---------------------------------------------------------- ---------------------------------------- Generating functions and Growth rates for the Enumerating Sequences for Como\ sitions Avoiding Each of the compositions of, 9 By Shalosh B. Ekhad The compositions of, 9, that yield polynomial growth (hence growth ratio 1, that is the lowest) are , [2, 1, 1, 1, 1, 1, 2] Theorem Number, 1, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 1, 1, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 47 45 44 43 42 41 40 39 ) a(n) x = - (x + 2 x + x + x + x + x - 3 x + 6 x / ----- n = 0 38 37 36 35 34 33 32 31 30 - 8 x + 5 x + x - 3 x - x + 8 x - 12 x + 7 x + 9 x 29 28 27 26 25 24 23 22 - 33 x + 55 x - 82 x + 126 x - 151 x + 100 x + 16 x - 87 x 21 20 19 18 17 16 15 - 52 x + 547 x - 1424 x + 2487 x - 3311 x + 3455 x - 2910 x 14 13 12 11 10 9 + 2437 x - 3293 x + 6353 x - 11267 x + 16328 x - 19272 x 8 7 6 5 4 3 2 + 18634 x - 14739 x + 9452 x - 4836 x + 1926 x - 575 x + 121 x / 3 3 2 - 16 x + 1) / ((x - x + 1) (x - x + 2 x - 1) / 9 6 5 4 3 2 6 4 3 2 (x - x + 2 x - x + x - 3 x + 3 x - 1) (x + x - x + x - 2 x + 1) 9 6 5 4 3 2 18 12 11 10 (x + x - 2 x + x + x - 3 x + 3 x - 1) (x + x - 5 x + 8 x 9 8 7 6 5 4 3 2 - 4 x - x + x + x - 6 x + 15 x - 20 x + 15 x - 6 x + 1)) and in Maple format -(x^47+2*x^45+x^44+x^43+x^42+x^41-3*x^40+6*x^39-8*x^38+5*x^37+x^36-3*x^35-x^34+ 8*x^33-12*x^32+7*x^31+9*x^30-33*x^29+55*x^28-82*x^27+126*x^26-151*x^25+100*x^24 +16*x^23-87*x^22-52*x^21+547*x^20-1424*x^19+2487*x^18-3311*x^17+3455*x^16-2910* x^15+2437*x^14-3293*x^13+6353*x^12-11267*x^11+16328*x^10-19272*x^9+18634*x^8-\ 14739*x^7+9452*x^6-4836*x^5+1926*x^4-575*x^3+121*x^2-16*x+1)/(x^3-x+1)/(x^3-x^2 +2*x-1)/(x^9-x^6+2*x^5-x^4+x^3-3*x^2+3*x-1)/(x^6+x^4-x^3+x^2-2*x+1)/(x^9+x^6-2* x^5+x^4+x^3-3*x^2+3*x-1)/(x^18+x^12-5*x^11+8*x^10-4*x^9-x^8+x^7+x^6-6*x^5+15*x^ 4-20*x^3+15*x^2-6*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 503, 977, 1862, 3477, 6363, 11426, 20170, 35082, 60283, 102654, 173816, 293648, 496576, 842854, 1438829, 2473334, 4282923, 7468452, 13102648, 23098217, 40858122] ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 2, -th largest growth, that is, 1.6180339887498948482, are , [1, 1, 1, 1, 1, 1, 3], [1, 1, 1, 1, 1, 3, 1], [1, 1, 1, 1, 3, 1, 1], [1, 1, 1, 3, 1, 1, 1], [1, 1, 3, 1, 1, 1, 1], [1, 3, 1, 1, 1, 1, 1], [3, 1, 1, 1, 1, 1, 1] Theorem Number, 2, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 1, 1, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 7 6 5 4 3 2 \ n 3 x - 5 x + x + 11 x - 19 x + 15 x - 6 x + 1 ) a(n) x = - -------------------------------------------------- / 2 6 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format -(3*x^7-5*x^6+x^5+11*x^4-19*x^3+15*x^2-6*x+1)/(x^2+x-1)/(-1+x)^6 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 503, 977, 1862, 3477, 6363, 11426, 20169, 35068, 60181, 102134, 171717, 286467, 474848, 783015, 1285759, 2104217, 3434528, 5594200, 9097134, 14775016, 23973734] The limit of a(n+1)/a(n) as n goes to infinity is 1.61803398875 a(n) is asymptotic to 12.9845971347*1.61803398875^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 3, -th largest growth, that is, 1.7548776662466927601, are , [1, 1, 2, 1, 1, 1, 2], [1, 2, 1, 1, 1, 2, 1], [2, 1, 1, 1, 2, 1, 1] Theorem Number, 3, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 2, 1, 1, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 22 21 20 19 18 17 16 ) a(n) x = - (2 x - 2 x + 5 x - 2 x + x + 3 x - 7 x / ----- n = 0 15 14 13 12 11 10 9 8 + 11 x - 14 x + 8 x + 7 x - 26 x + 39 x - 29 x - 18 x 7 6 5 4 3 2 / + 94 x - 163 x + 187 x - 155 x + 92 x - 37 x + 9 x - 1) / ( / 3 3 2 (x - x + 1) (x - x + 2 x - 1) 12 8 7 6 5 4 3 2 3 (x + x - x - x + x + x - 4 x + 6 x - 4 x + 1) (x + x - 1) 2 (-1 + x) ) and in Maple format -(2*x^22-2*x^21+5*x^20-2*x^19+x^18+3*x^17-7*x^16+11*x^15-14*x^14+8*x^13+7*x^12-\ 26*x^11+39*x^10-29*x^9-18*x^8+94*x^7-163*x^6+187*x^5-155*x^4+92*x^3-37*x^2+9*x-\ 1)/(x^3-x+1)/(x^3-x^2+2*x-1)/(x^12+x^8-x^7-x^6+x^5+x^4-4*x^3+6*x^2-4*x+1)/(x^3+ x-1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 503, 977, 1862, 3477, 6364, 11438, 20246, 35422, 61495, 106332, 183718, 317959, 552063, 962217, 1683298, 2953668, 5193624, 9142572, 16098739, 28338148, 49847442] The limit of a(n+1)/a(n) as n goes to infinity is 1.75487766625 a(n) is asymptotic to 2.34123928046*1.75487766625^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 4, -th largest growth, that is, 1.8392867552141611326, are , [1, 1, 1, 1, 1, 4], [1, 1, 1, 1, 4, 1], [1, 1, 1, 4, 1, 1], [1, 1, 4, 1, 1, 1], [1, 4, 1, 1, 1, 1], [4, 1, 1, 1, 1, 1] Theorem Number, 4, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 1, 1, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 8 7 6 5 4 3 2 \ n x - 2 x + 4 x - 4 x + 6 x - 10 x + 10 x - 5 x + 1 ) a(n) x = -------------------------------------------------------- / 3 2 5 ----- (x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^8-2*x^7+4*x^6-4*x^5+6*x^4-10*x^3+10*x^2-5*x+1)/(x^3+x^2+x-1)/(-1+x)^5 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 504, 986, 1908, 3654, 6934, 13058, 24440, 45525, 84494, 156400, 288936, 533044, 982428, 1809444, 3331112, 6130531, 11280196, 20752742, 38176420, 70224634, 129171698] The limit of a(n+1)/a(n) as n goes to infinity is 1.83928675521 a(n) is asymptotic to 1.48501767945*1.83928675521^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 5, -th largest growth, that is, 1.8667603991738620930, are , [2, 1, 1, 1, 1, 3], [3, 1, 1, 1, 1, 2] Theorem Number, 5, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 1, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 43 40 39 38 37 36 34 33 ) a(n) x = - (x + 2 x + x + x - x + 3 x - 5 x + 5 x / ----- n = 0 31 30 29 28 27 26 25 23 - 4 x + 3 x - 10 x + 17 x - 4 x - 16 x + 17 x - 4 x 22 21 20 19 18 17 16 15 - 15 x + 36 x - 47 x + 39 x + 11 x - 84 x + 101 x - 39 x 14 13 12 11 10 9 8 7 - 24 x + 35 x - 43 x + 110 x - 214 x + 260 x - 156 x - 90 x 6 5 4 3 2 / + 334 x - 416 x + 320 x - 164 x + 55 x - 11 x + 1) / ( / 4 3 (x - x + 2 x - 1) 20 13 12 10 9 5 4 3 2 20 (x - x + 2 x - 2 x + x + x - 5 x + 10 x - 10 x + 5 x - 1) (x 14 13 12 11 10 8 7 5 4 3 - x + 4 x - 3 x - 3 x + 3 x + x - x + x - 5 x + 10 x 2 - 10 x + 5 x - 1)) and in Maple format -(x^43+2*x^40+x^39+x^38-x^37+3*x^36-5*x^34+5*x^33-4*x^31+3*x^30-10*x^29+17*x^28 -4*x^27-16*x^26+17*x^25-4*x^23-15*x^22+36*x^21-47*x^20+39*x^19+11*x^18-84*x^17+ 101*x^16-39*x^15-24*x^14+35*x^13-43*x^12+110*x^11-214*x^10+260*x^9-156*x^8-90*x ^7+334*x^6-416*x^5+320*x^4-164*x^3+55*x^2-11*x+1)/(x^4-x^3+2*x-1)/(x^20-x^13+2* x^12-2*x^10+x^9+x^5-5*x^4+10*x^3-10*x^2+5*x-1)/(x^20-x^14+4*x^13-3*x^12-3*x^11+ 3*x^10+x^8-x^7+x^5-5*x^4+10*x^3-10*x^2+5*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 504, 986, 1908, 3654, 6934, 13058, 24441, 45537, 84574, 156788, 290464, 538232, 998199, 1853488, 3446216, 6415882, 11958159, 22308646, 41646747, 77783915, 145315977] The limit of a(n+1)/a(n) as n goes to infinity is 1.86676039917 a(n) is asymptotic to 1.07461638455*1.86676039917^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 6, -th largest growth, that is, 1.8908049048980778372, are , [1, 1, 2, 1, 2, 2], [1, 1, 2, 2, 1, 2], [1, 2, 1, 2, 2, 1], [1, 2, 2, 1, 2, 1], [2, 1, 2, 2, 1, 1], [2, 2, 1, 2, 1, 1] Theorem Number, 6, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 2, 1, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 9 8 7 6 5 4 3 2 2 x - 2 x + 2 x + 2 x - 7 x + 12 x - 14 x + 11 x - 5 x + 1 - ------------------------------------------------------------------ 8 7 6 5 4 3 2 2 (x - x + x + x - 3 x + 5 x - 6 x + 4 x - 1) (-1 + x) and in Maple format -(2*x^9-2*x^8+2*x^7+2*x^6-7*x^5+12*x^4-14*x^3+11*x^2-5*x+1)/(x^8-x^7+x^6+x^5-3* x^4+5*x^3-6*x^2+4*x-1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 504, 986, 1909, 3663, 6981, 13244, 25059, 47354, 89451, 168988, 319336, 603622, 1141239, 2157962, 4080694, 7716596, 14591772, 27591649, 52171819, 98647537, 186522776] The limit of a(n+1)/a(n) as n goes to infinity is 1.89080490490 a(n) is asymptotic to .936094451478*1.89080490490^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 7, -th largest growth, that is, 1.8922218871524161071, are , [2, 1, 1, 2, 1, 2], [2, 1, 2, 1, 1, 2] Theorem Number, 7, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 2, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 15 13 11 10 9 8 7 6 ) a(n) x = - (x + 2 x - x + 5 x - 8 x + 7 x - x - 9 x / ----- n = 0 5 4 3 2 / 16 15 14 13 + 19 x - 26 x + 25 x - 16 x + 6 x - 1) / (x - x + 2 x - 2 x / 12 11 10 9 8 7 6 5 4 3 + x + 3 x - 9 x + 13 x - 10 x - x + 16 x - 31 x + 40 x - 36 x 2 + 21 x - 7 x + 1) and in Maple format -(x^15+2*x^13-x^11+5*x^10-8*x^9+7*x^8-x^7-9*x^6+19*x^5-26*x^4+25*x^3-16*x^2+6*x -1)/(x^16-x^15+2*x^14-2*x^13+x^12+3*x^11-9*x^10+13*x^9-10*x^8-x^7+16*x^6-31*x^5 +40*x^4-36*x^3+21*x^2-7*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 504, 986, 1908, 3655, 6944, 13115, 24682, 46378, 87138, 163863, 308564, 581916, 1098958, 2077791, 3931797, 7444185, 14098240, 26702030, 50569995, 95757810, 181289613] The limit of a(n+1)/a(n) as n goes to infinity is 1.89222188715 a(n) is asymptotic to .889598891994*1.89222188715^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 8, -th largest growth, that is, 1.8922578866301683686, are , [2, 1, 1, 1, 2, 2], [2, 2, 1, 1, 1, 2] Theorem Number, 8, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 1, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 18 16 15 14 13 11 10 9 ) a(n) x = - (x + x - x + 3 x - x + 3 x - 6 x + 11 x / ----- n = 0 8 7 6 5 4 3 2 / - 17 x + 24 x - 34 x + 46 x - 51 x + 41 x - 22 x + 7 x - 1) / ( / 3 16 15 12 11 9 8 7 6 5 (x + x - 1) (x - x + 3 x - 4 x + 7 x - 8 x + 2 x - 2 x + 17 x 4 3 2 - 34 x + 35 x - 21 x + 7 x - 1)) and in Maple format -(x^18+x^16-x^15+3*x^14-x^13+3*x^11-6*x^10+11*x^9-17*x^8+24*x^7-34*x^6+46*x^5-\ 51*x^4+41*x^3-22*x^2+7*x-1)/(x^3+x-1)/(x^16-x^15+3*x^12-4*x^11+7*x^9-8*x^8+2*x^ 7-2*x^6+17*x^5-34*x^4+35*x^3-21*x^2+7*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 504, 986, 1909, 3662, 6972, 13198, 24885, 46812, 87978, 165368, 311096, 585953, 1105098, 2086758, 3944502, 7461996, 14123834, 26741645, 50638763, 95891469, 181568857] The limit of a(n+1)/a(n) as n goes to infinity is 1.89225788663 a(n) is asymptotic to .891218879754*1.89225788663^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 9, -th largest growth, that is, 1.8923110706522823122, are , [1, 2, 1, 1, 2, 2], [1, 2, 2, 1, 1, 2], [2, 1, 1, 2, 2, 1], [2, 2, 1, 1, 2, 1] Theorem Number, 9, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 1, 1, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 12 11 10 9 8 7 6 5 4 ) a(n) x = (x - x + 2 x - x + 2 x - x + 3 x - 7 x + 12 x / ----- n = 0 3 2 / - 14 x + 11 x - 5 x + 1) / ((-1 + x) / 11 10 9 8 7 6 5 4 3 2 (x - x + x - x + x - x + 4 x - 8 x + 11 x - 10 x + 5 x - 1)) and in Maple format (x^12-x^11+2*x^10-x^9+2*x^8-x^7+3*x^6-7*x^5+12*x^4-14*x^3+11*x^2-5*x+1)/(-1+x)/ (x^11-x^10+x^9-x^8+x^7-x^6+4*x^5-8*x^4+11*x^3-10*x^2+5*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 504, 986, 1909, 3662, 6973, 13208, 24938, 47015, 88614, 167107, 315408, 595889, 1126729, 2131781, 4034921, 7638532, 14461138, 27376121, 51820096, 98079193, 185615082] The limit of a(n+1)/a(n) as n goes to infinity is 1.89231107065 a(n) is asymptotic to .909398372273*1.89231107065^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 10, -th largest growth, that is, 1.9087907387871591034, are , [1, 2, 1, 2, 1, 2], [2, 1, 2, 1, 2, 1] Theorem Number, 10, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 1, 2, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 2 ) a(n) x = (x - x + 1) ( / ----- n = 0 14 12 10 9 8 7 6 4 3 2 x + x - x + 2 x - 2 x + x + 2 x - 11 x + 19 x - 15 x + 6 x - 1 / 5 4 3 2 ) / ((-1 + x) (x - x + 2 x - 3 x + 3 x - 1) / 10 8 7 5 4 3 2 (x - x + 2 x - 3 x + 2 x + 3 x - 6 x + 4 x - 1)) and in Maple format (x^2-x+1)*(x^14+x^12-x^10+2*x^9-2*x^8+x^7+2*x^6-11*x^4+19*x^3-15*x^2+6*x-1)/(-1 +x)/(x^5-x^4+2*x^3-3*x^2+3*x-1)/(x^10-x^8+2*x^7-3*x^5+2*x^4+3*x^3-6*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 504, 986, 1909, 3664, 6990, 13291, 25244, 47964, 91234, 173769, 331361, 632425, 1207657, 2306605, 4405581, 8413543, 16064967, 30669666, 58544033, 111743162, 213276372] The limit of a(n+1)/a(n) as n goes to infinity is 1.90879073879 a(n) is asymptotic to .805564293796*1.90879073879^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 11, -th largest growth, that is, 1.9275619754829253043, are , [1, 1, 1, 1, 5], [1, 1, 1, 5, 1], [1, 1, 5, 1, 1], [1, 5, 1, 1, 1], [5, 1, 1, 1, 1] Theorem Number, 11, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 1, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 3 4 3 2 \ n (x - x + 1) (2 x - 2 x + 3 x - 3 x + 1) ) a(n) x = - ------------------------------------------- / 4 3 2 4 ----- (x + x + x + x - 1) (-1 + x) n = 0 and in Maple format -(x^3-x+1)*(2*x^4-2*x^3+3*x^2-3*x+1)/(x^4+x^3+x^2+x-1)/(-1+x)^4 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 505, 994, 1946, 3793, 7368, 14277, 27616, 53353, 102992, 198708, 383245, 738995, 1424774, 2746710, 5294884, 10206714, 19674644, 37924746, 73103036, 140911465, 271616517] The limit of a(n+1)/a(n) as n goes to infinity is 1.92756197548 a(n) is asymptotic to .765081505018*1.92756197548^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 12, -th largest growth, that is, 1.9331849818995204468, are , [2, 1, 1, 1, 4], [4, 1, 1, 1, 2] Theorem Number, 12, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 1, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 34 30 29 28 27 26 25 24 23 ) a(n) x = - (x + 2 x + x - x + x + x + x - 3 x + x / ----- n = 0 21 20 19 18 17 16 15 14 13 12 + x + x - 3 x - x + x + 5 x - 7 x + 3 x - 2 x + 5 x 11 9 8 7 6 5 4 3 2 - 4 x + 7 x - 18 x + 24 x - 23 x + 27 x - 36 x + 35 x - 21 x / 2 5 3 2 + 7 x - 1) / ((x - x + 1) (x - x + 1) (x + x - 1) / 5 4 (x - x + 2 x - 1) 20 14 13 12 11 8 7 4 3 2 (x - x + x + 2 x - 2 x - x + x + x - 4 x + 6 x - 4 x + 1)) and in Maple format -(x^34+2*x^30+x^29-x^28+x^27+x^26+x^25-3*x^24+x^23+x^21+x^20-3*x^19-x^18+x^17+5 *x^16-7*x^15+3*x^14-2*x^13+5*x^12-4*x^11+7*x^9-18*x^8+24*x^7-23*x^6+27*x^5-36*x ^4+35*x^3-21*x^2+7*x-1)/(x^2-x+1)/(x^5-x+1)/(x^3+x^2-1)/(x^5-x^4+2*x-1)/(x^20-x ^14+x^13+2*x^12-2*x^11-x^8+x^7+x^4-4*x^3+6*x^2-4*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 505, 994, 1946, 3793, 7368, 14277, 27617, 53363, 103049, 198952, 384116, 741735, 1432633, 2767735, 5348192, 10336306, 19979327, 38622105, 74664785, 144347422, 279066317] The limit of a(n+1)/a(n) as n goes to infinity is 1.93318498190 a(n) is asymptotic to .720431945925*1.93318498190^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 13, -th largest growth, that is, 1.9407101328380924652, are , [1, 2, 1, 2, 3], [1, 2, 1, 3, 2], [1, 2, 2, 1, 3], [1, 2, 3, 1, 2], [1, 3, 1, 2, 2], [1, 3, 2, 1, 2], [2, 1, 2, 3, 1], [2, 1, 3, 2, 1], [2, 2, 1, 3, 1], [2, 3, 1, 2, 1], [3, 1, 2, 2, 1], [3, 2, 1, 2, 1] Theorem Number, 13, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 1, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 11 10 9 8 6 5 4 3 2 x - x + x + x - 2 x + 4 x - 2 x - 3 x + 6 x - 4 x + 1 ----------------------------------------------------------------- 10 9 7 6 5 4 3 2 (-1 + x) (x - x + x + x - 3 x + 3 x + x - 5 x + 4 x - 1) and in Maple format (x^11-x^10+x^9+x^8-2*x^6+4*x^5-2*x^4-3*x^3+6*x^2-4*x+1)/(-1+x)/(x^10-x^9+x^7+x^ 6-3*x^5+3*x^4+x^3-5*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 505, 994, 1946, 3794, 7376, 14316, 27764, 53833, 104389, 202467, 392783, 762135, 1478994, 2870335, 5570737, 10811733, 20983342, 40723827, 79034590, 153384570, 297675706] The limit of a(n+1)/a(n) as n goes to infinity is 1.94071013284 a(n) is asymptotic to .683783349401*1.94071013284^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 14, -th largest growth, that is, 1.9409751179367153000, are , [2, 1, 1, 2, 3], [2, 1, 1, 3, 2], [2, 2, 1, 1, 3], [2, 3, 1, 1, 2], [3, 1, 1, 2, 2], [3, 2, 1, 1, 2] Theorem Number, 14, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 13 11 10 7 6 5 4 3 2 x + x + x + x - 2 x + 4 x - 2 x - 3 x + 6 x - 4 x + 1 - ---------------------------------------------------------------------- 14 12 10 8 7 6 5 4 3 2 x + x - x + x - x + 4 x - 6 x + 2 x + 6 x - 9 x + 5 x - 1 and in Maple format -(x^13+x^11+x^10+x^7-2*x^6+4*x^5-2*x^4-3*x^3+6*x^2-4*x+1)/(x^14+x^12-x^10+x^8-x ^7+4*x^6-6*x^5+2*x^4+6*x^3-9*x^2+5*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 505, 994, 1946, 3794, 7375, 14308, 27728, 53712, 104049, 201620, 390846, 757976, 1470478, 2853511, 5538369, 10750613, 20869316, 40512499, 78643837, 152661544, 296334509] The limit of a(n+1)/a(n) as n goes to infinity is 1.94097511794 a(n) is asymptotic to .677953839844*1.94097511794^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 15, -th largest growth, that is, 1.9417130342786384772, are , [3, 1, 1, 1, 3] Theorem Number, 15, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 1, 1, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 33 32 31 30 29 28 27 ) a(n) x = - (x + 6 x + 15 x + 20 x + 17 x + 15 x + 18 x / ----- n = 0 26 25 24 23 22 21 20 19 18 + 17 x + 9 x + 2 x + 2 x + 3 x - 2 x - 5 x + x + 3 x 17 16 14 13 12 11 10 8 7 - 2 x - 3 x + 4 x + 2 x - 7 x + 3 x + 3 x - 2 x - 7 x 6 5 4 3 2 / 5 4 + 11 x + 6 x - 29 x + 34 x - 21 x + 7 x - 1) / ((x + x - x + 1) / 5 4 5 4 3 20 19 18 17 (x + x + x - 1) (x + x - x + 2 x - 1) (x + 4 x + 6 x + 4 x 16 12 10 8 6 4 3 2 + x + x - x - x + x + x - 4 x + 6 x - 4 x + 1)) and in Maple format -(x^33+6*x^32+15*x^31+20*x^30+17*x^29+15*x^28+18*x^27+17*x^26+9*x^25+2*x^24+2*x ^23+3*x^22-2*x^21-5*x^20+x^19+3*x^18-2*x^17-3*x^16+4*x^14+2*x^13-7*x^12+3*x^11+ 3*x^10-2*x^8-7*x^7+11*x^6+6*x^5-29*x^4+34*x^3-21*x^2+7*x-1)/(x^5+x^4-x+1)/(x^5+ x^4+x-1)/(x^5+x^4-x^3+2*x-1)/(x^20+4*x^19+6*x^18+4*x^17+x^16+x^12-x^10-x^8+x^6+ x^4-4*x^3+6*x^2-4*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 505, 994, 1946, 3793, 7368, 14278, 27627, 53420, 103293, 199823, 386856, 749593, 1453645, 2820949, 5477286, 10638840, 20668699, 40157379, 78020560, 151571392, 294425861] The limit of a(n+1)/a(n) as n goes to infinity is 1.94171303428 a(n) is asymptotic to .665971509149*1.94171303428^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 16, -th largest growth, that is, 1.9454365275632690792, are , [2, 1, 2, 1, 3], [2, 1, 3, 1, 2], [3, 1, 2, 1, 2] Theorem Number, 16, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 2, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 17 14 13 12 10 9 8 7 ) a(n) x = - (x + 2 x - 3 x + 2 x + 4 x - 10 x + 9 x - 4 x / ----- n = 0 6 4 3 2 / + 3 x - 11 x + 19 x - 15 x + 6 x - 1) / ( / 6 5 4 3 2 (x - x + 2 x - x - 2 x + 3 x - 1) 12 10 9 8 6 5 4 3 2 (x - x + x + x - 3 x + 3 x - 2 x + 4 x - 6 x + 4 x - 1)) and in Maple format -(x^17+2*x^14-3*x^13+2*x^12+4*x^10-10*x^9+9*x^8-4*x^7+3*x^6-11*x^4+19*x^3-15*x^ 2+6*x-1)/(x^6-x^5+2*x^4-x^3-2*x^2+3*x-1)/(x^12-x^10+x^9+x^8-3*x^6+3*x^5-2*x^4+4 *x^3-6*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 505, 994, 1946, 3794, 7377, 14323, 27795, 53944, 104738, 203470, 395480, 769021, 1495859, 2910260, 5662646, 11018497, 21439793, 41716082, 81164766, 157911440, 307216819] The limit of a(n+1)/a(n) as n goes to infinity is 1.94543652756 a(n) is asymptotic to .656001947608*1.94543652756^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 17, -th largest growth, that is, 1.9515637714286765859, are , [2, 1, 2, 2, 2], [2, 2, 2, 1, 2] Theorem Number, 17, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 2, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 5 3 2 4 2 (x - x + 3 x - 3 x + 1) (x + x - x + 1) - ----------------------------------------------------------------- 10 9 8 7 6 5 4 3 2 x - x + x + x - 3 x + 5 x - 8 x + 11 x - 10 x + 5 x - 1 and in Maple format -(x^5-x^3+3*x^2-3*x+1)*(x^4+x^2-x+1)/(x^10-x^9+x^8+x^7-3*x^6+5*x^5-8*x^4+11*x^3 -10*x^2+5*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 505, 994, 1947, 3802, 7413, 14446, 28152, 54876, 107003, 208709, 407186, 794549, 1550588, 3026198, 5906192, 11527032, 22496889, 43905714, 85686751, 167225164, 326352059] The limit of a(n+1)/a(n) as n goes to infinity is 1.95156377143 a(n) is asymptotic to .634159562704*1.95156377143^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 18, -th largest growth, that is, 1.9527971478516900544, are , [2, 2, 1, 2, 2] Theorem Number, 18, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 2, 1, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 3 ) a(n) x = - (x - x + 1) / ----- n = 0 8 7 6 5 4 3 2 / 12 11 (x - 2 x + 4 x - 6 x + 8 x - 8 x + 7 x - 4 x + 1) / (x - 2 x / 10 9 8 7 6 5 4 3 2 + 4 x - 5 x + 5 x - 2 x - 4 x + 12 x - 19 x + 21 x - 15 x + 6 x - 1) and in Maple format -(x^3-x+1)*(x^8-2*x^7+4*x^6-6*x^5+8*x^4-8*x^3+7*x^2-4*x+1)/(x^12-2*x^11+4*x^10-\ 5*x^9+5*x^8-2*x^7-4*x^6+12*x^5-19*x^4+21*x^3-15*x^2+6*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 505, 994, 1947, 3801, 7406, 14418, 28068, 54666, 106542, 207795, 405527, 791788, 1546434, 3020814, 5901168, 11527651, 22517214, 43979794, 85893053, 167739492, 327561567] The limit of a(n+1)/a(n) as n goes to infinity is 1.95279714785 a(n) is asymptotic to .624456866523*1.95279714785^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 19, -th largest growth, that is, 1.9611865309023902347, are , [1, 2, 2, 2, 2], [2, 2, 2, 2, 1] Theorem Number, 19, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 2, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 8 7 6 5 4 3 2 \ n x - x + 2 x - 3 x + 5 x - 7 x + 7 x - 4 x + 1 ) a(n) x = -------------------------------------------------------- / 7 6 5 4 3 2 ----- (-1 + x) (x - x + 2 x - 3 x + 5 x - 6 x + 4 x - 1) n = 0 and in Maple format (x^8-x^7+2*x^6-3*x^5+5*x^4-7*x^3+7*x^2-4*x+1)/(-1+x)/(x^7-x^6+2*x^5-3*x^4+5*x^3 -6*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 505, 995, 1954, 3831, 7507, 14711, 28836, 56539, 110878, 217463, 426518, 836536, 1640671, 3217723, 6310580, 12376181, 24271864, 47601426, 93354997, 183086318, 359066334] The limit of a(n+1)/a(n) as n goes to infinity is 1.96118653090 a(n) is asymptotic to .602016227947*1.96118653090^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 20, -th largest growth, that is, 1.9659482366454853372, are , [1, 1, 1, 6], [1, 1, 6, 1], [1, 6, 1, 1], [6, 1, 1, 1] Theorem Number, 20, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 6] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 8 7 6 3 2 \ n x - x + x - x + 3 x - 3 x + 1 ) a(n) x = ------------------------------------- / 5 4 3 2 3 ----- (x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^8-x^7+x^6-x^3+3*x^2-3*x+1)/(x^5+x^4+x^3+x^2+x-1)/(-1+x)^3 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 506, 1001, 1976, 3895, 7670, 15094, 29692, 58394, 114824, 225766, 443876, 872673, 1715670, 3372963, 6631120, 13036493, 25629130, 50385608, 99055568, 194738196, 382845296] The limit of a(n+1)/a(n) as n goes to infinity is 1.96594823665 a(n) is asymptotic to .596844324123*1.96594823665^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 21, -th largest growth, that is, 1.9671682128139660358, are , [2, 1, 1, 5], [5, 1, 1, 2] Theorem Number, 21, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 23 18 17 16 13 12 11 9 8 ) a(n) x = - (x + 2 x - x + x + 2 x - 3 x + x - x + 4 x / ----- n = 0 7 6 5 4 3 2 / 6 5 - 5 x + 3 x - x - x + 4 x - 6 x + 4 x - 1) / ((x - x + 2 x - 1) / 18 12 11 8 7 3 2 (x - x + x + x - x + x - 3 x + 3 x - 1)) and in Maple format -(x^23+2*x^18-x^17+x^16+2*x^13-3*x^12+x^11-x^9+4*x^8-5*x^7+3*x^6-x^5-x^4+4*x^3-\ 6*x^2+4*x-1)/(x^6-x^5+2*x-1)/(x^18-x^12+x^11+x^8-x^7+x^3-3*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 506, 1001, 1976, 3895, 7670, 15094, 29693, 58402, 114862, 225906, 444319, 873939, 1719036, 3381450, 6651679, 13084782, 25739844, 50634627, 99607111, 195944622, 385457266] The limit of a(n+1)/a(n) as n goes to infinity is 1.96716821281 a(n) is asymptotic to .589839993458*1.96716821281^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 22, -th largest growth, that is, 1.9691817825046685829, are , [2, 1, 2, 4], [2, 1, 4, 2], [2, 2, 1, 4], [2, 4, 1, 2], [4, 1, 2, 2], [4, 2, 1, 2] Theorem Number, 22, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 2, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 2 9 8 6 5 4 3 (x - x + 1) (x + x - x - x + 2 x + x - 2 x + 1) - -------------------------------------------------------------- 12 11 8 7 6 5 4 3 2 x - x + x + x - 3 x + 3 x - x + 2 x - 5 x + 4 x - 1 and in Maple format -(x^2-x+1)*(x^9+x^8-x^6-x^5+2*x^4+x^3-2*x+1)/(x^12-x^11+x^8+x^7-3*x^6+3*x^5-x^4 +2*x^3-5*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 506, 1001, 1976, 3895, 7671, 15101, 29723, 58504, 115164, 226722, 446387, 878943, 1730737, 3408114, 6711252, 13215856, 26024779, 51248095, 100917625, 198726161, 391329035] The limit of a(n+1)/a(n) as n goes to infinity is 1.96918178250 a(n) is asymptotic to .580721127112*1.96918178250^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 23, -th largest growth, that is, 1.9693144732632464526, are , [3, 1, 1, 4], [4, 1, 1, 3] Theorem Number, 23, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 1, 1, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 22 21 20 19 17 16 13 12 ) a(n) x = - (x + 3 x + 3 x + x + 2 x + 3 x + x + 2 x / ----- n = 0 11 10 8 7 6 5 4 3 2 - x - 2 x + 2 x + x - 4 x + 3 x - 2 x + 4 x - 6 x + 4 x - 1) / / ( / 18 17 16 15 11 9 8 6 3 2 (x + 3 x + 3 x + x - x + x + x - x + x - 3 x + 3 x - 1) 6 5 4 (x + x - x + 2 x - 1)) and in Maple format -(x^22+3*x^21+3*x^20+x^19+2*x^17+3*x^16+x^13+2*x^12-x^11-2*x^10+2*x^8+x^7-4*x^6 +3*x^5-2*x^4+4*x^3-6*x^2+4*x-1)/(x^18+3*x^17+3*x^16+x^15-x^11+x^9+x^8-x^6+x^3-3 *x^2+3*x-1)/(x^6+x^5-x^4+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 506, 1001, 1976, 3895, 7670, 15095, 29701, 58440, 115002, 226349, 445585, 877304, 1727512, 3401942, 6699668, 13194388, 25985277, 51175626, 100784702, 198482047, 390880020] The limit of a(n+1)/a(n) as n goes to infinity is 1.96931447326 a(n) is asymptotic to .578881635909*1.96931447326^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 24, -th largest growth, that is, 1.9703230372932668084, are , [1, 2, 2, 4], [1, 2, 4, 2], [1, 4, 2, 2], [2, 2, 4, 1], [2, 4, 2, 1], [4, 2, 2, 1] Theorem Number, 24, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 2, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 8 7 6 5 4 3 2 \ n x - x + 2 x - 2 x + x - x + 3 x - 3 x + 1 ) a(n) x = ------------------------------------------------ / 7 6 5 4 2 ----- (-1 + x) (x - x + 2 x - x - 2 x + 3 x - 1) n = 0 and in Maple format (x^8-x^7+2*x^6-2*x^5+x^4-x^3+3*x^2-3*x+1)/(-1+x)/(x^7-x^6+2*x^5-x^4-2*x^2+3*x-1 ) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 506, 1001, 1976, 3896, 7677, 15124, 29794, 58696, 115641, 227842, 448918, 884516, 1742794, 3433889, 6765901, 13331045, 26266496, 51753500, 101971108, 200915987, 395869328] The limit of a(n+1)/a(n) as n goes to infinity is 1.97032303729 a(n) is asymptotic to .577336084970*1.97032303729^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 25, -th largest growth, that is, 1.9706560177668563263, are , [2, 1, 3, 3], [3, 3, 1, 2] Theorem Number, 25, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 3, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 8 7 5 3 \ n x + x - x + 2 x - 2 x + 1 ) a(n) x = - ----------------------------------------- / 9 7 5 4 3 2 ----- x - x + 2 x + x - 3 x - x + 3 x - 1 n = 0 and in Maple format -(x^8+x^7-x^5+2*x^3-2*x+1)/(x^9-x^7+2*x^5+x^4-3*x^3-x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 506, 1001, 1976, 3896, 7677, 15124, 29794, 58697, 115648, 227873, 449027, 884848, 1743712, 3436259, 6771721, 13344816, 26298182, 51824871, 102129247, 201261897, 396618234] The limit of a(n+1)/a(n) as n goes to infinity is 1.97065601777 a(n) is asymptotic to .575503355863*1.97065601777^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 26, -th largest growth, that is, 1.9708395870474530685, are , [2, 3, 1, 3], [3, 1, 3, 2] Theorem Number, 26, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 3, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 2 3 6 5 4 2 (x - x + 1) (x - x + 1) (x + 2 x + x - x - x + 1) - ---------------------------------------------------------------- 12 10 9 7 6 5 4 3 2 x - x + 2 x - x + 2 x - 3 x + 3 x + x - 5 x + 4 x - 1 and in Maple format -(x^2-x+1)*(x^3-x+1)*(x^6+2*x^5+x^4-x^2-x+1)/(x^12-x^10+2*x^9-x^7+2*x^6-3*x^5+3 *x^4+x^3-5*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 506, 1001, 1976, 3895, 7672, 15108, 29753, 58605, 115459, 227507, 448346, 883614, 1741513, 3432379, 6764915, 13332930, 26277535, 51789306, 102068807, 201161313, 396456130] The limit of a(n+1)/a(n) as n goes to infinity is 1.97083958705 a(n) is asymptotic to .573659392112*1.97083958705^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 27, -th largest growth, that is, 1.9708817901785482789, are , [3, 1, 2, 3], [3, 2, 1, 3] Theorem Number, 27, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 1, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 18 17 15 14 12 11 10 9 7 ) a(n) x = - (x + x + x + 2 x - x + 2 x + x - 3 x + 3 x / ----- n = 0 6 5 4 3 2 / 20 19 16 15 - x - 3 x + 2 x + 3 x - 6 x + 4 x - 1) / (x + 2 x + 3 x + x / 14 13 12 11 10 9 8 7 5 4 - 2 x + x + 3 x - 3 x - 3 x + 5 x + x - 5 x + 5 x - 2 x 3 2 - 6 x + 9 x - 5 x + 1) and in Maple format -(x^18+x^17+x^15+2*x^14-x^12+2*x^11+x^10-3*x^9+3*x^7-x^6-3*x^5+2*x^4+3*x^3-6*x^ 2+4*x-1)/(x^20+2*x^19+3*x^16+x^15-2*x^14+x^13+3*x^12-3*x^11-3*x^10+5*x^9+x^8-5* x^7+5*x^5-2*x^4-6*x^3+9*x^2-5*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 506, 1001, 1976, 3895, 7671, 15102, 29731, 58542, 115303, 227156, 447607, 882129, 1738622, 3426860, 6754481, 13313247, 26240296, 51718450, 101933109, 200899907, 395950321] The limit of a(n+1)/a(n) as n goes to infinity is 1.97088179018 a(n) is asymptotic to .572549404359*1.97088179018^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 28, -th largest growth, that is, 1.9717270001741243154, are , [1, 3, 2, 3], [3, 2, 3, 1] Theorem Number, 28, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 3, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 3 8 5 2 (x - x + 1) (x + x + x - 2 x + 1) -------------------------------------------------------------------- 10 9 8 7 6 5 4 3 2 (-1 + x) (x + x - x + x + x - x + 2 x - x - 2 x + 3 x - 1) and in Maple format (x^3-x+1)*(x^8+x^5+x^2-2*x+1)/(-1+x)/(x^10+x^9-x^8+x^7+x^6-x^5+2*x^4-x^3-2*x^2+ 3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 506, 1001, 1976, 3896, 7678, 15131, 29823, 58791, 115912, 228549, 450654, 888603, 1752135, 3454790, 6811947, 13431304, 26482802, 52216709, 102956870, 203002578, 400265446] The limit of a(n+1)/a(n) as n goes to infinity is 1.97172700017 a(n) is asymptotic to .571406042186*1.97172700017^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 29, -th largest growth, that is, 1.9735704833094816886, are , [1, 2, 3, 3], [1, 3, 3, 2], [2, 3, 3, 1], [3, 3, 2, 1] Theorem Number, 29, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 3, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 8 7 6 5 4 2 \ n x - x + x + x - 2 x + 3 x - 3 x + 1 ) a(n) x = ----------------------------------------------- / 7 6 4 3 2 ----- (-1 + x) (x - x + 2 x - x - 2 x + 3 x - 1) n = 0 and in Maple format (x^8-x^7+x^6+x^5-2*x^4+3*x^2-3*x+1)/(-1+x)/(x^7-x^6+2*x^4-x^3-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 506, 1001, 1977, 3902, 7700, 15195, 29987, 59181, 116799, 230514, 454940, 897861, 1771996, 3497161, 6901893, 13621369, 26882727, 53054952, 104707685, 206647998, 407834395] The limit of a(n+1)/a(n) as n goes to infinity is 1.97357048331 a(n) is asymptotic to .566114946302*1.97357048331^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 30, -th largest growth, that is, 1.9756564557792322769, are , [2, 2, 2, 3], [2, 2, 3, 2], [2, 3, 2, 2], [3, 2, 2, 2] Theorem Number, 30, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 2, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 7 6 5 4 2 \ n x - x + 2 x - 2 x + 3 x - 3 x + 1 ) a(n) x = - -------------------------------------------------- / 8 7 6 5 4 3 2 ----- x - x + 2 x - 3 x + 3 x + x - 5 x + 4 x - 1 n = 0 and in Maple format -(x^7-x^6+2*x^5-2*x^4+3*x^2-3*x+1)/(x^8-x^7+2*x^6-3*x^5+3*x^4+x^3-5*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 506, 1001, 1977, 3902, 7701, 15202, 30018, 59289, 117124, 231400, 457194, 903318, 1784742, 3526162, 6966615, 13763729, 27192387, 53722620, 106136993, 209689501, 414273443] The limit of a(n+1)/a(n) as n goes to infinity is 1.97565645578 a(n) is asymptotic to .557113786055*1.97565645578^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 31, -th largest growth, that is, 1.9835828434243263304, are , [1, 1, 7], [1, 7, 1], [7, 1, 1] Theorem Number, 31, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 7] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 2 5 4 2 \ n (x - x + 1) (x + x - x - x + 1) ) a(n) x = - ------------------------------------------ / 6 5 4 3 2 2 ----- (x + x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format -(x^2-x+1)*(x^5+x^4-x^2-x+1)/(x^6+x^5+x^4+x^3+x^2+x-1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 507, 1007, 1999, 3967, 7871, 15615, 30976, 61446, 121886, 241774, 479582, 951294, 1886974, 3742973, 7424501, 14727117, 29212461, 57945341, 114939389, 227991805, 452240638] The limit of a(n+1)/a(n) as n goes to infinity is 1.98358284342 a(n) is asymptotic to .539335853149*1.98358284342^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 32, -th largest growth, that is, 1.9838613961621262283, are , [2, 1, 6], [6, 1, 2] Theorem Number, 32, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 6] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 13 6 2 \ n x + x + x - 2 x + 1 ) a(n) x = - -------------------------------- / 7 6 7 ----- (x - x + 2 x - 1) (x - x + 1) n = 0 and in Maple format -(x^13+x^6+x^2-2*x+1)/(x^7-x^6+2*x-1)/(x^7-x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 507, 1007, 1999, 3967, 7871, 15615, 30977, 61452, 121909, 241846, 479783, 951816, 1888264, 3746048, 7431636, 14743335, 29248737, 58025450, 115114467, 228371171, 453056780] The limit of a(n+1)/a(n) as n goes to infinity is 1.98386139616 a(n) is asymptotic to .538037894604*1.98386139616^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 33, -th largest growth, that is, 1.9843858253440954550, are , [3, 1, 5], [5, 1, 3] Theorem Number, 33, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 1, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 12 11 5 2 \ n x + x + x + x - 2 x + 1 ) a(n) x = - ------------------------------------------ / 7 6 7 6 5 ----- (x + x - x + 1) (x + x - x + 2 x - 1) n = 0 and in Maple format -(x^12+x^11+x^5+x^2-2*x+1)/(x^7+x^6-x+1)/(x^7+x^6-x^5+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 507, 1007, 1999, 3967, 7871, 15616, 30983, 61475, 121981, 242047, 480305, 953105, 1891331, 3753146, 7447722, 14779206, 29327719, 58197597, 115486576, 229169995, 454761712] The limit of a(n+1)/a(n) as n goes to infinity is 1.98438582534 a(n) is asymptotic to .535796930829*1.98438582534^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 34, -th largest growth, that is, 1.9846407398915826487, are , [2, 2, 5], [2, 5, 2], [5, 2, 2] Theorem Number, 34, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 2, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 7 6 5 2 \ n x - x + x + x - 2 x + 1 ) a(n) x = - ------------------------------------ / 8 7 6 5 2 ----- x - x + 2 x - x - 2 x + 3 x - 1 n = 0 and in Maple format -(x^7-x^6+x^5+x^2-2*x+1)/(x^8-x^7+2*x^6-x^5-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 507, 1007, 1999, 3967, 7872, 15621, 30999, 61518, 122087, 242295, 480866, 954346, 1894037, 3758990, 7460256, 14805943, 29384495, 58317683, 115739663, 229701656, 455875258] The limit of a(n+1)/a(n) as n goes to infinity is 1.98464073989 a(n) is asymptotic to .535043291827*1.98464073989^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 35, -th largest growth, that is, 1.9850654703526320630, are , [3, 2, 4], [4, 2, 3] Theorem Number, 35, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 2, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 10 6 5 4 2 x + x - x + x + x - 2 x + 1 - ----------------------------------------------------------- 12 11 10 8 7 6 5 4 2 x + x - x + x + x - x + 2 x - x - 2 x + 3 x - 1 and in Maple format -(x^10+x^6-x^5+x^4+x^2-2*x+1)/(x^12+x^11-x^10+x^8+x^7-x^6+2*x^5-x^4-2*x^2+3*x-1 ) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 507, 1007, 1999, 3967, 7872, 15622, 31005, 61541, 122158, 242490, 481363, 955550, 1896852, 3765407, 7474616, 14837636, 29453698, 58467508, 116061775, 230390109, 457339279] The limit of a(n+1)/a(n) as n goes to infinity is 1.98506547035 a(n) is asymptotic to .533326675988*1.98506547035^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 36, -th largest growth, that is, 1.9853288885629234253, are , [4, 1, 4] Theorem Number, 36, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [4, 1, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 11 10 9 4 2 \ n x + x + x + x + x - 2 x + 1 ) a(n) x = - ---------------------------------------------------- / 7 6 5 4 7 6 5 ----- (x + x + x - x + 2 x - 1) (x + x + x - x + 1) n = 0 and in Maple format -(x^11+x^10+x^9+x^4+x^2-2*x+1)/(x^7+x^6+x^5-x^4+2*x-1)/(x^7+x^6+x^5-x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 507, 1007, 1999, 3967, 7872, 15622, 31006, 61547, 122182, 242568, 481586, 956135, 1898297, 3768827, 7482467, 14855267, 29492665, 58552621, 116246024, 230786160, 458185704] The limit of a(n+1)/a(n) as n goes to infinity is 1.98532888856 a(n) is asymptotic to .532190286380*1.98532888856^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 37, -th largest growth, that is, 1.9855529777414181545, are , [2, 3, 4], [2, 4, 3], [3, 4, 2], [4, 3, 2] Theorem Number, 37, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 3, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 7 5 4 2 \ n x - x + x + x - 2 x + 1 ) a(n) x = - ------------------------------------ / 8 7 5 4 2 ----- x - x + 2 x - x - 2 x + 3 x - 1 n = 0 and in Maple format -(x^7-x^5+x^4+x^2-2*x+1)/(x^8-x^7+2*x^5-x^4-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 507, 1007, 1999, 3968, 7877, 15638, 31048, 61646, 122401, 243035, 482562, 958157, 1902477, 3777474, 7500378, 14892398, 29569641, 58712080, 116575933, 231467677, 459591323] The limit of a(n+1)/a(n) as n goes to infinity is 1.98555297774 a(n) is asymptotic to .532019371341*1.98555297774^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 38, -th largest growth, that is, 1.9874108030247649893, are , [3, 3, 3] Theorem Number, 38, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 3, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 2 4 3 2 \ n (x - x + 1) (x + x - x - x + 1) ) a(n) x = - ----------------------------------------- / 8 7 6 4 3 2 ----- x + x - x + 2 x - x - 2 x + 3 x - 1 n = 0 and in Maple format -(x^2-x+1)*(x^4+x^3-x^2-x+1)/(x^8+x^7-x^6+2*x^4-x^3-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 507, 1007, 2000, 3973, 7894, 15687, 31176, 61961, 123146, 244748, 486422, 966726, 1921283, 3818373, 7588662, 15081768, 29973645, 59569927, 118389912, 235289409, 467616760] The limit of a(n+1)/a(n) as n goes to infinity is 1.98741080302 a(n) is asymptotic to .526333163485*1.98741080302^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 39, -th largest growth, that is, 1.9919641966050350211, are , [1, 8], [8, 1] Theorem Number, 39, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 8] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 2 6 5 3 2 \ n (x - x + 1) (x + x - x - x + 1) ) a(n) x = ---------------------------------------------- / 7 6 5 4 3 2 ----- (-1 + x) (x + x + x + x + x + x + x - 1) n = 0 and in Maple format (x^2-x+1)*(x^6+x^5-x^3-x^2+1)/(-1+x)/(x^7+x^6+x^5+x^4+x^3+x^2+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 508, 1012, 2016, 4016, 8000, 15936, 31744, 63233, 125958, 250904, 499792, 995568, 1983136, 3950336, 7868928, 15674623, 31223288, 62195672, 123891552, 246787536, 491591936] The limit of a(n+1)/a(n) as n goes to infinity is 1.99196419661 a(n) is asymptotic to .516605356304*1.99196419661^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 40, -th largest growth, that is, 1.9920300868462484222, are , [2, 7], [7, 2] Theorem Number, 40, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 7] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 7 \ n x - x + 1 ) a(n) x = - ----------------- / 8 7 ----- x - x + 2 x - 1 n = 0 and in Maple format -(x^7-x+1)/(x^8-x^7+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 508, 1012, 2016, 4016, 8000, 15936, 31745, 63237, 125970, 250936, 499872, 995760, 1983584, 3951359, 7871226, 15679719, 31234472, 62220008, 123944128, 246900432, 491833089] The limit of a(n+1)/a(n) as n goes to infinity is 1.99203008685 a(n) is asymptotic to .516346142451*1.99203008685^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 41, -th largest growth, that is, 1.9921580953553798820, are , [3, 6], [6, 3] Theorem Number, 41, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 6] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 6 \ n x - x + 1 ) a(n) x = - ---------------------- / 8 7 6 ----- x + x - x + 2 x - 1 n = 0 and in Maple format -(x^6-x+1)/(x^8+x^7-x^6+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 508, 1012, 2016, 4016, 8000, 15937, 31749, 63249, 126002, 251016, 500064, 996207, 1984602, 3953641, 7876278, 15690791, 31258536, 62271945, 124055559, 247138286, 492338537] The limit of a(n+1)/a(n) as n goes to infinity is 1.99215809536 a(n) is asymptotic to .515881333788*1.99215809536^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 42, -th largest growth, that is, 1.9924010004614550874, are , [4, 5], [5, 4] Theorem Number, 42, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [4, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 5 \ n x - x + 1 ) a(n) x = - --------------------------- / 8 7 6 5 ----- x + x + x - x + 2 x - 1 n = 0 and in Maple format -(x^5-x+1)/(x^8+x^7+x^6-x^5+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 508, 1012, 2016, 4016, 8001, 15941, 31761, 63281, 126082, 251207, 500506, 997209, 1986840, 3958581, 7887079, 15714222, 31309030, 62380142, 124286258, 247628067, 493374412] The limit of a(n+1)/a(n) as n goes to infinity is 1.99240100046 a(n) is asymptotic to .515079291766*1.99240100046^n ---------------------------------------------------------------------------- The compositions of, 9, that yield the, 43, -th largest growth, that is, 1.9960311797354145898, are , [9] Theorem Number, 43, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [9] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 1 ) a(n) x = - ---------------------------------------- / 8 7 6 5 4 3 2 ----- x + x + x + x + x + x + x + x - 1 n = 0 and in Maple format -1/(x^8+x^7+x^6+x^5+x^4+x^3+x^2+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, 1016, 2028, 4048, 8080, 16128, 32192, 64256, 128257, 256005, 510994, 1019960, 2035872, 4063664, 8111200, 16190208, 32316160, 64504063, 128752121, 256993248, 512966536] The limit of a(n+1)/a(n) as n goes to infinity is 1.99603117974 a(n) is asymptotic to .507071734457*1.99603117974^n ---------------------------------------------------------------------------- This ends this article, that took, 6.727, seconds to generate. ---------------------------------------------------------- ---------------------------------------- Generating functions and Growth rates for the Enumerating Sequences for Como\ sitions Avoiding Each of the compositions of, 10 By Shalosh B. Ekhad The compositions of, 10, that yield polynomial growth (hence growth ratio 1, that is the lowest) are , [2, 1, 1, 1, 1, 1, 1, 2] Theorem Number, 1, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 1, 1, 1, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 65 63 62 61 60 59 57 56 ) a(n) x = - (x + 2 x + x + x + x + x - x + 6 x / ----- n = 0 55 54 53 51 50 49 48 47 - 12 x + 14 x - 9 x + 3 x - x + 8 x - 38 x + 85 x 46 45 44 43 42 41 40 - 119 x + 101 x - 19 x - 84 x + 177 x - 297 x + 435 x 39 38 37 36 35 34 33 - 432 x + 128 x + 393 x - 791 x + 814 x - 802 x + 1746 x 32 31 30 29 28 27 - 4517 x + 8872 x - 13127 x + 14839 x - 12325 x + 5846 x 26 25 24 23 22 21 + 3334 x - 15840 x + 35735 x - 67411 x + 108916 x - 147678 x 20 19 18 17 16 + 163952 x - 140026 x + 66452 x + 60424 x - 243380 x 15 14 13 12 11 + 478967 x - 740707 x + 969352 x - 1090740 x + 1057036 x 10 9 8 7 6 5 - 879887 x + 626040 x - 378048 x + 191879 x - 80770 x + 27683 x 4 3 2 / 3 2 21 - 7526 x + 1561 x - 232 x + 22 x - 1) / ((x - x + 2 x - 1) (x / 14 13 12 11 10 9 8 7 6 5 4 - x + x + 5 x - 10 x + 5 x + x - x + x - 7 x + 21 x - 35 x 3 2 21 14 13 12 11 10 + 35 x - 21 x + 7 x - 1) (x + 2 x - 7 x + 8 x - 2 x - 2 x 9 7 6 5 4 3 2 21 14 + x + x - 7 x + 21 x - 35 x + 35 x - 21 x + 7 x - 1) (x - x 13 12 11 10 7 6 5 4 3 2 + 4 x - 6 x + 4 x - x + x - 7 x + 21 x - 35 x + 35 x - 21 x + 7 x - 1)) and in Maple format -(x^65+2*x^63+x^62+x^61+x^60+x^59-x^57+6*x^56-12*x^55+14*x^54-9*x^53+3*x^51-x^ 50+8*x^49-38*x^48+85*x^47-119*x^46+101*x^45-19*x^44-84*x^43+177*x^42-297*x^41+ 435*x^40-432*x^39+128*x^38+393*x^37-791*x^36+814*x^35-802*x^34+1746*x^33-4517*x ^32+8872*x^31-13127*x^30+14839*x^29-12325*x^28+5846*x^27+3334*x^26-15840*x^25+ 35735*x^24-67411*x^23+108916*x^22-147678*x^21+163952*x^20-140026*x^19+66452*x^ 18+60424*x^17-243380*x^16+478967*x^15-740707*x^14+969352*x^13-1090740*x^12+ 1057036*x^11-879887*x^10+626040*x^9-378048*x^8+191879*x^7-80770*x^6+27683*x^5-\ 7526*x^4+1561*x^3-232*x^2+22*x-1)/(x^3-x^2+2*x-1)/(x^21-x^14+x^13+5*x^12-10*x^ 11+5*x^10+x^9-x^8+x^7-7*x^6+21*x^5-35*x^4+35*x^3-21*x^2+7*x-1)/(x^21+2*x^14-7*x ^13+8*x^12-2*x^11-2*x^10+x^9+x^7-7*x^6+21*x^5-35*x^4+35*x^3-21*x^2+7*x-1)/(x^21 -x^14+4*x^13-6*x^12+4*x^11-x^10+x^7-7*x^6+21*x^5-35*x^4+35*x^3-21*x^2+7*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1014, 1991, 3853, 7330, 13693, 25119, 45288, 80357, 140553, 242803, 415142, 704228, 1188356, 2000280, 3367607, 5684752, 9641648, 16454525, 28280058, 48958244] ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 2, -th largest growth, that is, 1.6180339887498948482, are , [1, 1, 1, 1, 1, 1, 1, 3], [1, 1, 1, 1, 1, 1, 3, 1], [1, 1, 1, 1, 1, 3, 1, 1], [1, 1, 1, 1, 3, 1, 1, 1], [1, 1, 1, 3, 1, 1, 1, 1], [1, 1, 3, 1, 1, 1, 1, 1], [1, 3, 1, 1, 1, 1, 1, 1], [3, 1, 1, 1, 1, 1, 1, 1] Theorem Number, 2, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 1, 1, 1, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 9 8 7 6 5 4 3 2 x - 3 x + 8 x - 6 x - 10 x + 30 x - 34 x + 21 x - 7 x + 1 ----------------------------------------------------------------- 2 7 (x + x - 1) (-1 + x) and in Maple format (x^9-3*x^8+8*x^7-6*x^6-10*x^5+30*x^4-34*x^3+21*x^2-7*x+1)/(x^2+x-1)/(-1+x)^7 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1014, 1991, 3853, 7330, 13693, 25119, 45288, 80356, 140537, 242671, 414388, 700855, 1175703, 1958718, 3244477, 5348694, 8783222, 14377422, 23474556, 38249572] The limit of a(n+1)/a(n) as n goes to infinity is 1.61803398875 a(n) is asymptotic to 21.0095194942*1.61803398875^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 3, -th largest growth, that is, 1.7548776662466927601, are , [1, 1, 1, 1, 2, 1, 1, 2], [1, 1, 1, 2, 1, 1, 2, 1], [1, 1, 2, 1, 1, 2, 1, 1], [1, 2, 1, 1, 2, 1, 1, 1], [2, 1, 1, 2, 1, 1, 1, 1] Theorem Number, 3, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 1, 2, 1, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 15 14 13 12 11 10 9 ) a(n) x = - (3 x - 6 x + 13 x - 17 x + 24 x - 34 x + 50 x / ----- n = 0 8 7 6 5 4 3 2 / - 68 x + 83 x - 96 x + 103 x - 93 x + 63 x - 29 x + 8 x - 1) / ( / 3 2 9 6 5 4 3 2 4 (x - x + 2 x - 1) (x - x + 2 x - x + x - 3 x + 3 x - 1) (-1 + x) ) and in Maple format -(3*x^15-6*x^14+13*x^13-17*x^12+24*x^11-34*x^10+50*x^9-68*x^8+83*x^7-96*x^6+103 *x^5-93*x^4+63*x^3-29*x^2+8*x-1)/(x^3-x^2+2*x-1)/(x^9-x^6+2*x^5-x^4+x^3-3*x^2+3 *x-1)/(-1+x)^4 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1014, 1991, 3853, 7331, 13705, 25197, 45654, 81747, 145088, 255975, 450012, 789786, 1385413, 2430613, 4265962, 7489722, 13152028, 23095240, 40550770, 71186142] The limit of a(n+1)/a(n) as n goes to infinity is 1.75487766625 a(n) is asymptotic to 3.34834608804*1.75487766625^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 4, -th largest growth, that is, 1.8392867552141611326, are , [1, 1, 1, 1, 1, 1, 4], [1, 1, 1, 1, 1, 4, 1], [1, 1, 1, 1, 4, 1, 1], [1, 1, 1, 4, 1, 1, 1], [1, 1, 4, 1, 1, 1, 1], [1, 4, 1, 1, 1, 1, 1], [4, 1, 1, 1, 1, 1, 1] Theorem Number, 4, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 1, 1, 1, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 8 7 6 5 4 3 2 3 x - 6 x + 8 x - 10 x + 16 x - 20 x + 15 x - 6 x + 1 - ------------------------------------------------------------ 3 2 6 (x + x + x - 1) (-1 + x) and in Maple format -(3*x^8-6*x^7+8*x^6-10*x^5+16*x^4-20*x^3+15*x^2-6*x+1)/(x^3+x^2+x-1)/(-1+x)^6 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1015, 2001, 3909, 7563, 14497, 27555, 51995, 97520, 182014, 338414, 627350, 1160394, 2142822, 3952266, 7283378, 13413909, 24694105, 45446847, 83623267, 153847901] The limit of a(n+1)/a(n) as n goes to infinity is 1.83928675521 a(n) is asymptotic to 1.76938057252*1.83928675521^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 5, -th largest growth, that is, 1.8667603991738620930, are , [2, 1, 1, 1, 1, 1, 3], [3, 1, 1, 1, 1, 1, 2] Theorem Number, 5, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 1, 1, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 63 60 59 58 57 56 55 54 ) a(n) x = - (x + 2 x + x + x + x - x + 4 x - 2 x / ----- n = 0 53 52 50 49 48 47 46 45 - 6 x + 9 x - 10 x + 2 x + 10 x - 17 x + 31 x - 23 x 44 43 42 41 40 39 38 37 - 26 x + 62 x - 30 x - 27 x + 24 x + 15 x + 12 x - 112 x 36 35 34 33 32 31 30 + 176 x - 108 x - 102 x + 300 x - 243 x - 16 x + 122 x 29 28 27 26 25 24 23 + 27 x - 175 x + 248 x - 462 x + 760 x - 646 x - 146 x 22 21 20 19 18 17 16 + 1077 x - 1243 x + 464 x + 396 x - 490 x + 17 x + 168 x 15 14 13 12 11 10 9 + 489 x - 2017 x + 3977 x - 5184 x + 4002 x + 165 x - 5675 x 8 7 6 5 4 3 2 + 9571 x - 9996 x + 7540 x - 4262 x + 1805 x - 559 x + 120 x - 16 x / 2 12 8 7 6 5 3 2 + 1) / ((x - x + 1) (x + x - x - x + x + x - 3 x + 3 x - 1) / 4 3 12 8 7 6 5 3 2 (x - x + 2 x - 1) (x - x + x + x - x + x - 3 x + 3 x - 1) 6 5 2 24 17 16 15 14 13 12 (x + x - x - x + 1) (x - x + 5 x - 6 x - 2 x + 4 x + 4 x 11 9 8 6 5 4 3 2 - 4 x - x + x + x - 6 x + 15 x - 20 x + 15 x - 6 x + 1) 4 (x - x + 1)) and in Maple format -(x^63+2*x^60+x^59+x^58+x^57-x^56+4*x^55-2*x^54-6*x^53+9*x^52-10*x^50+2*x^49+10 *x^48-17*x^47+31*x^46-23*x^45-26*x^44+62*x^43-30*x^42-27*x^41+24*x^40+15*x^39+ 12*x^38-112*x^37+176*x^36-108*x^35-102*x^34+300*x^33-243*x^32-16*x^31+122*x^30+ 27*x^29-175*x^28+248*x^27-462*x^26+760*x^25-646*x^24-146*x^23+1077*x^22-1243*x^ 21+464*x^20+396*x^19-490*x^18+17*x^17+168*x^16+489*x^15-2017*x^14+3977*x^13-\ 5184*x^12+4002*x^11+165*x^10-5675*x^9+9571*x^8-9996*x^7+7540*x^6-4262*x^5+1805* x^4-559*x^3+120*x^2-16*x+1)/(x^2-x+1)/(x^12+x^8-x^7-x^6+x^5+x^3-3*x^2+3*x-1)/(x ^4-x^3+2*x-1)/(x^12-x^8+x^7+x^6-x^5+x^3-3*x^2+3*x-1)/(x^6+x^5-x^2-x+1)/(x^24-x^ 17+5*x^16-6*x^15-2*x^14+4*x^13+4*x^12-4*x^11-x^9+x^8+x^6-6*x^5+15*x^4-20*x^3+15 *x^2-6*x+1)/(x^4-x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1015, 2001, 3909, 7563, 14497, 27555, 51995, 97521, 182028, 338521, 627937, 1162977, 2152509, 3984440, 7380556, 13686022, 25410854, 47242840, 87942397, 163886779] The limit of a(n+1)/a(n) as n goes to infinity is 1.86676039917 a(n) is asymptotic to 1.21492409182*1.86676039917^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 6, -th largest growth, that is, 1.8908049048980778372, are , [2, 1, 1, 1, 2, 1, 2], [2, 1, 2, 1, 1, 1, 2] Theorem Number, 6, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 1, 2, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 37 36 35 34 33 32 31 29 28 ) a(n) x = - (x + x + x + 2 x - x + x - 2 x + x - x / ----- n = 0 26 25 24 23 22 21 20 19 + 5 x - x - 19 x + 36 x - 22 x - 38 x + 105 x - 77 x 18 17 16 15 14 13 12 - 98 x + 324 x - 428 x + 318 x - 105 x + 53 x - 283 x 11 10 9 8 7 6 5 + 519 x - 256 x - 731 x + 2039 x - 2881 x + 2772 x - 1935 x 4 3 2 / + 989 x - 363 x + 91 x - 14 x + 1) / ( / 8 7 6 5 4 3 2 30 29 27 26 (x - x + x + x - 3 x + 5 x - 6 x + 4 x - 1) (x + x - x - x 23 22 21 20 19 18 17 16 15 + 3 x + x - 3 x - 7 x + 10 x - x - 8 x + 15 x + 3 x 14 13 12 11 10 9 8 7 - 55 x + 77 x - 20 x - 60 x + 67 x + 17 x - 86 x + 12 x 6 5 4 3 2 + 177 x - 307 x + 276 x - 154 x + 54 x - 11 x + 1)) and in Maple format -(x^37+x^36+x^35+2*x^34-x^33+x^32-2*x^31+x^29-x^28+5*x^26-x^25-19*x^24+36*x^23-\ 22*x^22-38*x^21+105*x^20-77*x^19-98*x^18+324*x^17-428*x^16+318*x^15-105*x^14+53 *x^13-283*x^12+519*x^11-256*x^10-731*x^9+2039*x^8-2881*x^7+2772*x^6-1935*x^5+ 989*x^4-363*x^3+91*x^2-14*x+1)/(x^8-x^7+x^6+x^5-3*x^4+5*x^3-6*x^2+4*x-1)/(x^30+ x^29-x^27-x^26+3*x^23+x^22-3*x^21-7*x^20+10*x^19-x^18-8*x^17+15*x^16+3*x^15-55* x^14+77*x^13-20*x^12-60*x^11+67*x^10+17*x^9-86*x^8+12*x^7+177*x^6-307*x^5+276*x ^4-154*x^3+54*x^2-11*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1015, 2001, 3909, 7564, 14507, 27613, 52249, 98450, 185017, 347249, 651554, 1223092, 2298116, 4323124, 8142720, 15355531, 28988376, 54772869, 103561954, 195902705] The limit of a(n+1)/a(n) as n goes to infinity is 1.89080490490 a(n) is asymptotic to .988014882613*1.89080490490^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 7, -th largest growth, that is, 1.8922218871524161071, are , [1, 2, 1, 1, 2, 1, 2], [1, 2, 1, 2, 1, 1, 2], [2, 1, 1, 2, 1, 2, 1], [2, 1, 2, 1, 1, 2, 1] Theorem Number, 7, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 1, 1, 2, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 17 16 15 14 13 12 11 10 ) a(n) x = (x - x + 3 x - 2 x + 3 x + x - 7 x + 14 x / ----- n = 0 9 8 7 6 5 4 3 2 - 16 x + 8 x + 8 x - 28 x + 45 x - 51 x + 41 x - 22 x + 7 x - 1) / 16 15 14 13 12 11 10 9 / ((-1 + x) (x - x + 2 x - 2 x + x + 3 x - 9 x + 13 x / 8 7 6 5 4 3 2 - 10 x - x + 16 x - 31 x + 40 x - 36 x + 21 x - 7 x + 1)) and in Maple format (x^17-x^16+3*x^15-2*x^14+3*x^13+x^12-7*x^11+14*x^10-16*x^9+8*x^8+8*x^7-28*x^6+ 45*x^5-51*x^4+41*x^3-22*x^2+7*x-1)/(-1+x)/(x^16-x^15+2*x^14-2*x^13+x^12+3*x^11-\ 9*x^10+13*x^9-10*x^8-x^7+16*x^6-31*x^5+40*x^4-36*x^3+21*x^2-7*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1015, 2001, 3909, 7564, 14508, 27623, 52305, 98683, 185821, 349684, 658248, 1240164, 2339122, 4416913, 8348710, 15792895, 29891135, 56593165, 107163160, 202920970] The limit of a(n+1)/a(n) as n goes to infinity is 1.89222188715 a(n) is asymptotic to .997060153762*1.89222188715^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 8, -th largest growth, that is, 1.8922512945970379670, are , [2, 1, 1, 1, 1, 2, 2], [2, 2, 1, 1, 1, 1, 2] Theorem Number, 8, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 1, 1, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 34 32 29 28 27 26 25 24 ) a(n) x = - (x + x + 4 x - 3 x + x + 6 x - 14 x + 18 x / ----- n = 0 23 22 21 20 19 18 17 16 - 13 x - 4 x + 34 x - 67 x + 89 x - 95 x + 70 x + 21 x 15 14 13 12 11 10 9 - 195 x + 412 x - 565 x + 498 x - 75 x - 707 x + 1647 x 8 7 6 5 4 3 2 - 2394 x + 2625 x - 2256 x + 1517 x - 782 x + 298 x - 79 x + 13 x / 35 34 33 32 30 29 28 27 26 - 1) / (x - x + x - x + 4 x - 7 x + 6 x + 2 x - 15 x / 25 24 23 22 21 20 19 18 + 26 x - 29 x + 17 x + 18 x - 69 x + 115 x - 146 x + 157 x 17 16 15 14 13 12 11 - 114 x - 38 x + 315 x - 642 x + 843 x - 676 x - 50 x 10 9 8 7 6 5 4 + 1296 x - 2706 x + 3729 x - 3915 x + 3222 x - 2068 x + 1013 x 3 2 - 365 x + 91 x - 14 x + 1) and in Maple format -(x^34+x^32+4*x^29-3*x^28+x^27+6*x^26-14*x^25+18*x^24-13*x^23-4*x^22+34*x^21-67 *x^20+89*x^19-95*x^18+70*x^17+21*x^16-195*x^15+412*x^14-565*x^13+498*x^12-75*x^ 11-707*x^10+1647*x^9-2394*x^8+2625*x^7-2256*x^6+1517*x^5-782*x^4+298*x^3-79*x^2 +13*x-1)/(x^35-x^34+x^33-x^32+4*x^30-7*x^29+6*x^28+2*x^27-15*x^26+26*x^25-29*x^ 24+17*x^23+18*x^22-69*x^21+115*x^20-146*x^19+157*x^18-114*x^17-38*x^16+315*x^15 -642*x^14+843*x^13-676*x^12-50*x^11+1296*x^10-2706*x^9+3729*x^8-3915*x^7+3222*x ^6-2068*x^5+1013*x^4-365*x^3+91*x^2-14*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1015, 2001, 3910, 7572, 14544, 27741, 52615, 99363, 187076, 351560, 660114, 1239498, 2328916, 4380445, 8249451, 15555646, 29367633, 55500462, 104974373, 198672263] The limit of a(n+1)/a(n) as n goes to infinity is 1.89225129460 a(n) is asymptotic to .979774130645*1.89225129460^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 9, -th largest growth, that is, 1.8922578866301683686, are , [1, 2, 1, 1, 1, 2, 2], [1, 2, 2, 1, 1, 1, 2], [2, 1, 1, 1, 2, 2, 1], [2, 2, 1, 1, 1, 2, 1] Theorem Number, 9, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 1, 1, 1, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 20 19 18 17 16 15 14 13 ) a(n) x = (x - x + 2 x - 2 x + 5 x - 5 x + 6 x - 3 x / ----- n = 0 12 11 10 9 8 7 6 5 4 - x + 7 x - 15 x + 27 x - 41 x + 58 x - 80 x + 97 x - 92 x 3 2 / 3 16 15 12 + 63 x - 29 x + 8 x - 1) / ((-1 + x) (x + x - 1) (x - x + 3 x / 11 9 8 7 6 5 4 3 2 - 4 x + 7 x - 8 x + 2 x - 2 x + 17 x - 34 x + 35 x - 21 x + 7 x - 1)) and in Maple format (x^20-x^19+2*x^18-2*x^17+5*x^16-5*x^15+6*x^14-3*x^13-x^12+7*x^11-15*x^10+27*x^9 -41*x^8+58*x^7-80*x^6+97*x^5-92*x^4+63*x^3-29*x^2+8*x-1)/(-1+x)/(x^3+x-1)/(x^16 -x^15+3*x^12-4*x^11+7*x^9-8*x^8+2*x^7-2*x^6+17*x^5-34*x^4+35*x^3-21*x^2+7*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1015, 2001, 3910, 7572, 14544, 27742, 52627, 99439, 187417, 352785, 663881, 1249834, 2354932, 4441690, 8386192, 15848188, 29972022, 56713667, 107352430, 203243899] The limit of a(n+1)/a(n) as n goes to infinity is 1.89225788663 a(n) is asymptotic to .998835530746*1.89225788663^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 10, -th largest growth, that is, 1.8923110706522823122, are , [1, 1, 2, 1, 1, 2, 2], [1, 1, 2, 2, 1, 1, 2], [1, 2, 1, 1, 2, 2, 1], [1, 2, 2, 1, 1, 2, 1], [2, 1, 1, 2, 2, 1, 1], [2, 2, 1, 1, 2, 1, 1] Theorem Number, 10, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 2, 1, 1, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 12 11 10 9 8 7 6 ) a(n) x = - (2 x - 2 x + 3 x - 2 x + 3 x - 4 x + 10 x / ----- n = 0 5 4 3 2 / - 19 x + 26 x - 25 x + 16 x - 6 x + 1) / ( / 11 10 9 8 7 6 5 4 3 2 (x - x + x - x + x - x + 4 x - 8 x + 11 x - 10 x + 5 x - 1) 2 (-1 + x) ) and in Maple format -(2*x^12-2*x^11+3*x^10-2*x^9+3*x^8-4*x^7+10*x^6-19*x^5+26*x^4-25*x^3+16*x^2-6*x +1)/(x^11-x^10+x^9-x^8+x^7-x^6+4*x^5-8*x^4+11*x^3-10*x^2+5*x-1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1015, 2001, 3910, 7572, 14545, 27753, 52691, 99706, 188320, 355427, 670835, 1266724, 2393453, 4525234, 8560155, 16198687, 30659825, 58035946, 109856042, 207935235] The limit of a(n+1)/a(n) as n goes to infinity is 1.89231107065 a(n) is asymptotic to 1.01914948966*1.89231107065^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 11, -th largest growth, that is, 1.9087907387871591034, are , [2, 1, 1, 2, 1, 1, 2] Theorem Number, 11, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 2, 1, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 39 37 36 35 32 31 30 29 ) a(n) x = - (x + 2 x - x + x - 2 x + 6 x - 10 x + 11 x / ----- n = 0 28 27 26 25 24 23 22 - 4 x - 16 x + 56 x - 121 x + 198 x - 256 x + 262 x 21 20 19 18 17 16 15 - 178 x - 15 x + 255 x - 401 x + 358 x - 260 x + 567 x 14 13 12 11 10 9 - 1956 x + 4989 x - 9657 x + 15057 x - 19527 x + 21338 x 8 7 6 5 4 3 2 - 19647 x + 15104 x - 9543 x + 4850 x - 1927 x + 575 x - 121 x / 5 4 3 2 30 27 26 + 16 x - 1) / ((x - x + 2 x - 3 x + 3 x - 1) (x + 2 x - 3 x / 24 23 22 20 19 18 17 16 + 4 x - 10 x + 11 x - 21 x + 47 x - 52 x + 13 x + 55 x 15 14 13 12 11 10 9 8 - 111 x + 103 x - 3 x - 152 x + 264 x - 213 x - 58 x + 466 x 7 6 5 4 3 2 - 822 x + 949 x - 800 x + 496 x - 220 x + 66 x - 12 x + 1) 5 2 (x - x + 2 x - 1)) and in Maple format -(x^39+2*x^37-x^36+x^35-2*x^32+6*x^31-10*x^30+11*x^29-4*x^28-16*x^27+56*x^26-\ 121*x^25+198*x^24-256*x^23+262*x^22-178*x^21-15*x^20+255*x^19-401*x^18+358*x^17 -260*x^16+567*x^15-1956*x^14+4989*x^13-9657*x^12+15057*x^11-19527*x^10+21338*x^ 9-19647*x^8+15104*x^7-9543*x^6+4850*x^5-1927*x^4+575*x^3-121*x^2+16*x-1)/(x^5-x ^4+2*x^3-3*x^2+3*x-1)/(x^30+2*x^27-3*x^26+4*x^24-10*x^23+11*x^22-21*x^20+47*x^ 19-52*x^18+13*x^17+55*x^16-111*x^15+103*x^14-3*x^13-152*x^12+264*x^11-213*x^10-\ 58*x^9+466*x^8-822*x^7+949*x^6-800*x^5+496*x^4-220*x^3+66*x^2-12*x+1)/(x^5-x^2+ 2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1015, 2001, 3909, 7564, 14509, 27634, 52373, 98992, 186972, 353412, 669139, 1269583, 2413920, 4598297, 8772449, 16753831, 32018822, 61213409, 117036204, 223738591] The limit of a(n+1)/a(n) as n goes to infinity is 1.90879073879 a(n) is asymptotic to .846805878447*1.90879073879^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 12, -th largest growth, that is, 1.9275619754829253043, are , [1, 1, 1, 1, 1, 5], [1, 1, 1, 1, 5, 1], [1, 1, 1, 5, 1, 1], [1, 1, 5, 1, 1, 1], [1, 5, 1, 1, 1, 1], [5, 1, 1, 1, 1, 1] Theorem Number, 12, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 1, 1, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 4 3 2 5 4 3 2 \ n (x - x + 2 x - 2 x + 1) (x - x + x + 2 x - 3 x + 1) ) a(n) x = ---------------------------------------------------------- / 4 3 2 5 ----- (x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^4-x^3+2*x^2-2*x+1)*(x^5-x^4+x^3+2*x^2-3*x+1)/(x^4+x^3+x^2+x-1)/(-1+x)^5 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1016, 2010, 3956, 7749, 15117, 29394, 57010, 110363, 213355, 412063, 795308, 1534303, 2959077, 5705787, 11000671, 21207385, 40882029, 78806775, 151909811, 292821276] The limit of a(n+1)/a(n) as n goes to infinity is 1.92756197548 a(n) is asymptotic to .824830604574*1.92756197548^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 13, -th largest growth, that is, 1.9331849818995204468, are , [2, 1, 1, 1, 1, 4], [4, 1, 1, 1, 1, 2] Theorem Number, 13, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 1, 1, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 54 50 49 48 47 46 45 43 42 ) a(n) x = - (x + 2 x + x + x - x + x + 2 x - 3 x - x / ----- n = 0 41 39 37 36 35 34 33 32 31 + 4 x - 2 x - 3 x - 2 x + 7 x + 4 x - 8 x - 4 x + 9 x 30 28 27 26 25 23 22 21 20 - x - x - 8 x + 12 x - 7 x + 8 x - 6 x + 3 x - 26 x 19 18 17 16 15 14 13 12 + 52 x - 47 x + 31 x - 23 x + 8 x + 19 x - 58 x + 129 x 11 10 9 8 7 6 5 4 - 233 x + 327 x - 376 x + 405 x - 458 x + 508 x - 472 x + 331 x 3 2 / 5 4 25 16 15 - 165 x + 55 x - 11 x + 1) / ((x - x + 2 x - 1) (x - x + 2 x / 14 13 12 11 5 4 3 2 25 - x + x - 2 x + x + x - 5 x + 10 x - 10 x + 5 x - 1) (x 18 17 16 15 13 12 9 8 5 4 3 - x + x + 3 x - 3 x - 3 x + 3 x + x - x + x - 5 x + 10 x 2 - 10 x + 5 x - 1)) and in Maple format -(x^54+2*x^50+x^49+x^48-x^47+x^46+2*x^45-3*x^43-x^42+4*x^41-2*x^39-3*x^37-2*x^ 36+7*x^35+4*x^34-8*x^33-4*x^32+9*x^31-x^30-x^28-8*x^27+12*x^26-7*x^25+8*x^23-6* x^22+3*x^21-26*x^20+52*x^19-47*x^18+31*x^17-23*x^16+8*x^15+19*x^14-58*x^13+129* x^12-233*x^11+327*x^10-376*x^9+405*x^8-458*x^7+508*x^6-472*x^5+331*x^4-165*x^3+ 55*x^2-11*x+1)/(x^5-x^4+2*x-1)/(x^25-x^16+2*x^15-x^14+x^13-2*x^12+x^11+x^5-5*x^ 4+10*x^3-10*x^2+5*x-1)/(x^25-x^18+x^17+3*x^16-3*x^15-3*x^13+3*x^12+x^9-x^8+x^5-\ 5*x^4+10*x^3-10*x^2+5*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1016, 2010, 3956, 7749, 15117, 29394, 57010, 110364, 213367, 412143, 795700, 1535877, 2964563, 5722987, 11050367, 21342026, 41228546, 79662370, 153952283, 297563937] The limit of a(n+1)/a(n) as n goes to infinity is 1.93318498190 a(n) is asymptotic to .768338440425*1.93318498190^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 14, -th largest growth, that is, 1.9407101328380924652, are , [1, 1, 2, 1, 2, 3], [1, 1, 2, 1, 3, 2], [1, 1, 2, 2, 1, 3], [1, 1, 2, 3, 1, 2], [1, 1, 3, 1, 2, 2], [1, 1, 3, 2, 1, 2], [1, 2, 1, 2, 3, 1], [1, 2, 1, 3, 2, 1], [1, 2, 2, 1, 3, 1], [1, 2, 3, 1, 2, 1], [1, 3, 1, 2, 2, 1], [1, 3, 2, 1, 2, 1], [2, 1, 2, 3, 1, 1], [2, 1, 3, 2, 1, 1], [2, 2, 1, 3, 1, 1], [2, 3, 1, 2, 1, 1], [3, 1, 2, 2, 1, 1], [3, 2, 1, 2, 1, 1] Theorem Number, 14, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 2, 1, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = - ( / ----- n = 0 11 10 9 8 7 6 5 4 3 2 2 x - 2 x + x + x + 2 x - 6 x + 6 x + x - 9 x + 10 x - 5 x + 1) / 10 9 7 6 5 4 3 2 2 / ((x - x + x + x - 3 x + 3 x + x - 5 x + 4 x - 1) (-1 + x) ) / and in Maple format -(2*x^11-2*x^10+x^9+x^8+2*x^7-6*x^6+6*x^5+x^4-9*x^3+10*x^2-5*x+1)/(x^10-x^9+x^7 +x^6-3*x^5+3*x^4+x^3-5*x^2+4*x-1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1016, 2010, 3956, 7750, 15126, 29442, 57206, 111039, 215428, 417895, 810678, 1572813, 3051807, 5922142, 11492879, 22304612, 43287954, 84011781, 163046371, 316430941] The limit of a(n+1)/a(n) as n goes to infinity is 1.94071013284 a(n) is asymptotic to .726879966030*1.94071013284^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 15, -th largest growth, that is, 1.9409607910644739216, are , [2, 1, 1, 2, 1, 3], [2, 1, 1, 3, 1, 2], [2, 1, 2, 1, 1, 3], [2, 1, 3, 1, 1, 2], [3, 1, 1, 2, 1, 2], [3, 1, 2, 1, 1, 2] Theorem Number, 15, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 2, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 22 20 19 18 17 16 15 14 ) a(n) x = - (x - x + 2 x + x - 4 x + 6 x - x - 5 x / ----- n = 0 13 12 11 10 9 8 7 6 5 4 + 3 x + 7 x - 9 x + x + 2 x + 8 x - 20 x + 17 x + 5 x - 29 x 3 2 / 23 22 21 20 19 18 + 34 x - 21 x + 7 x - 1) / (x - x - x + 3 x - x - 3 x / 17 16 15 14 13 12 11 10 9 + 8 x - 8 x - x + 9 x - x - 13 x + 11 x + 2 x - 2 x 8 7 6 5 4 3 2 - 16 x + 32 x - 22 x - 15 x + 48 x - 49 x + 27 x - 8 x + 1) and in Maple format -(x^22-x^20+2*x^19+x^18-4*x^17+6*x^16-x^15-5*x^14+3*x^13+7*x^12-9*x^11+x^10+2*x ^9+8*x^8-20*x^7+17*x^6+5*x^5-29*x^4+34*x^3-21*x^2+7*x-1)/(x^23-x^22-x^21+3*x^20 -x^19-3*x^18+8*x^17-8*x^16-x^15+9*x^14-x^13-13*x^12+11*x^11+2*x^10-2*x^9-16*x^8 +32*x^7-22*x^6-15*x^5+48*x^4-49*x^3+27*x^2-8*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1016, 2010, 3956, 7749, 15118, 29404, 57067, 110608, 214238, 414882, 803545, 1556803, 3017378, 5850608, 11348275, 22018512, 42731049, 82940412, 161001402, 312545508] The limit of a(n+1)/a(n) as n goes to infinity is 1.94096079106 a(n) is asymptotic to .715631497849*1.94096079106^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 16, -th largest growth, that is, 1.9409620467295949022, are , [2, 1, 1, 1, 2, 3], [2, 1, 1, 1, 3, 2], [2, 2, 1, 1, 1, 3], [2, 3, 1, 1, 1, 2], [3, 1, 1, 1, 2, 2], [3, 2, 1, 1, 1, 2] Theorem Number, 16, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 1, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 23 22 21 20 19 18 16 15 14 ) a(n) x = - (x + x + x + x - x + 2 x + 3 x - 2 x - x / ----- n = 0 13 12 11 10 9 8 7 6 5 + 5 x - 6 x + 5 x + x - 12 x + 22 x - 26 x + 18 x + 5 x 4 3 2 / 24 23 20 19 17 - 29 x + 34 x - 21 x + 7 x - 1) / (x + x - x + 2 x + x / 16 15 14 13 12 11 10 9 8 - 4 x + 4 x + 2 x - 7 x + 9 x - 8 x - 3 x + 21 x - 35 x 7 6 5 4 3 2 + 39 x - 23 x - 15 x + 48 x - 49 x + 27 x - 8 x + 1) and in Maple format -(x^23+x^22+x^21+x^20-x^19+2*x^18+3*x^16-2*x^15-x^14+5*x^13-6*x^12+5*x^11+x^10-\ 12*x^9+22*x^8-26*x^7+18*x^6+5*x^5-29*x^4+34*x^3-21*x^2+7*x-1)/(x^24+x^23-x^20+2 *x^19+x^17-4*x^16+4*x^15+2*x^14-7*x^13+9*x^12-8*x^11-3*x^10+21*x^9-35*x^8+39*x^ 7-23*x^6-15*x^5+48*x^4-49*x^3+27*x^2-8*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1016, 2010, 3956, 7750, 15125, 29432, 57151, 110817, 214693, 415778, 805177, 1559595, 3021911, 5857640, 11358751, 22033593, 42752282, 82970386, 161045718, 312617885] The limit of a(n+1)/a(n) as n goes to infinity is 1.94096204673 a(n) is asymptotic to .715931280313*1.94096204673^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 17, -th largest growth, that is, 1.9409751179367153000, are , [1, 2, 1, 1, 2, 3], [1, 2, 1, 1, 3, 2], [1, 2, 2, 1, 1, 3], [1, 2, 3, 1, 1, 2], [1, 3, 1, 1, 2, 2], [1, 3, 2, 1, 1, 2], [2, 1, 1, 2, 3, 1], [2, 1, 1, 3, 2, 1], [2, 2, 1, 1, 3, 1], [2, 3, 1, 1, 2, 1], [3, 1, 1, 2, 2, 1], [3, 2, 1, 1, 2, 1] Theorem Number, 17, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 1, 1, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 15 13 10 9 8 7 6 5 4 3 ) a(n) x = (x + x + x + x - x + 3 x - 6 x + 6 x + x - 9 x / ----- n = 0 2 / + 10 x - 5 x + 1) / ((-1 + x) / 14 12 10 8 7 6 5 4 3 2 (x + x - x + x - x + 4 x - 6 x + 2 x + 6 x - 9 x + 5 x - 1)) and in Maple format (x^15+x^13+x^10+x^9-x^8+3*x^7-6*x^6+6*x^5+x^4-9*x^3+10*x^2-5*x+1)/(-1+x)/(x^14+ x^12-x^10+x^8-x^7+4*x^6-6*x^5+2*x^4+6*x^3-9*x^2+5*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1016, 2010, 3956, 7750, 15125, 29433, 57161, 110873, 214922, 416542, 807388, 1565364, 3035842, 5889353, 11427722, 22178335, 43047651, 83560150, 162203987, 314865531] The limit of a(n+1)/a(n) as n goes to infinity is 1.94097511794 a(n) is asymptotic to .720480092322*1.94097511794^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 18, -th largest growth, that is, 1.9417130342786384772, are , [3, 1, 1, 1, 1, 3] Theorem Number, 18, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 1, 1, 1, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 53 52 51 50 49 48 ) a(n) x = - (x + 10 x + 45 x + 120 x + 212 x + 269 x / ----- n = 0 47 46 45 44 43 42 41 + 275 x + 267 x + 263 x + 233 x + 164 x + 90 x + 41 x 40 39 38 37 36 35 34 33 32 + 12 x - 7 x - 12 x - 7 x - 3 x + x + 2 x - 2 x + 4 x 31 30 29 28 27 26 25 24 23 + 15 x + 5 x - 10 x - 3 x + 4 x - 4 x - 2 x + 6 x - 7 x 22 21 20 19 18 17 16 15 - 8 x + 22 x - 4 x - 32 x + 34 x + 4 x - 40 x + 45 x 14 13 12 11 10 9 8 7 - 13 x - 40 x + 75 x - 52 x - 31 x + 106 x - 64 x - 127 x 6 5 4 3 2 / 2 + 343 x - 417 x + 320 x - 164 x + 55 x - 11 x + 1) / ((x + 1) / 5 4 3 25 24 23 22 21 20 (x + x - x + 2 x - 1) (x + 5 x + 10 x + 10 x + 5 x + x 15 13 11 9 7 5 4 3 2 + 2 x - 5 x + 3 x + x - x + x - 5 x + 10 x - 10 x + 5 x - 1) ( 23 22 21 20 19 18 17 16 15 14 x + 5 x + 9 x + 5 x - 4 x - 4 x + 4 x + 4 x - 4 x - 4 x 13 12 10 9 8 7 6 5 4 3 + 3 x + 4 x - 4 x - 3 x + 4 x + 4 x - 4 x - 4 x + 4 x + 5 x 2 - 9 x + 5 x - 1)) and in Maple format -(x^53+10*x^52+45*x^51+120*x^50+212*x^49+269*x^48+275*x^47+267*x^46+263*x^45+ 233*x^44+164*x^43+90*x^42+41*x^41+12*x^40-7*x^39-12*x^38-7*x^37-3*x^36+x^35+2*x ^34-2*x^33+4*x^32+15*x^31+5*x^30-10*x^29-3*x^28+4*x^27-4*x^26-2*x^25+6*x^24-7*x ^23-8*x^22+22*x^21-4*x^20-32*x^19+34*x^18+4*x^17-40*x^16+45*x^15-13*x^14-40*x^ 13+75*x^12-52*x^11-31*x^10+106*x^9-64*x^8-127*x^7+343*x^6-417*x^5+320*x^4-164*x ^3+55*x^2-11*x+1)/(x^2+1)/(x^5+x^4-x^3+2*x-1)/(x^25+5*x^24+10*x^23+10*x^22+5*x^ 21+x^20+2*x^15-5*x^13+3*x^11+x^9-x^7+x^5-5*x^4+10*x^3-10*x^2+5*x-1)/(x^23+5*x^ 22+9*x^21+5*x^20-4*x^19-4*x^18+4*x^17+4*x^16-4*x^15-4*x^14+3*x^13+4*x^12-4*x^10 -3*x^9+4*x^8+4*x^7-4*x^6-4*x^5+4*x^4+5*x^3-9*x^2+5*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1016, 2010, 3956, 7749, 15117, 29394, 57011, 110376, 213447, 412535, 797274, 1541363, 2981763, 5772683, 11185005, 21688496, 42083745, 81702459, 158683471, 308280590] The limit of a(n+1)/a(n) as n goes to infinity is 1.94171303428 a(n) is asymptotic to .699765183073*1.94171303428^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 19, -th largest growth, that is, 1.9454365275632690792, are , [1, 2, 1, 2, 1, 3], [1, 2, 1, 3, 1, 2], [1, 3, 1, 2, 1, 2], [2, 1, 2, 1, 3, 1], [2, 1, 3, 1, 2, 1], [3, 1, 2, 1, 2, 1] Theorem Number, 19, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 1, 2, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 19 18 17 15 14 13 12 11 ) a(n) x = (x - x + 2 x - 3 x + 6 x - 4 x + x - 5 x / ----- n = 0 10 9 8 7 6 5 4 3 2 + 16 x - 20 x + 13 x - 7 x + 3 x + 11 x - 30 x + 34 x - 21 x / 6 5 4 3 2 + 7 x - 1) / ((-1 + x) (x - x + 2 x - x - 2 x + 3 x - 1) / 12 10 9 8 6 5 4 3 2 (x - x + x + x - 3 x + 3 x - 2 x + 4 x - 6 x + 4 x - 1)) and in Maple format (x^19-x^18+2*x^17-3*x^15+6*x^14-4*x^13+x^12-5*x^11+16*x^10-20*x^9+13*x^8-7*x^7+ 3*x^6+11*x^5-30*x^4+34*x^3-21*x^2+7*x-1)/(-1+x)/(x^6-x^5+2*x^4-x^3-2*x^2+3*x-1) /(x^12-x^10+x^9+x^8-3*x^6+3*x^5-2*x^4+4*x^3-6*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1016, 2010, 3956, 7750, 15127, 29450, 57245, 111189, 215927, 419397, 814877, 1583898, 3079757, 5990017, 11652663, 22671160, 44110953, 85827035, 166991801, 324903241] The limit of a(n+1)/a(n) as n goes to infinity is 1.94543652756 a(n) is asymptotic to .693861437001*1.94543652756^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 20, -th largest growth, that is, 1.9501414915693502232, are , [2, 1, 1, 2, 2, 2], [2, 2, 2, 1, 1, 2] Theorem Number, 20, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 2, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = - ( / ----- n = 0 12 10 8 7 6 5 4 3 2 / x + x + x - 2 x + 5 x - 8 x + 12 x - 14 x + 11 x - 5 x + 1) / / 13 12 11 10 9 8 7 6 5 4 3 (x - x + x - x + x - x + 4 x - 8 x + 13 x - 19 x + 21 x 2 - 15 x + 6 x - 1) and in Maple format -(x^12+x^10+x^8-2*x^7+5*x^6-8*x^5+12*x^4-14*x^3+11*x^2-5*x+1)/(x^13-x^12+x^11-x ^10+x^9-x^8+4*x^7-8*x^6+13*x^5-19*x^4+21*x^3-15*x^2+6*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1016, 2010, 3957, 7758, 15164, 29581, 57639, 112251, 218579, 425672, 829170, 1615595, 3148748, 6138228, 11968110, 23337963, 45512975, 88761905, 173111414, 337617555] The limit of a(n+1)/a(n) as n goes to infinity is 1.95014149157 a(n) is asymptotic to .670690304686*1.95014149157^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 21, -th largest growth, that is, 1.9505475001908849807, are , [2, 1, 2, 1, 2, 2], [2, 2, 1, 2, 1, 2] Theorem Number, 21, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 2, 1, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 14 13 10 9 7 6 5 4 ) a(n) x = - (x + x - x + x + 3 x - 7 x + 10 x - 16 x / ----- n = 0 3 2 2 / 17 16 15 13 12 + 20 x - 15 x + 6 x - 1) (x - x + 1) / (x - x + x - x + 3 x / 11 10 9 8 7 6 5 4 3 - 4 x + 4 x - 7 x + 17 x - 33 x + 53 x - 72 x + 76 x - 57 x 2 + 28 x - 8 x + 1) and in Maple format -(x^14+x^13-x^10+x^9+3*x^7-7*x^6+10*x^5-16*x^4+20*x^3-15*x^2+6*x-1)*(x^2-x+1)/( x^17-x^16+x^15-x^13+3*x^12-4*x^11+4*x^10-7*x^9+17*x^8-33*x^7+53*x^6-72*x^5+76*x ^4-57*x^3+28*x^2-8*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1016, 2010, 3956, 7751, 15135, 29488, 57384, 111623, 217149, 422603, 822877, 1603140, 3124764, 6092997, 11884101, 23183529, 45230752, 88247263, 172172176, 335898465] The limit of a(n+1)/a(n) as n goes to infinity is 1.95054750019 a(n) is asymptotic to .663150369868*1.95054750019^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 22, -th largest growth, that is, 1.9515637714286765859, are , [1, 2, 1, 2, 2, 2], [1, 2, 2, 2, 1, 2], [2, 1, 2, 2, 2, 1], [2, 2, 2, 1, 2, 1] Theorem Number, 22, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 1, 2, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 2 ) a(n) x = (x - x + 1) / ----- n = 0 9 7 6 5 4 3 2 / (x + x + x - 2 x + 2 x - 4 x + 6 x - 4 x + 1) / ((-1 + x) / 10 9 8 7 6 5 4 3 2 (x - x + x + x - 3 x + 5 x - 8 x + 11 x - 10 x + 5 x - 1)) and in Maple format (x^2-x+1)*(x^9+x^7+x^6-2*x^5+2*x^4-4*x^3+6*x^2-4*x+1)/(-1+x)/(x^10-x^9+x^8+x^7-\ 3*x^6+5*x^5-8*x^4+11*x^3-10*x^2+5*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1016, 2010, 3957, 7759, 15172, 29618, 57770, 112646, 219649, 428358, 835544, 1630093, 3180681, 6206879, 12113071, 23640103, 46136992, 90042706, 175729457, 342954621] The limit of a(n+1)/a(n) as n goes to infinity is 1.95156377143 a(n) is asymptotic to .666439372478*1.95156377143^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 23, -th largest growth, that is, 1.9527971478516900544, are , [1, 2, 2, 1, 2, 2], [2, 2, 1, 2, 2, 1] Theorem Number, 23, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 2, 1, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 4 3 2 ) a(n) x = (x - x + 2 x - 2 x + 1) / ----- n = 0 9 8 7 6 5 4 3 2 / 12 (x - x + x - x + x + x - 4 x + 6 x - 4 x + 1) / ((-1 + x) (x / 11 10 9 8 7 6 5 4 3 - 2 x + 4 x - 5 x + 5 x - 2 x - 4 x + 12 x - 19 x + 21 x 2 - 15 x + 6 x - 1)) and in Maple format (x^4-x^3+2*x^2-2*x+1)*(x^9-x^8+x^7-x^6+x^5+x^4-4*x^3+6*x^2-4*x+1)/(-1+x)/(x^12-\ 2*x^11+4*x^10-5*x^9+5*x^8-2*x^7-4*x^6+12*x^5-19*x^4+21*x^3-15*x^2+6*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1016, 2010, 3957, 7758, 15164, 29582, 57650, 112316, 218858, 426653, 832180, 1623968, 3170402, 6191216, 12092384, 23620035, 46137249, 90117043, 176010096, 343749588] The limit of a(n+1)/a(n) as n goes to infinity is 1.95279714785 a(n) is asymptotic to .655393299540*1.95279714785^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 24, -th largest growth, that is, 1.9539877574581293211, are , [2, 1, 2, 2, 1, 2] Theorem Number, 24, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 2, 2, 1, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 18 16 14 13 12 11 9 8 ) a(n) x = - (x + x - x + 2 x - 3 x + 3 x - x - 10 x / ----- n = 0 7 6 5 4 3 2 / 19 18 + 38 x - 73 x + 96 x - 92 x + 63 x - 29 x + 8 x - 1) / (x - x / 17 16 15 14 13 12 11 9 8 7 + 2 x - 2 x + x + x - 4 x + 7 x - 6 x + x + 20 x - 66 x 6 5 4 3 2 + 118 x - 147 x + 133 x - 85 x + 36 x - 9 x + 1) and in Maple format -(x^18+x^16-x^14+2*x^13-3*x^12+3*x^11-x^9-10*x^8+38*x^7-73*x^6+96*x^5-92*x^4+63 *x^3-29*x^2+8*x-1)/(x^19-x^18+2*x^17-2*x^16+x^15+x^14-4*x^13+7*x^12-6*x^11+x^9+ 20*x^8-66*x^7+118*x^6-147*x^5+133*x^4-85*x^3+36*x^2-9*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1016, 2010, 3956, 7751, 15136, 29498, 57439, 111846, 217898, 424818, 828859, 1618251, 3161068, 6176960, 12072735, 23598041, 46126394, 90157907, 176208878, 344364833] The limit of a(n+1)/a(n) as n goes to infinity is 1.95398775746 a(n) is asymptotic to .644669574312*1.95398775746^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 25, -th largest growth, that is, 1.9543423291595747005, are , [2, 2, 1, 1, 2, 2] Theorem Number, 25, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 2, 1, 1, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 19 18 17 16 15 14 13 ) a(n) x = - (x - x + 2 x - 3 x + 5 x - 6 x + 10 x / ----- n = 0 12 11 10 9 8 7 6 5 4 - 16 x + 24 x - 32 x + 39 x - 46 x + 59 x - 80 x + 97 x - 92 x 3 2 / 20 19 18 17 16 15 + 63 x - 29 x + 8 x - 1) / (x - x + 3 x - 4 x + 6 x - 8 x / 14 13 12 11 10 9 8 7 + 12 x - 18 x + 28 x - 40 x + 51 x - 60 x + 71 x - 93 x 6 5 4 3 2 + 126 x - 148 x + 133 x - 85 x + 36 x - 9 x + 1) and in Maple format -(x^19-x^18+2*x^17-3*x^16+5*x^15-6*x^14+10*x^13-16*x^12+24*x^11-32*x^10+39*x^9-\ 46*x^8+59*x^7-80*x^6+97*x^5-92*x^4+63*x^3-29*x^2+8*x-1)/(x^20-x^19+3*x^18-4*x^ 17+6*x^16-8*x^15+12*x^14-18*x^13+28*x^12-40*x^11+51*x^10-60*x^9+71*x^8-93*x^7+ 126*x^6-148*x^5+133*x^4-85*x^3+36*x^2-9*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1016, 2010, 3957, 7758, 15163, 29573, 57605, 112151, 218362, 425358, 829146, 1617472, 3157585, 6167927, 12053912, 23564443, 46075242, 90096995, 176176663, 344474692] The limit of a(n+1)/a(n) as n goes to infinity is 1.95434232916 a(n) is asymptotic to .641734600337*1.95434232916^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 26, -th largest growth, that is, 1.9611865309023902347, are , [1, 1, 2, 2, 2, 2], [1, 2, 2, 2, 2, 1], [2, 2, 2, 2, 1, 1] Theorem Number, 26, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 2, 2, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 8 7 6 5 4 3 2 2 x - 3 x + 5 x - 8 x + 12 x - 14 x + 11 x - 5 x + 1 - ----------------------------------------------------------- 7 6 5 4 3 2 2 (x - x + 2 x - 3 x + 5 x - 6 x + 4 x - 1) (-1 + x) and in Maple format -(2*x^8-3*x^7+5*x^6-8*x^5+12*x^4-14*x^3+11*x^2-5*x+1)/(x^7-x^6+2*x^5-3*x^4+5*x^ 3-6*x^2+4*x-1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1016, 2011, 3965, 7796, 15303, 30014, 58850, 115389, 226267, 443730, 870248, 1706784, 3347455, 6565178, 12875758, 25251939, 49523803, 97125229, 190480226, 373566544] The limit of a(n+1)/a(n) as n goes to infinity is 1.96118653090 a(n) is asymptotic to .626326117348*1.96118653090^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 27, -th largest growth, that is, 1.9659482366454853372, are , [1, 1, 1, 1, 6], [1, 1, 1, 6, 1], [1, 1, 6, 1, 1], [1, 6, 1, 1, 1], [6, 1, 1, 1, 1] Theorem Number, 27, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 1, 6] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 8 7 6 4 3 2 \ n 2 x - 2 x + x + x - 4 x + 6 x - 4 x + 1 ) a(n) x = - --------------------------------------------- / 5 4 3 2 4 ----- (x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format -(2*x^8-2*x^7+x^6+x^4-4*x^3+6*x^2-4*x+1)/(x^5+x^4+x^3+x^2+x-1)/(-1+x)^4 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2018, 3994, 7889, 15559, 30653, 60345, 118739, 233563, 459329, 903205, 1775878, 3491548, 6864511, 13495631, 26532124, 52161254, 102546862, 201602430, 396340626] The limit of a(n+1)/a(n) as n goes to infinity is 1.96594823665 a(n) is asymptotic to .617884376698*1.96594823665^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 28, -th largest growth, that is, 1.9671682128139660358, are , [2, 1, 1, 1, 5], [5, 1, 1, 1, 2] Theorem Number, 28, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 1, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 41 36 35 34 33 31 30 29 28 ) a(n) x = - (x + 2 x + x - x + x + x + x - 3 x + x / ----- n = 0 25 24 23 22 21 20 19 18 17 + x + x - 4 x + 4 x - 5 x + 2 x + 5 x - 8 x + 6 x 16 15 14 13 12 11 10 9 8 7 - 3 x - x + 5 x - 4 x + x - x + 7 x - 18 x + 23 x - 15 x 6 5 4 3 2 / 6 - x + 20 x - 35 x + 35 x - 21 x + 7 x - 1) / ((x + x - 1) / 6 5 (x - x + 2 x - 1) 24 17 16 14 13 9 8 4 3 2 (x - x + x + 2 x - 2 x - x + x + x - 4 x + 6 x - 4 x + 1) 6 (x - x + 1)) and in Maple format -(x^41+2*x^36+x^35-x^34+x^33+x^31+x^30-3*x^29+x^28+x^25+x^24-4*x^23+4*x^22-5*x^ 21+2*x^20+5*x^19-8*x^18+6*x^17-3*x^16-x^15+5*x^14-4*x^13+x^12-x^11+7*x^10-18*x^ 9+23*x^8-15*x^7-x^6+20*x^5-35*x^4+35*x^3-21*x^2+7*x-1)/(x^6+x-1)/(x^6-x^5+2*x-1 )/(x^24-x^17+x^16+2*x^14-2*x^13-x^9+x^8+x^4-4*x^3+6*x^2-4*x+1)/(x^6-x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2018, 3994, 7889, 15559, 30653, 60345, 118740, 233573, 459386, 903449, 1776752, 3494317, 6872533, 13517356, 26587980, 52299133, 102876388, 202369818, 398090514] The limit of a(n+1)/a(n) as n goes to infinity is 1.96716821281 a(n) is asymptotic to .609183935429*1.96716821281^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 29, -th largest growth, that is, 1.9691817825046685829, are , [1, 2, 1, 2, 4], [1, 2, 1, 4, 2], [1, 2, 2, 1, 4], [1, 2, 4, 1, 2], [1, 4, 1, 2, 2], [1, 4, 2, 1, 2], [2, 1, 2, 4, 1], [2, 1, 4, 2, 1], [2, 2, 1, 4, 1], [2, 4, 1, 2, 1], [4, 1, 2, 2, 1], [4, 2, 1, 2, 1] Theorem Number, 29, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 1, 2, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 13 12 11 9 7 6 5 4 3 2 x - x + x + x - 2 x + 4 x - 3 x + 2 x - 4 x + 6 x - 4 x + 1 ------------------------------------------------------------------------- 12 11 8 7 6 5 4 3 2 (-1 + x) (x - x + x + x - 3 x + 3 x - x + 2 x - 5 x + 4 x - 1) and in Maple format (x^13-x^12+x^11+x^9-2*x^7+4*x^6-3*x^5+2*x^4-4*x^3+6*x^2-4*x+1)/(-1+x)/(x^12-x^ 11+x^8+x^7-3*x^6+3*x^5-x^4+2*x^3-5*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2018, 3994, 7889, 15560, 30661, 60384, 118888, 234052, 460774, 907161, 1786104, 3516841, 6924955, 13636207, 26852063, 52876842, 104124937, 205042562, 403768723] The limit of a(n+1)/a(n) as n goes to infinity is 1.96918178250 a(n) is asymptotic to .599187002474*1.96918178250^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 30, -th largest growth, that is, 1.9692165211230598361, are , [2, 1, 1, 2, 4], [2, 1, 1, 4, 2], [2, 2, 1, 1, 4], [2, 4, 1, 1, 2], [4, 1, 1, 2, 2], [4, 2, 1, 1, 2] Theorem Number, 30, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 2, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 16 14 13 12 8 7 6 5 4 ) a(n) x = - (x + x + x + x + x - 2 x + 4 x - 3 x + 2 x / ----- n = 0 3 2 / 17 15 12 9 8 7 6 - 4 x + 6 x - 4 x + 1) / (x + 2 x - x + x - x + 4 x - 6 x / 5 4 3 2 + 4 x - 3 x + 7 x - 9 x + 5 x - 1) and in Maple format -(x^16+x^14+x^13+x^12+x^8-2*x^7+4*x^6-3*x^5+2*x^4-4*x^3+6*x^2-4*x+1)/(x^17+2*x^ 15-x^12+x^9-x^8+4*x^7-6*x^6+4*x^5-3*x^4+7*x^3-9*x^2+5*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2018, 3994, 7889, 15560, 30660, 60376, 118851, 233923, 460397, 906183, 1783772, 3511611, 6913745, 13612975, 26805099, 52783595, 103942089, 204686894, 403080085] The limit of a(n+1)/a(n) as n goes to infinity is 1.96921652112 a(n) is asymptotic to .597887221874*1.96921652112^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 31, -th largest growth, that is, 1.9693144732632464526, are , [3, 1, 1, 1, 4], [4, 1, 1, 1, 3] Theorem Number, 31, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 1, 1, 1, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 40 39 38 37 36 35 34 ) a(n) x = - (x + 6 x + 15 x + 20 x + 15 x + 8 x + 10 x / ----- n = 0 33 32 31 30 29 28 27 26 25 + 17 x + 17 x + 8 x + x + 2 x + 4 x + 2 x - 3 x - 4 x 24 23 22 20 19 18 17 16 14 13 - x + x + 4 x - 7 x + x + x + 5 x - 4 x + x - 2 x 12 11 10 9 8 7 6 5 4 + 5 x - 5 x + 5 x - 3 x - 8 x + 19 x - 22 x + 27 x - 36 x 3 2 / 2 24 23 22 21 + 35 x - 21 x + 7 x - 1) / ((x + 1) (x + 4 x + 6 x + 4 x / 20 15 14 13 12 9 7 4 3 2 + x - x + 2 x + x - 2 x - x + x + x - 4 x + 6 x - 4 x + 1) 6 5 4 4 3 2 6 5 (x + x - x + 2 x - 1) (x + x - x - x + 1) (x + x + x - 1)) and in Maple format -(x^40+6*x^39+15*x^38+20*x^37+15*x^36+8*x^35+10*x^34+17*x^33+17*x^32+8*x^31+x^ 30+2*x^29+4*x^28+2*x^27-3*x^26-4*x^25-x^24+x^23+4*x^22-7*x^20+x^19+x^18+5*x^17-\ 4*x^16+x^14-2*x^13+5*x^12-5*x^11+5*x^10-3*x^9-8*x^8+19*x^7-22*x^6+27*x^5-36*x^4 +35*x^3-21*x^2+7*x-1)/(x^2+1)/(x^24+4*x^23+6*x^22+4*x^21+x^20-x^15+2*x^14+x^13-\ 2*x^12-x^9+x^7+x^4-4*x^3+6*x^2-4*x+1)/(x^6+x^5-x^4+2*x-1)/(x^4+x^3-x^2-x+1)/(x^ 6+x^5+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2018, 3994, 7889, 15559, 30653, 60346, 118750, 233630, 459630, 904323, 1779521, 3502339, 6894257, 13573198, 26725754, 52628093, 103641308, 204110443, 401981759] The limit of a(n+1)/a(n) as n goes to infinity is 1.96931447326 a(n) is asymptotic to .595489612304*1.96931447326^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 32, -th largest growth, that is, 1.9703230372932668084, are , [2, 1, 2, 1, 4], [2, 1, 4, 1, 2], [4, 1, 2, 1, 2] Theorem Number, 32, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 2, 1, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 20 17 16 14 12 11 9 8 ) a(n) x = - (x + 2 x - 2 x + x + 3 x - 5 x + 4 x - 3 x / ----- n = 0 7 6 5 4 3 2 / 3 + 4 x - 7 x + 10 x - 16 x + 20 x - 15 x + 6 x - 1) / ((x - x + 1) / 7 6 5 4 2 (x - x + 2 x - x - 2 x + 3 x - 1) 11 6 4 3 2 (x + x - 2 x + 2 x - 3 x + 3 x - 1)) and in Maple format -(x^20+2*x^17-2*x^16+x^14+3*x^12-5*x^11+4*x^9-3*x^8+4*x^7-7*x^6+10*x^5-16*x^4+ 20*x^3-15*x^2+6*x-1)/(x^3-x+1)/(x^7-x^6+2*x^5-x^4-2*x^2+3*x-1)/(x^11+x^6-2*x^4+ 2*x^3-3*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2018, 3994, 7889, 15560, 30662, 60391, 118918, 234156, 461095, 908079, 1788586, 3523261, 6940967, 13674936, 26943332, 53087228, 104600923, 206102629, 406098598] The limit of a(n+1)/a(n) as n goes to infinity is 1.97032303729 a(n) is asymptotic to .592288780184*1.97032303729^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 33, -th largest growth, that is, 1.9706560177668563263, are , [1, 2, 1, 3, 3], [1, 3, 3, 1, 2], [2, 1, 3, 3, 1], [3, 3, 1, 2, 1] Theorem Number, 33, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 1, 3, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 10 7 6 5 4 3 2 \ n x + x + x - x - 2 x + 2 x + 2 x - 3 x + 1 ) a(n) x = ---------------------------------------------------- / 9 7 5 4 3 2 ----- (-1 + x) (x - x + 2 x + x - 3 x - x + 3 x - 1) n = 0 and in Maple format (x^10+x^7+x^6-x^5-2*x^4+2*x^3+2*x^2-3*x+1)/(-1+x)/(x^9-x^7+2*x^5+x^4-3*x^3-x^2+ 3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2018, 3994, 7890, 15567, 30691, 60485, 119182, 234830, 462703, 911730, 1796578, 3540290, 6976549, 13748270, 27093086, 53391268, 105216139, 207345386, 408607283] The limit of a(n+1)/a(n) as n goes to infinity is 1.97065601777 a(n) is asymptotic to .592901445341*1.97065601777^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 34, -th largest growth, that is, 1.9708395870474530685, are , [1, 2, 3, 1, 3], [1, 3, 1, 3, 2], [2, 3, 1, 3, 1], [3, 1, 3, 2, 1] Theorem Number, 34, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 3, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = ( / ----- n = 0 13 10 9 7 6 5 4 3 2 / x + 2 x - x + 2 x - 3 x + 4 x - 2 x - 3 x + 6 x - 4 x + 1) / ( / 12 10 9 7 6 5 4 3 2 (-1 + x) (x - x + 2 x - x + 2 x - 3 x + 3 x + x - 5 x + 4 x - 1) ) and in Maple format (x^13+2*x^10-x^9+2*x^7-3*x^6+4*x^5-2*x^4-3*x^3+6*x^2-4*x+1)/(-1+x)/(x^12-x^10+2 *x^9-x^7+2*x^6-3*x^5+3*x^4+x^3-5*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2018, 3994, 7889, 15561, 30669, 60422, 119027, 234486, 461993, 910339, 1793953, 3535466, 6967845, 13732760, 27065690, 53343225, 105132531, 207201338, 408362651] The limit of a(n+1)/a(n) as n goes to infinity is 1.97083958705 a(n) is asymptotic to .590889988177*1.97083958705^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 35, -th largest growth, that is, 1.9708817901785482789, are , [1, 3, 1, 2, 3], [1, 3, 2, 1, 3], [3, 1, 2, 3, 1], [3, 2, 1, 3, 1] Theorem Number, 35, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 3, 1, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 21 20 19 18 17 16 15 14 13 ) a(n) x = (x + x - x + x + 3 x - x - x + 3 x + x / ----- n = 0 12 11 10 9 8 7 6 5 4 3 - 4 x + x + 5 x - 4 x - 3 x + 4 x + 2 x - 5 x - x + 9 x 2 / 20 19 16 15 14 13 - 10 x + 5 x - 1) / ((-1 + x) (x + 2 x + 3 x + x - 2 x + x / 12 11 10 9 8 7 5 4 3 2 + 3 x - 3 x - 3 x + 5 x + x - 5 x + 5 x - 2 x - 6 x + 9 x - 5 x + 1)) and in Maple format (x^21+x^20-x^19+x^18+3*x^17-x^16-x^15+3*x^14+x^13-4*x^12+x^11+5*x^10-4*x^9-3*x^ 8+4*x^7+2*x^6-5*x^5-x^4+9*x^3-10*x^2+5*x-1)/(-1+x)/(x^20+2*x^19+3*x^16+x^15-2*x ^14+x^13+3*x^12-3*x^11-3*x^10+5*x^9+x^8-5*x^7+5*x^5-2*x^4-6*x^3+9*x^2-5*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2018, 3994, 7889, 15560, 30662, 60393, 118935, 234238, 461394, 909001, 1791130, 3529752, 6956612, 13711093, 27024340, 53264636, 104983086, 206916195, 407816102] The limit of a(n+1)/a(n) as n goes to infinity is 1.97088179018 a(n) is asymptotic to .589721024898*1.97088179018^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 36, -th largest growth, that is, 1.9709013528640663101, are , [3, 1, 1, 2, 3], [3, 2, 1, 1, 3] Theorem Number, 36, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 1, 1, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 38 37 36 35 34 33 32 ) a(n) x = - (x + 3 x + 3 x + 2 x + 3 x + 3 x + 3 x / ----- n = 0 31 30 29 28 27 25 24 23 22 21 + 2 x - 2 x + x + 3 x - 3 x + 3 x - x + 2 x - x - 5 x 20 19 18 17 16 15 14 13 12 + 6 x + 4 x - 9 x + 3 x + 5 x - 11 x + 12 x + x - 22 x 11 9 8 7 6 5 4 3 2 + 25 x - 31 x + 48 x - 44 x + 13 x + 34 x - 63 x + 55 x - 28 x / 40 39 38 37 36 35 34 + 8 x - 1) / (x + 4 x + 5 x + 2 x + 2 x + 4 x + 3 x / 33 32 31 30 29 28 27 26 25 + 2 x - 2 x - 2 x + 5 x - x - 5 x + 6 x + 3 x - 4 x 24 23 21 20 19 18 17 16 15 + x - 4 x + 13 x - 8 x - 10 x + 14 x - 4 x - 8 x + 19 x 14 13 12 11 10 9 8 7 6 - 19 x - 7 x + 41 x - 37 x - 8 x + 53 x - 74 x + 62 x - 8 x 5 4 3 2 - 63 x + 97 x - 76 x + 35 x - 9 x + 1) and in Maple format -(x^38+3*x^37+3*x^36+2*x^35+3*x^34+3*x^33+3*x^32+2*x^31-2*x^30+x^29+3*x^28-3*x^ 27+3*x^25-x^24+2*x^23-x^22-5*x^21+6*x^20+4*x^19-9*x^18+3*x^17+5*x^16-11*x^15+12 *x^14+x^13-22*x^12+25*x^11-31*x^9+48*x^8-44*x^7+13*x^6+34*x^5-63*x^4+55*x^3-28* x^2+8*x-1)/(x^40+4*x^39+5*x^38+2*x^37+2*x^36+4*x^35+3*x^34+2*x^33-2*x^32-2*x^31 +5*x^30-x^29-5*x^28+6*x^27+3*x^26-4*x^25+x^24-4*x^23+13*x^21-8*x^20-10*x^19+14* x^18-4*x^17-8*x^16+19*x^15-19*x^14-7*x^13+41*x^12-37*x^11-8*x^10+53*x^9-74*x^8+ 62*x^7-8*x^6-63*x^5+97*x^4-76*x^3+35*x^2-9*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2018, 3994, 7889, 15560, 30660, 60377, 118861, 233980, 460640, 907046, 1786474, 3519332, 6934352, 13665181, 26932105, 53082840, 104629347, 206233088, 406501207] The limit of a(n+1)/a(n) as n goes to infinity is 1.97090135286 a(n) is asymptotic to .587757686511*1.97090135286^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 37, -th largest growth, that is, 1.9709021410209757284, are , [2, 3, 1, 1, 3], [3, 1, 1, 3, 2] Theorem Number, 37, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 3, 1, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 21 20 17 16 15 13 12 ) a(n) x = - (x + 2 x + 3 x + 2 x - 2 x + 4 x - 3 x / ----- n = 0 11 10 9 8 7 6 5 4 3 2 - 2 x + 5 x - 2 x - 4 x + 4 x + 2 x - 5 x - x + 9 x - 10 x / 4 2 16 15 12 10 + 5 x - 1) / ((x - x + 1) (x - x + 1) (x + 2 x + 3 x - 2 x / 9 7 6 5 4 3 2 + 2 x - 4 x + 2 x + 2 x - 5 x + 2 x + 4 x - 4 x + 1)) and in Maple format -(x^21+2*x^20+3*x^17+2*x^16-2*x^15+4*x^13-3*x^12-2*x^11+5*x^10-2*x^9-4*x^8+4*x^ 7+2*x^6-5*x^5-x^4+9*x^3-10*x^2+5*x-1)/(x^4-x+1)/(x^2-x+1)/(x^16+2*x^15+3*x^12-2 *x^10+2*x^9-4*x^7+2*x^6+2*x^5-5*x^4+2*x^3+4*x^2-4*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2018, 3994, 7889, 15560, 30661, 60385, 118898, 234110, 461024, 908052, 1788892, 3524789, 6946103, 13689603, 26981513, 53180821, 104820955, 206604525, 407218156] The limit of a(n+1)/a(n) as n goes to infinity is 1.97090214102 a(n) is asymptotic to .588714283570*1.97090214102^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 38, -th largest growth, that is, 1.9709232598962379131, are , [2, 1, 1, 3, 3], [3, 3, 1, 1, 2] Theorem Number, 38, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 3, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 14 11 8 7 5 4 3 2 ) a(n) x = - (x + x + x - x + 3 x - 2 x - 3 x + 6 x - 4 x + 1 / ----- n = 0 / 15 14 12 11 9 8 7 6 5 4 3 ) / (x - x + x - x + x - x + 2 x + x - 5 x + 2 x + 6 x / 2 - 9 x + 5 x - 1) and in Maple format -(x^14+x^11+x^8-x^7+3*x^5-2*x^4-3*x^3+6*x^2-4*x+1)/(x^15-x^14+x^12-x^11+x^9-x^8 +2*x^7+x^6-5*x^5+2*x^4+6*x^3-9*x^2+5*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2018, 3994, 7890, 15567, 30690, 60478, 119153, 234739, 462464, 911177, 1795417, 3538041, 6972502, 13741532, 27082926, 53378271, 105205374, 207354351, 408684842] The limit of a(n+1)/a(n) as n goes to infinity is 1.97092325990 a(n) is asymptotic to .590624128719*1.97092325990^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 39, -th largest growth, that is, 1.9717270001741243154, are , [3, 1, 2, 1, 3] Theorem Number, 39, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 1, 2, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 38 37 36 35 34 33 32 ) a(n) x = - (x + 3 x + 3 x + 2 x + 3 x + 4 x + 4 x / ----- n = 0 31 30 29 28 26 25 24 23 22 20 + 2 x - x + x + 2 x - 2 x - 2 x + 4 x + x - 6 x + 4 x 19 17 16 15 14 13 11 10 9 + x - 6 x - x + 11 x - 2 x - 9 x + 10 x - x - 11 x 8 7 6 5 4 3 2 / + 13 x - 9 x - 8 x + 41 x - 64 x + 55 x - 28 x + 8 x - 1) / ( / 10 9 8 7 6 5 4 3 2 30 29 (x + x - x + x + x - x + 2 x - x - 2 x + 3 x - 1) (x + 3 x 28 27 25 24 23 22 21 20 19 18 16 + 3 x + x + x + x - x - x + x + x - 2 x - 2 x + 4 x 15 14 12 10 9 8 7 6 5 4 - x - 3 x + 4 x - 4 x + 2 x - 3 x + 9 x - 11 x + 11 x - 16 x 3 2 + 20 x - 15 x + 6 x - 1)) and in Maple format -(x^38+3*x^37+3*x^36+2*x^35+3*x^34+4*x^33+4*x^32+2*x^31-x^30+x^29+2*x^28-2*x^26 -2*x^25+4*x^24+x^23-6*x^22+4*x^20+x^19-6*x^17-x^16+11*x^15-2*x^14-9*x^13+10*x^ 11-x^10-11*x^9+13*x^8-9*x^7-8*x^6+41*x^5-64*x^4+55*x^3-28*x^2+8*x-1)/(x^10+x^9- x^8+x^7+x^6-x^5+2*x^4-x^3-2*x^2+3*x-1)/(x^30+3*x^29+3*x^28+x^27+x^25+x^24-x^23- x^22+x^21+x^20-2*x^19-2*x^18+4*x^16-x^15-3*x^14+4*x^12-4*x^10+2*x^9-3*x^8+9*x^7 -11*x^6+11*x^5-16*x^4+20*x^3-15*x^2+6*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2018, 3994, 7889, 15560, 30662, 60392, 118928, 234212, 461329, 908896, 1791110, 3530396, 6959833, 13722323, 27057686, 53354627, 105210778, 207466154, 409099187] The limit of a(n+1)/a(n) as n goes to infinity is 1.97172700017 a(n) is asymptotic to .584082883820*1.97172700017^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 40, -th largest growth, that is, 1.9728837077631717755, are , [2, 1, 2, 2, 3], [2, 1, 2, 3, 2], [2, 1, 3, 2, 2], [2, 2, 2, 1, 3], [2, 2, 3, 1, 2], [2, 3, 2, 1, 2], [3, 1, 2, 2, 2], [3, 2, 2, 1, 2] Theorem Number, 40, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 2, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 11 7 6 5 4 3 2 x + 2 x - 3 x + 4 x - 2 x - 3 x + 6 x - 4 x + 1 - ----------------------------------------------------------------------- 12 11 9 8 7 6 5 4 3 2 x - x + x + x - 3 x + 5 x - 6 x + 2 x + 6 x - 9 x + 5 x - 1 and in Maple format -(x^11+2*x^7-3*x^6+4*x^5-2*x^4-3*x^3+6*x^2-4*x+1)/(x^12-x^11+x^9+x^8-3*x^7+5*x^ 6-6*x^5+2*x^4+6*x^3-9*x^2+5*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2018, 3994, 7890, 15568, 30699, 60523, 119322, 235273, 463970, 915104, 1805103, 3560988, 7025271, 13860194, 27345239, 53950507, 106440784, 209999459, 414310242] The limit of a(n+1)/a(n) as n goes to infinity is 1.97288370776 a(n) is asymptotic to .581145958125*1.97288370776^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 41, -th largest growth, that is, 1.9730623001088685615, are , [2, 2, 1, 2, 3], [2, 2, 1, 3, 2], [2, 3, 1, 2, 2], [3, 2, 1, 2, 2] Theorem Number, 41, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 2, 1, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 13 12 11 10 8 7 6 5 ) a(n) x = - (x - 2 x + 3 x - 2 x + x + 2 x - 6 x + 6 x / ----- n = 0 4 3 2 / 14 13 12 11 10 + x - 9 x + 10 x - 5 x + 1) / (x - 2 x + 4 x - 4 x + 2 x / 9 8 7 6 5 4 3 2 + x - x - 4 x + 10 x - 8 x - 4 x + 15 x - 14 x + 6 x - 1) and in Maple format -(x^13-2*x^12+3*x^11-2*x^10+x^8+2*x^7-6*x^6+6*x^5+x^4-9*x^3+10*x^2-5*x+1)/(x^14 -2*x^13+4*x^12-4*x^11+2*x^10+x^9-x^8-4*x^7+10*x^6-8*x^5-4*x^4+15*x^3-14*x^2+6*x -1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2018, 3994, 7890, 15567, 30692, 60494, 119230, 235026, 463379, 913804, 1802420, 3555718, 7015316, 13841965, 27312680, 53893508, 106342634, 209832870, 414031474] The limit of a(n+1)/a(n) as n goes to infinity is 1.97306230011 a(n) is asymptotic to .579193495541*1.97306230011^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 42, -th largest growth, that is, 1.9735704833094816886, are , [2, 1, 3, 1, 3], [3, 1, 3, 1, 2] Theorem Number, 42, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 3, 1, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 20 16 13 12 11 10 9 8 ) a(n) x = - (x + x - 2 x + 3 x + 2 x - 6 x + 3 x + x / ----- n = 0 7 6 4 3 2 / - 2 x + 3 x - 11 x + 19 x - 15 x + 6 x - 1) / ( / 7 6 4 3 2 (x - x + 2 x - x - 2 x + 3 x - 1) 14 11 10 9 7 5 4 3 2 (x - x + x + x - 2 x + 2 x - 2 x + 4 x - 6 x + 4 x - 1)) and in Maple format -(x^20+x^16-2*x^13+3*x^12+2*x^11-6*x^10+3*x^9+x^8-2*x^7+3*x^6-11*x^4+19*x^3-15* x^2+6*x-1)/(x^7-x^6+2*x^4-x^3-2*x^2+3*x-1)/(x^14-x^11+x^10+x^9-2*x^7+2*x^5-2*x^ 4+4*x^3-6*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2018, 3994, 7890, 15568, 30700, 60531, 119360, 235412, 464402, 916306, 1808197, 3568521, 7042891, 13900230, 27434310, 54145622, 106863271, 210906206, 416242784] The limit of a(n+1)/a(n) as n goes to infinity is 1.97357048331 a(n) is asymptotic to .577784303828*1.97357048331^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 43, -th largest growth, that is, 1.9756564557792322769, are , [1, 2, 2, 2, 3], [1, 2, 2, 3, 2], [1, 2, 3, 2, 2], [1, 3, 2, 2, 2], [2, 2, 2, 3, 1], [2, 2, 3, 2, 1], [2, 3, 2, 2, 1], [3, 2, 2, 2, 1] Theorem Number, 43, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 2, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 9 8 7 6 5 4 3 2 x - x + 2 x - 3 x + 4 x - 2 x - 3 x + 6 x - 4 x + 1 ------------------------------------------------------------- 8 7 6 5 4 3 2 (-1 + x) (x - x + 2 x - 3 x + 3 x + x - 5 x + 4 x - 1) and in Maple format (x^9-x^8+2*x^7-3*x^6+4*x^5-2*x^4-3*x^3+6*x^2-4*x+1)/(-1+x)/(x^8-x^7+2*x^6-3*x^5 +3*x^4+x^3-5*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2018, 3995, 7897, 15598, 30800, 60818, 120107, 237231, 468631, 925825, 1829143, 3613885, 7140047, 14106662, 27870391, 55062778, 108785398, 214922391, 424611892] The limit of a(n+1)/a(n) as n goes to infinity is 1.97565645578 a(n) is asymptotic to .571014297866*1.97565645578^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 44, -th largest growth, that is, 1.9822984210734063491, are , [2, 2, 2, 2, 2] Theorem Number, 44, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 2, 2, 2, 2] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 8 7 6 5 4 3 2 x - x + 2 x - 3 x + 5 x - 7 x + 7 x - 4 x + 1 - ------------------------------------------------------------- 9 8 7 6 5 4 3 2 x - x + 2 x - 3 x + 5 x - 8 x + 11 x - 10 x + 5 x - 1 and in Maple format -(x^8-x^7+2*x^6-3*x^5+5*x^4-7*x^3+7*x^2-4*x+1)/(x^9-x^8+2*x^7-3*x^6+5*x^5-8*x^4 +11*x^3-10*x^2+5*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1017, 2019, 4002, 7927, 15699, 31095, 61604, 122075, 241949, 479595, 950727, 1884732, 3736324, 7406869, 14683110, 29106845, 57698887, 114376475, 226727625, 449439958] The limit of a(n+1)/a(n) as n goes to infinity is 1.98229842107 a(n) is asymptotic to .546508435333*1.98229842107^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 45, -th largest growth, that is, 1.9835828434243263304, are , [1, 1, 1, 7], [1, 1, 7, 1], [1, 7, 1, 1], [7, 1, 1, 1] Theorem Number, 45, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1, 7] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 9 8 7 3 2 \ n x - x + x - x + 3 x - 3 x + 1 ) a(n) x = ------------------------------------------ / 6 5 4 3 2 3 ----- (x + x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^9-x^8+x^7-x^3+3*x^2-3*x+1)/(x^6+x^5+x^4+x^3+x^2+x-1)/(-1+x)^3 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1018, 2025, 4024, 7991, 15862, 31477, 62453, 123899, 245785, 487559, 967141, 1918435, 3805409, 7548382, 14972883, 29700000, 58912461, 116857802, 231797191, 459788996] The limit of a(n+1)/a(n) as n goes to infinity is 1.98358284342 a(n) is asymptotic to .548338004018*1.98358284342^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 46, -th largest growth, that is, 1.9838613961621262283, are , [2, 1, 1, 6], [6, 1, 1, 2] Theorem Number, 46, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 1, 6] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 27 21 20 19 15 14 13 10 ) a(n) x = - (x + 2 x - x + x + 2 x - 3 x + x - x / ----- n = 0 9 8 7 6 4 3 2 / + 4 x - 5 x + 3 x - x - x + 4 x - 6 x + 4 x - 1) / ( / 21 14 13 9 8 3 2 7 6 (x - x + x + x - x + x - 3 x + 3 x - 1) (x - x + 2 x - 1)) and in Maple format -(x^27+2*x^21-x^20+x^19+2*x^15-3*x^14+x^13-x^10+4*x^9-5*x^8+3*x^7-x^6-x^4+4*x^3 -6*x^2+4*x-1)/(x^21-x^14+x^13+x^9-x^8+x^3-3*x^2+3*x-1)/(x^7-x^6+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1018, 2025, 4024, 7991, 15862, 31477, 62453, 123900, 245793, 487597, 967281, 1918878, 3806677, 7551764, 14981449, 29720868, 58961799, 116971749, 232055497, 460365831] The limit of a(n+1)/a(n) as n goes to infinity is 1.98386139616 a(n) is asymptotic to .546718776972*1.98386139616^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 47, -th largest growth, that is, 1.9843693628442243022, are , [2, 1, 2, 5], [2, 1, 5, 2], [2, 2, 1, 5], [2, 5, 1, 2], [5, 1, 2, 2], [5, 2, 1, 2] Theorem Number, 47, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 2, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 13 7 6 5 3 2 x + 2 x - 2 x + x - x + 3 x - 3 x + 1 - -------------------------------------------------------------- 14 13 9 8 7 6 5 3 2 x - x + x + x - 3 x + 3 x - x + 2 x - 5 x + 4 x - 1 and in Maple format -(x^13+2*x^7-2*x^6+x^5-x^3+3*x^2-3*x+1)/(x^14-x^13+x^9+x^8-3*x^7+3*x^6-x^5+2*x^ 3-5*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1018, 2025, 4024, 7991, 15862, 31478, 62460, 123930, 245895, 487900, 968105, 1920985, 3811829, 7563949, 15009547, 29784402, 59103259, 117282855, 232732901, 461828768] The limit of a(n+1)/a(n) as n goes to infinity is 1.98436936284 a(n) is asymptotic to .544260748124*1.98436936284^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 48, -th largest growth, that is, 1.9843858253440954550, are , [1, 1, 3, 5], [1, 1, 5, 3], [1, 3, 5, 1], [1, 5, 3, 1], [3, 5, 1, 1], [5, 3, 1, 1] Theorem Number, 48, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 3, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 3 5 4 3 2 \ n (x - x + 1) (x + x - x + x - 2 x + 1) ) a(n) x = - ------------------------------------------ / 7 6 5 2 ----- (x + x - x + 2 x - 1) (-1 + x) n = 0 and in Maple format -(x^3-x+1)*(x^5+x^4-x^3+x^2-2*x+1)/(x^7+x^6-x^5+2*x-1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1018, 2025, 4024, 7991, 15863, 31484, 62482, 123994, 246058, 488281, 968946, 1922772, 3815532, 7571499, 15024787, 29814986, 59164447, 117405101, 232977029, 462316325] The limit of a(n+1)/a(n) as n goes to infinity is 1.98438582534 a(n) is asymptotic to .544697969901*1.98438582534^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 49, -th largest growth, that is, 1.9846407398915826487, are , [1, 2, 2, 5], [1, 2, 5, 2], [1, 5, 2, 2], [2, 2, 5, 1], [2, 5, 2, 1], [5, 2, 2, 1] Theorem Number, 49, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 2, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 9 8 7 6 5 3 2 \ n x - x + 2 x - 2 x + x - x + 3 x - 3 x + 1 ) a(n) x = ------------------------------------------------ / 8 7 6 5 2 ----- (-1 + x) (x - x + 2 x - x - 2 x + 3 x - 1) n = 0 and in Maple format (x^9-x^8+2*x^7-2*x^6+x^5-x^3+3*x^2-3*x+1)/(-1+x)/(x^8-x^7+2*x^6-x^5-2*x^2+3*x-1 ) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1018, 2025, 4024, 7991, 15863, 31484, 62483, 124001, 246088, 488383, 969249, 1923595, 3817632, 7576622, 15036878, 29842821, 59227316, 117544999, 233284662, 462986318] The limit of a(n+1)/a(n) as n goes to infinity is 1.98464073989 a(n) is asymptotic to .543389350197*1.98464073989^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 50, -th largest growth, that is, 1.9848029699614000332, are , [2, 1, 3, 4], [2, 1, 4, 3], [3, 4, 1, 2], [4, 3, 1, 2] Theorem Number, 50, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 3, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 12 7 6 5 4 3 2 x + x + x - 2 x + x - x + 3 x - 3 x + 1 - ------------------------------------------------------------------- 13 12 9 8 7 6 5 4 3 2 x - x + x + x - x - 2 x + 3 x - x + 2 x - 5 x + 4 x - 1 and in Maple format -(x^12+x^7+x^6-2*x^5+x^4-x^3+3*x^2-3*x+1)/(x^13-x^12+x^9+x^8-x^7-2*x^6+3*x^5-x^ 4+2*x^3-5*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1018, 2025, 4024, 7991, 15863, 31484, 62483, 124002, 246095, 488414, 969358, 1923928, 3818558, 7579032, 15042854, 29857113, 59260567, 117620727, 233454270, 463361173] The limit of a(n+1)/a(n) as n goes to infinity is 1.98480296996 a(n) is asymptotic to .542498019435*1.98480296996^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 51, -th largest growth, that is, 1.9848266528671481993, are , [2, 3, 1, 4], [2, 4, 1, 3], [3, 1, 4, 2], [4, 1, 3, 2] Theorem Number, 51, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 3, 1, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = - ( / ----- n = 0 13 12 11 10 8 7 6 5 4 3 2 x + x - x + x + x - x + 2 x - 2 x + x - x + 3 x - 3 x + 1) / 14 12 11 10 9 8 7 6 5 4 3 / (x - x + 2 x - x + x - x + 2 x - 3 x + 3 x - x + 2 x / 2 - 5 x + 4 x - 1) and in Maple format -(x^13+x^12-x^11+x^10+x^8-x^7+2*x^6-2*x^5+x^4-x^3+3*x^2-3*x+1)/(x^14-x^12+2*x^ 11-x^10+x^9-x^8+2*x^7-3*x^6+3*x^5-x^4+2*x^3-5*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1018, 2025, 4024, 7991, 15862, 31479, 62467, 123960, 245997, 488202, 968922, 1923061, 3816871, 7575792, 15036674, 29845357, 59238208, 117578158, 233373115, 463206289] The limit of a(n+1)/a(n) as n goes to infinity is 1.98482665287 a(n) is asymptotic to .542122764500*1.98482665287^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 52, -th largest growth, that is, 1.9848333098768844858, are , [3, 1, 2, 4], [3, 2, 1, 4], [4, 1, 2, 3], [4, 2, 1, 3] Theorem Number, 52, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 1, 2, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 22 21 18 17 15 14 13 12 11 ) a(n) x = - (x + x + x + 2 x - x + x + x + x - 3 x / ----- n = 0 10 9 8 7 6 5 4 3 2 / + x - x + 3 x - x - 3 x + 3 x - 2 x + 4 x - 6 x + 4 x - 1) / ( / 24 23 21 20 19 18 17 15 13 12 x + 2 x - x + x + 3 x + x - 2 x + 3 x - 2 x - 3 x 11 9 8 6 5 4 3 2 + 4 x + 2 x - 5 x + 5 x - 4 x + 3 x - 7 x + 9 x - 5 x + 1) and in Maple format -(x^22+x^21+x^18+2*x^17-x^15+x^14+x^13+x^12-3*x^11+x^10-x^9+3*x^8-x^7-3*x^6+3*x ^5-2*x^4+4*x^3-6*x^2+4*x-1)/(x^24+2*x^23-x^21+x^20+3*x^19+x^18-2*x^17+3*x^15-2* x^13-3*x^12+4*x^11+2*x^9-5*x^8+5*x^6-4*x^5+3*x^4-7*x^3+9*x^2-5*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1018, 2025, 4024, 7991, 15862, 31478, 62461, 123938, 245933, 488040, 968547, 1922244, 3815165, 7572334, 15029804, 29831878, 59211938, 117527083, 233273778, 463012719] The limit of a(n+1)/a(n) as n goes to infinity is 1.98483330988 a(n) is asymptotic to .541844937295*1.98483330988^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 53, -th largest growth, that is, 1.9850654703526320630, are , [1, 3, 2, 4], [1, 4, 2, 3], [3, 2, 4, 1], [4, 2, 3, 1] Theorem Number, 53, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 3, 2, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 13 11 10 9 7 6 5 4 3 2 x - x + x + x - x + 2 x - 2 x + x - x + 3 x - 3 x + 1 ---------------------------------------------------------------------- 12 11 10 8 7 6 5 4 2 (-1 + x) (x + x - x + x + x - x + 2 x - x - 2 x + 3 x - 1) and in Maple format (x^13-x^11+x^10+x^9-x^7+2*x^6-2*x^5+x^4-x^3+3*x^2-3*x+1)/(-1+x)/(x^12+x^11-x^10 +x^8+x^7-x^6+2*x^5-x^4-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1018, 2025, 4024, 7991, 15863, 31485, 62490, 124031, 246189, 488679, 970042, 1925592, 3822444, 7587851, 15062467, 29900103, 59353801, 117821309, 233883084, 464273193] The limit of a(n+1)/a(n) as n goes to infinity is 1.98506547035 a(n) is asymptotic to .541412415763*1.98506547035^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 54, -th largest growth, that is, 1.9853288885629234253, are , [4, 1, 1, 4] Theorem Number, 54, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [4, 1, 1, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 25 24 23 22 21 20 19 ) a(n) x = - (x + 3 x + 6 x + 7 x + 6 x + 3 x + 3 x / ----- n = 0 18 17 16 15 13 12 10 9 8 7 + 3 x + 4 x + x + x + 3 x - x - 3 x + 3 x - x + 2 x 6 5 4 3 2 / - 4 x + 3 x - 2 x + 4 x - 6 x + 4 x - 1) / ( / 7 6 5 4 21 20 19 18 17 16 (x + x + x - x + 2 x - 1) (x + 3 x + 6 x + 7 x + 6 x + 3 x 15 12 9 6 3 2 + x - x + 2 x - x + x - 3 x + 3 x - 1)) and in Maple format -(x^25+3*x^24+6*x^23+7*x^22+6*x^21+3*x^20+3*x^19+3*x^18+4*x^17+x^16+x^15+3*x^13 -x^12-3*x^10+3*x^9-x^8+2*x^7-4*x^6+3*x^5-2*x^4+4*x^3-6*x^2+4*x-1)/(x^7+x^6+x^5- x^4+2*x-1)/(x^21+3*x^20+6*x^19+7*x^18+6*x^17+3*x^16+x^15-x^12+2*x^9-x^6+x^3-3*x ^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1018, 2025, 4024, 7991, 15862, 31478, 62462, 123946, 245971, 488180, 968992, 1923527, 3818613, 7581119, 15051276, 29882668, 59328998, 117791372, 233860753, 464299424] The limit of a(n+1)/a(n) as n goes to infinity is 1.98532888856 a(n) is asymptotic to .539298326604*1.98532888856^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 55, -th largest growth, that is, 1.9855197870868743418, are , [3, 1, 3, 3], [3, 3, 1, 3] Theorem Number, 55, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 1, 3, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 3 8 7 6 2 (x - x + 1) (x + x + x + x - 2 x + 1) - ------------------------------------------------------------------------- 13 12 10 9 8 7 6 5 4 3 2 x + 2 x - x + x + x + x - x - 2 x + 3 x + x - 5 x + 4 x - 1 and in Maple format -(x^3-x+1)*(x^8+x^7+x^6+x^2-2*x+1)/(x^13+2*x^12-x^10+x^9+x^8+x^7-x^6-2*x^5+3*x^ 4+x^3-5*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1018, 2025, 4024, 7991, 15863, 31485, 62491, 124039, 246227, 488818, 970478, 1926830, 3825722, 7596093, 15082388, 29946803, 59460703, 118061507, 234414938, 465437241] The limit of a(n+1)/a(n) as n goes to infinity is 1.98551978709 a(n) is asymptotic to .539056792066*1.98551978709^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 56, -th largest growth, that is, 1.9855529777414181545, are , [1, 2, 3, 4], [1, 2, 4, 3], [1, 3, 4, 2], [1, 4, 3, 2], [2, 3, 4, 1], [2, 4, 3, 1], [3, 4, 2, 1], [4, 3, 2, 1] Theorem Number, 56, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 2, 3, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 3 6 5 4 3 2 \ n (x - x + 1) (x - x + 2 x - x + x - 2 x + 1) ) a(n) x = ------------------------------------------------- / 8 7 5 4 2 ----- (-1 + x) (x - x + 2 x - x - 2 x + 3 x - 1) n = 0 and in Maple format (x^3-x+1)*(x^6-x^5+2*x^4-x^3+x^2-2*x+1)/(-1+x)/(x^8-x^7+2*x^5-x^4-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1018, 2025, 4024, 7992, 15869, 31507, 62555, 124201, 246602, 489637, 972199, 1930356, 3832833, 7610307, 15110685, 30003083, 59572724, 118284804, 234860737, 466328414] The limit of a(n+1)/a(n) as n goes to infinity is 1.98555297774 a(n) is asymptotic to .539818135967*1.98555297774^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 57, -th largest growth, that is, 1.9861840427096552271, are , [2, 2, 2, 4], [2, 2, 4, 2], [2, 4, 2, 2], [4, 2, 2, 2] Theorem Number, 57, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 2, 2, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 8 7 6 5 4 3 2 \ n x - x + 2 x - 2 x + x - x + 3 x - 3 x + 1 ) a(n) x = - --------------------------------------------------------- / 9 8 7 6 5 4 3 2 ----- x - x + 2 x - 3 x + 3 x - x + 2 x - 5 x + 4 x - 1 n = 0 and in Maple format -(x^8-x^7+2*x^6-2*x^5+x^4-x^3+3*x^2-3*x+1)/(x^9-x^8+2*x^7-3*x^6+3*x^5-x^4+2*x^3 -5*x^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1018, 2025, 4024, 7992, 15869, 31508, 62562, 124232, 246712, 489976, 973148, 1932836, 3838993, 7625037, 15144909, 30080871, 59746556, 118668153, 235697297, 468138578] The limit of a(n+1)/a(n) as n goes to infinity is 1.98618404271 a(n) is asymptotic to .536770920616*1.98618404271^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 58, -th largest growth, that is, 1.9864180586156033445, are , [3, 2, 2, 3] Theorem Number, 58, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 2, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 2 16 15 12 11 10 8 7 6 ) a(n) x = - (x - x + 1) (x + x + x + x - x + x - x - x / ----- n = 0 5 4 3 2 / 20 19 18 17 16 + x + 2 x - 2 x - 2 x + 3 x - 1) / (x + x - x + x + 2 x / 14 13 12 11 9 8 7 6 5 4 3 - x + x + 2 x - 3 x + 3 x - x - 2 x - x + 5 x - 2 x - 6 x 2 + 9 x - 5 x + 1) and in Maple format -(x^2-x+1)*(x^16+x^15+x^12+x^11-x^10+x^8-x^7-x^6+x^5+2*x^4-2*x^3-2*x^2+3*x-1)/( x^20+x^19-x^18+x^17+2*x^16-x^14+x^13+2*x^12-3*x^11+3*x^9-x^8-2*x^7-x^6+5*x^5-2* x^4-6*x^3+9*x^2-5*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1018, 2025, 4024, 7992, 15869, 31508, 62563, 124240, 246748, 490100, 973514, 1933815, 3841445, 7630904, 15158497, 30111604, 59814863, 118817998, 236022743, 468839898] The limit of a(n+1)/a(n) as n goes to infinity is 1.98641805862 a(n) is asymptotic to .535676599493*1.98641805862^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 59, -th largest growth, that is, 1.9867708871840811416, are , [2, 3, 2, 3], [3, 2, 3, 2] Theorem Number, 59, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 3, 2, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 3 8 5 2 (x - x + 1) (x + x + x - 2 x + 1) - ------------------------------------------------------------------ 12 11 9 8 6 5 4 3 2 x - x + 3 x - 2 x + 2 x - 3 x + 3 x + x - 5 x + 4 x - 1 and in Maple format -(x^3-x+1)*(x^8+x^5+x^2-2*x+1)/(x^12-x^11+3*x^9-2*x^8+2*x^6-3*x^5+3*x^4+x^3-5*x ^2+4*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1018, 2025, 4024, 7992, 15870, 31515, 62591, 124327, 246983, 490683, 974884, 1936923, 3848334, 7645940, 15190977, 30181266, 59963490, 119133778, 236691252, 470250505] The limit of a(n+1)/a(n) as n goes to infinity is 1.98677088718 a(n) is asymptotic to .534434032726*1.98677088718^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 60, -th largest growth, that is, 1.9874108030247649893, are , [1, 3, 3, 3], [3, 3, 3, 1] Theorem Number, 60, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 3, 3, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 3 6 2 \ n (x - x + 1) (x + x - 2 x + 1) ) a(n) x = ---------------------------------------------------- / 8 7 6 4 3 2 ----- (-1 + x) (x + x - x + 2 x - x - 2 x + 3 x - 1) n = 0 and in Maple format (x^3-x+1)*(x^6+x^2-2*x+1)/(-1+x)/(x^8+x^7-x^6+2*x^4-x^3-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1018, 2025, 4025, 7998, 15892, 31579, 62755, 124716, 247862, 492610, 979032, 1945758, 3867041, 7685414, 15274076, 30355844, 60329489, 119899416, 238289328, 473578737] The limit of a(n+1)/a(n) as n goes to infinity is 1.98741080302 a(n) is asymptotic to .533043756330*1.98741080302^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 61, -th largest growth, that is, 1.9875763672979512663, are , [2, 2, 3, 3], [2, 3, 3, 2], [3, 3, 2, 2] Theorem Number, 61, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 2, 3, 3] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 8 7 6 5 4 2 \ n x - x + x + x - 2 x + 3 x - 3 x + 1 ) a(n) x = - ------------------------------------------------------- / 9 8 7 6 5 4 3 2 ----- x - x + 2 x - x - 2 x + 3 x + x - 5 x + 4 x - 1 n = 0 and in Maple format -(x^8-x^7+x^6+x^5-2*x^4+3*x^2-3*x+1)/(x^9-x^8+2*x^7-x^6-2*x^5+3*x^4+x^3-5*x^2+4 *x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1018, 2025, 4025, 7998, 15892, 31579, 62755, 124717, 247870, 492648, 979170, 1946183, 3868215, 7688425, 15281410, 30373069, 60368903, 119987901, 238485167, 474007445] The limit of a(n+1)/a(n) as n goes to infinity is 1.98757636730 a(n) is asymptotic to .532194205807*1.98757636730^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 62, -th largest growth, that is, 1.9919641966050350211, are , [1, 1, 8], [1, 8, 1], [8, 1, 1] Theorem Number, 62, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 8] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 8 2 \ n x + x - 2 x + 1 ) a(n) x = - ----------------------------------------------- / 7 6 5 4 3 2 2 ----- (x + x + x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format -(x^8+x^2-2*x+1)/(x^7+x^6+x^5+x^4+x^3+x^2+x-1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1019, 2031, 4047, 8063, 16063, 31999, 63743, 126976, 252934, 503838, 1003630, 1999198, 3982334, 7932670, 15801598, 31476221, 62699509, 124895181, 248786733, 495574269] The limit of a(n+1)/a(n) as n goes to infinity is 1.99196419661 a(n) is asymptotic to .520790324966*1.99196419661^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 63, -th largest growth, that is, 1.9920300868462484222, are , [2, 1, 7], [7, 1, 2] Theorem Number, 63, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 1, 7] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 15 7 2 \ n x + x + x - 2 x + 1 ) a(n) x = - -------------------------------------------------------- / 6 5 3 2 8 7 2 ----- (x + x - x - x + 1) (x - x + 2 x - 1) (x - x + 1) n = 0 and in Maple format -(x^15+x^7+x^2-2*x+1)/(x^6+x^5-x^3-x^2+1)/(x^8-x^7+2*x-1)/(x^2-x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1019, 2031, 4047, 8063, 16063, 31999, 63743, 126977, 252940, 503861, 1003702, 1999399, 3982856, 7933961, 15804681, 31483393, 62715861, 124931880, 248868066, 495752683] The limit of a(n+1)/a(n) as n goes to infinity is 1.99203008685 a(n) is asymptotic to .520461112883*1.99203008685^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 64, -th largest growth, that is, 1.9921580953553798820, are , [3, 1, 6], [6, 1, 3] Theorem Number, 64, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 1, 6] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 14 13 6 2 \ n x + x + x + x - 2 x + 1 ) a(n) x = - ------------------------------------------ / 8 7 6 8 7 ----- (x + x - x + 2 x - 1) (x + x - x + 1) n = 0 and in Maple format -(x^14+x^13+x^6+x^2-2*x+1)/(x^8+x^7-x^6+2*x-1)/(x^8+x^7-x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1019, 2031, 4047, 8063, 16063, 31999, 63744, 126983, 252963, 503933, 1003903, 1999921, 3984147, 7937043, 15811845, 31499708, 62752428, 125012808, 249045352, 496137813] The limit of a(n+1)/a(n) as n goes to infinity is 1.99215809536 a(n) is asymptotic to .519862388220*1.99215809536^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 65, -th largest growth, that is, 1.9922212637540336675, are , [2, 2, 6], [2, 6, 2], [6, 2, 2] Theorem Number, 65, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 2, 6] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 8 7 6 2 \ n x - x + x + x - 2 x + 1 ) a(n) x = - ------------------------------------ / 9 8 7 6 2 ----- x - x + 2 x - x - 2 x + 3 x - 1 n = 0 and in Maple format -(x^8-x^7+x^6+x^2-2*x+1)/(x^9-x^8+2*x^7-x^6-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1019, 2031, 4047, 8063, 16063, 32000, 63749, 126999, 253006, 504039, 1004152, 2000488, 3985411, 7939819, 15817878, 31512719, 62780320, 125072305, 249171727, 496405238] The limit of a(n+1)/a(n) as n goes to infinity is 1.99222126375 a(n) is asymptotic to .519647972626*1.99222126375^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 66, -th largest growth, that is, 1.9923365951207343993, are , [3, 2, 5], [5, 2, 3] Theorem Number, 66, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 2, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 12 7 6 5 2 x + x - x + x + x - 2 x + 1 - ----------------------------------------------------------- 14 13 12 9 8 7 6 5 2 x + x - x + x + x - x + 2 x - x - 2 x + 3 x - 1 and in Maple format -(x^12+x^7-x^6+x^5+x^2-2*x+1)/(x^14+x^13-x^12+x^9+x^8-x^7+2*x^6-x^5-2*x^2+3*x-1 ) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1019, 2031, 4047, 8063, 16063, 32000, 63750, 127005, 253029, 504111, 1004352, 2001004, 3986677, 7942816, 15824789, 31528346, 62815130, 125148943, 249338879, 496767018] The limit of a(n+1)/a(n) as n goes to infinity is 1.99233659512 a(n) is asymptotic to .519124274617*1.99233659512^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 67, -th largest growth, that is, 1.9924010004614550874, are , [4, 1, 5], [5, 1, 4] Theorem Number, 67, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [4, 1, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 13 12 11 5 2 \ n x + x + x + x + x - 2 x + 1 ) a(n) x = - ---------------------------------------------------- / 8 7 6 8 7 6 5 ----- (x + x + x - x + 1) (x + x + x - x + 2 x - 1) n = 0 and in Maple format -(x^13+x^12+x^11+x^5+x^2-2*x+1)/(x^8+x^7+x^6-x+1)/(x^8+x^7+x^6-x^5+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1019, 2031, 4047, 8063, 16063, 32000, 63750, 127006, 253035, 504134, 1004425, 2001211, 3987221, 7944170, 15828028, 31535870, 62832228, 125187155, 249423186, 496951151] The limit of a(n+1)/a(n) as n goes to infinity is 1.99240100046 a(n) is asymptotic to .518813178100*1.99240100046^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 68, -th largest growth, that is, 1.9924602335260952454, are , [2, 3, 5], [2, 5, 3], [3, 5, 2], [5, 3, 2] Theorem Number, 68, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 3, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 8 6 5 2 \ n x - x + x + x - 2 x + 1 ) a(n) x = - ------------------------------------ / 9 8 6 5 2 ----- x - x + 2 x - x - 2 x + 3 x - 1 n = 0 and in Maple format -(x^8-x^6+x^5+x^2-2*x+1)/(x^9-x^8+2*x^6-x^5-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1019, 2031, 4047, 8063, 16064, 32005, 63766, 127048, 253135, 504359, 1004914, 2002252, 3989411, 7948749, 15837575, 31555749, 62873586, 125273129, 249601733, 497321526] The limit of a(n+1)/a(n) as n goes to infinity is 1.99246023353 a(n) is asymptotic to .518737176056*1.99246023353^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 69, -th largest growth, that is, 1.9925574403170594250, are , [4, 2, 4] Theorem Number, 69, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [4, 2, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 11 10 6 5 4 2 / ) a(n) x = - (x + x + x - x + x + x - 2 x + 1) / ( / / ----- n = 0 14 13 12 10 9 8 7 6 5 4 2 x + 2 x + 2 x - x + x + x + x - x + 2 x - x - 2 x + 3 x - 1) and in Maple format -(x^11+x^10+x^6-x^5+x^4+x^2-2*x+1)/(x^14+2*x^13+2*x^12-x^10+x^9+x^8+x^7-x^6+2*x ^5-x^4-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1019, 2031, 4047, 8063, 16063, 32001, 63756, 127028, 253101, 504311, 1004868, 2002269, 3989666, 7949691, 15840291, 31562782, 62890751, 125313504, 249694359, 497530239] The limit of a(n+1)/a(n) as n goes to infinity is 1.99255744032 a(n) is asymptotic to .518195458821*1.99255744032^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 70, -th largest growth, that is, 1.9928945584015041909, are , [2, 4, 4], [4, 4, 2] Theorem Number, 70, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 4, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 8 5 4 2 \ n x - x + x + x - 2 x + 1 ) a(n) x = - ------------------------------------ / 9 8 5 4 2 ----- x - x + 2 x - x - 2 x + 3 x - 1 n = 0 and in Maple format -(x^8-x^5+x^4+x^2-2*x+1)/(x^9-x^8+2*x^5-x^4-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1019, 2031, 4047, 8064, 16069, 32022, 63815, 127176, 253449, 505099, 1006612, 2006075, 3997899, 7967393, 15878174, 31643524, 63062202, 125676313, 250459634, 499139637] The limit of a(n+1)/a(n) as n goes to infinity is 1.99289455840 a(n) is asymptotic to .517240378759*1.99289455840^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 71, -th largest growth, that is, 1.9929967428173323177, are , [3, 3, 4], [3, 4, 3], [4, 3, 3] Theorem Number, 71, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 3, 4] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 7 5 4 2 \ n x - x + x + x - 2 x + 1 ) a(n) x = - ----------------------------------------- / 9 8 7 5 4 2 ----- x + x - x + 2 x - x - 2 x + 3 x - 1 n = 0 and in Maple format -(x^7-x^5+x^4+x^2-2*x+1)/(x^9+x^8-x^7+2*x^5-x^4-2*x^2+3*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1019, 2031, 4047, 8064, 16069, 32022, 63816, 127182, 253472, 505171, 1006811, 2006583, 3999130, 7970272, 15884743, 31658250, 63094783, 125747669, 250614641, 499474085] The limit of a(n+1)/a(n) as n goes to infinity is 1.99299674282 a(n) is asymptotic to .516791351539*1.99299674282^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 72, -th largest growth, that is, 1.9960311797354145898, are , [1, 9], [9, 1] Theorem Number, 72, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 9] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 9 \ n x - x + 1 ) a(n) x = --------------------------------------------------- / 8 7 6 5 4 3 2 ----- (-1 + x) (x + x + x + x + x + x + x + x - 1) n = 0 and in Maple format (x^9-x+1)/(-1+x)/(x^8+x^7+x^6+x^5+x^4+x^3+x^2+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1020, 2036, 4064, 8112, 16192, 32320, 64512, 128768, 257025, 513030, 1024024, 2043984, 4079856, 8143520, 16254720, 32444928, 64761088, 129265151, 258017272, 515010520] The limit of a(n+1)/a(n) as n goes to infinity is 1.99603117974 a(n) is asymptotic to .509092230016*1.99603117974^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 73, -th largest growth, that is, 1.9960471205602957908, are , [2, 8], [8, 2] Theorem Number, 73, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [2, 8] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 2 6 5 3 2 \ n (x - x + 1) (x + x - x - x + 1) ) a(n) x = - ------------------------------------ / 9 8 ----- x - x + 2 x - 1 n = 0 and in Maple format -(x^2-x+1)*(x^6+x^5-x^3-x^2+1)/(x^9-x^8+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1020, 2036, 4064, 8112, 16192, 32320, 64512, 128769, 257029, 513042, 1024056, 2044064, 4080048, 8143968, 16255744, 32447231, 64766202, 129276391, 258041768, 515063528] The limit of a(n+1)/a(n) as n goes to infinity is 1.99604712056 a(n) is asymptotic to .509022658976*1.99604712056^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 74, -th largest growth, that is, 1.9960785538884684114, are , [3, 7], [7, 3] Theorem Number, 74, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [3, 7] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 7 \ n x - x + 1 ) a(n) x = - ---------------------- / 9 8 7 ----- x + x - x + 2 x - 1 n = 0 and in Maple format -(x^7-x+1)/(x^9+x^8-x^7+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1020, 2036, 4064, 8112, 16192, 32320, 64513, 128773, 257041, 513074, 1024136, 2044256, 4080496, 8144991, 16258042, 32452329, 64777398, 129300775, 258094504, 515176904] The limit of a(n+1)/a(n) as n goes to infinity is 1.99607855389 a(n) is asymptotic to .508894231203*1.99607855389^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 75, -th largest growth, that is, 1.9961398080808123357, are , [4, 6], [6, 4] Theorem Number, 75, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [4, 6] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 6 \ n x - x + 1 ) a(n) x = - --------------------------- / 9 8 7 6 ----- x + x + x - x + 2 x - 1 n = 0 and in Maple format -(x^6-x+1)/(x^9+x^8+x^7-x^6+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1020, 2036, 4064, 8112, 16192, 32321, 64517, 128785, 257073, 513154, 1024328, 2044703, 4081514, 8147273, 16263096, 32463413, 64801510, 129352872, 258206415, 515416102] The limit of a(n+1)/a(n) as n goes to infinity is 1.99613980808 a(n) is asymptotic to .508662020641*1.99613980808^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 76, -th largest growth, that is, 1.9962564778474358596, are , [5, 5] Theorem Number, 76, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [5, 5] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 5 \ n x - x + 1 ) a(n) x = - -------------------------------- / 9 8 7 6 5 ----- x + x + x + x - x + 2 x - 1 n = 0 and in Maple format -(x^5-x+1)/(x^9+x^8+x^7+x^6-x^5+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1020, 2036, 4064, 8112, 16193, 32325, 64529, 128817, 257153, 513345, 1024770, 2045705, 4083752, 8152215, 16273909, 32486892, 64852164, 129461549, 258438455, 515909443] The limit of a(n+1)/a(n) as n goes to infinity is 1.99625647785 a(n) is asymptotic to .508256955954*1.99625647785^n ---------------------------------------------------------------------------- The compositions of, 10, that yield the, 77, -th largest growth, that is, 1.9980294702622866987, are , [10] Theorem Number, 77, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [10] We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 1 ) a(n) x = - --------------------------------------------- / 9 8 7 6 5 4 3 2 ----- x + x + x + x + x + x + x + x + x - 1 n = 0 and in Maple format -1/(x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1021, 2040, 4076, 8144, 16272, 32512, 64960, 129792, 259328, 518145, 1035269, 2068498, 4132920, 8257696, 16499120, 32965728, 65866496, 131603200, 262947072, 525375999] The limit of a(n+1)/a(n) as n goes to infinity is 1.99802947026 a(n) is asymptotic to .503980275734*1.99802947026^n ---------------------------------------------------------------------------- This ends this article, that took, 68.732, seconds to generate. ---------------------------------------------------------- --------------------- altogether it took, 78.261, seconds.