Generating functions and Growth rates for the Enumerating Sequences for Comositions Avoiding Each of the Possible subsets of the set of compositions of, 2, into exactly, 1, parts By Shalosh B. Ekhad The subsets of the set of compositions of, 2, with exactly, 1, parts, that yield the 1, -th largest growth, that is, 1., are , {[2]} 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, 0.059, seconds to generate. ---------------------------------------------------------- ---------------------------------------- Generating functions and Growth rates for the Enumerating Sequences for Como\ sitions Avoiding Each of the Possible subsets of the set of compositions of, 2, into exactly, 2, parts By Shalosh B. Ekhad The subsets of the set of compositions of, 2, with exactly, 2, parts, that yield the 1, -th largest growth, that is, 1., are , {[1, 1]} Theorem Number, 1, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1] 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, 0.004, seconds to generate. ---------------------------------------------------------- ---------------------------------------- Generating functions and Growth rates for the Enumerating Sequences for Como\ sitions Avoiding Each of the Possible subsets of the set of compositions of, 3, into exactly, 1, parts By Shalosh B. Ekhad The subsets of the set of compositions of, 3, with exactly, 1, parts, that yield the 1, -th largest growth, that is, 1.6180339887498948482, are , {[3]} Theorem Number, 1, : 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, 0.007, seconds to generate. ---------------------------------------------------------- ---------------------------------------- Generating functions and Growth rates for the Enumerating Sequences for Como\ sitions Avoiding Each of the Possible subsets of the set of compositions of, 3, into exactly, 2, parts By Shalosh B. Ekhad The subsets of the set of compositions of, 3, with exactly, 2, parts, that yield the 1, -th largest growth, that is, 1, are , {[1, 2]}, {[2, 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 subsets of the set of compositions of, 3, with exactly, 2, parts, that yield the 2, -th largest growth, that is, 1., are , {[1, 2], [2, 1]} Theorem Number, 2, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, {[1, 2], [2, 1]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 2 \ n x + 1 ) a(n) x = - ------ / -1 + x ----- n = 0 and in Maple format -(x^2+1)/(-1+x) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2] The limit of a(n+1)/a(n) as n goes to infinity is 1. a(n) is asymptotic to 2.00000000000*1.^n ---------------------------------------------------------------------------- This ends this article, that took, 0.038, seconds to generate. ---------------------------------------------------------- ---------------------------------------- Generating functions and Growth rates for the Enumerating Sequences for Como\ sitions Avoiding Each of the Possible subsets of the set of compositions of, 3, into exactly, 3, parts By Shalosh B. Ekhad The subsets of the set of compositions of, 3, with exactly, 3, parts, that yield the 1, -th largest growth, that is, 1, are , {[1, 1, 1]} Theorem Number, 1, : Let a(n) be the number of compositions of n avoiding, as a subcomposition , [1, 1, 1] 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] ---------------------------------------------------------------------------- This ends this article, that took, 0.002, seconds to generate. ---------------------------------------------------------- ---------------------------------------- Generating functions and Growth rates for the Enumerating Sequences for Como\ sitions Avoiding Each of the Possible subsets of the set of compositions of, 4, into exactly, 1, parts By Shalosh B. Ekhad The subsets of the set of compositions of, 4, with exactly, 1, parts, that yield the 1, -th largest growth, that is, 1.8392867552141611326, are , {[4]} Theorem Number, 1, : 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, 0.008, seconds to generate. ---------------------------------------------------------- ---------------------------------------- Generating functions and Growth rates for the Enumerating Sequences for Como\ sitions Avoiding Each of the Possible subsets of the set of compositions of, 4, into exactly, 2, parts By Shalosh B. Ekhad The subsets of the set of compositions of, 4, with exactly, 2, parts, that yield the 1, -th largest growth, that is, 1.4655712318767680267, are , {[1, 3], [2, 2], [3, 1]} Theorem Number, 1, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, {[1, 3], [2, 2], [3, 1]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 6 4 2 \ n x + x - x + x - 1 ) a(n) x = - --------------------- / 3 ----- (-1 + x) (x + x - 1) n = 0 and in Maple format -(x^6+x^4-x^2+x-1)/(-1+x)/(x^3+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 5, 7, 10, 14, 20, 29, 42, 61, 89, 130, 190, 278, 407, 596, 873, 1279, 1874, 2746, 4024, 5897, 8642, 12665, 18561, 27202, 39866, 58426, 85627] The limit of a(n+1)/a(n) as n goes to infinity is 1.46557123188 a(n) is asymptotic to .896185071926*1.46557123188^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 4, with exactly, 2, parts, 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 \ subcompositions, the members of the set, {[1, 3], [3, 1]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 5 4 3 \ n x + x - x + x - 1 ) a(n) x = - --------------------- / 2 ----- (-1 + x) (x + x - 1) n = 0 and in Maple format -(x^5+x^4-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, 6, 9, 14, 22, 35, 56, 90, 145, 234, 378, 611, 988, 1598, 2585, 4182, 6766, 10947, 17712, 28658, 46369, 75026, 121394, 196419, 317812, 514230, 832041, 1346270] 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 ---------------------------------------------------------------------------- The subsets of the set of compositions of, 4, with exactly, 2, parts, 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 ---------------------------------------------------------------------------- This ends this article, that took, 0.038, seconds to generate. ---------------------------------------------------------- ---------------------------------------- Generating functions and Growth rates for the Enumerating Sequences for Como\ sitions Avoiding Each of the Possible subsets of the set of compositions of, 4, into exactly, 3, parts By Shalosh B. Ekhad The subsets of the set of compositions of, 4, with exactly, 3, parts, that yield the 1, -th largest growth, that is, 1, are , {[1, 1, 2], [2, 1, 1]} Theorem Number, 1, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, {[1, 1, 2], [2, 1, 1]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 5 3 2 \ n x - x - x + x - 1 ) a(n) x = - -------------------- / 2 ----- (-1 + x) n = 0 and in Maple format -(x^5-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, 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, 31, 32] ---------------------------------------------------------------------------- This ends this article, that took, 0.041, seconds to generate. ---------------------------------------------------------- ---------------------------------------- Generating functions and Growth rates for the Enumerating Sequences for Como\ sitions Avoiding Each of the Possible subsets of the set of compositions of, 4, into exactly, 4, parts By Shalosh B. Ekhad The subsets of the set of compositions of, 4, with exactly, 4, parts, that yield the 1, -th largest growth, that is, 1, are , {[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] 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] ---------------------------------------------------------------------------- This ends this article, that took, 0.001, seconds to generate. ---------------------------------------------------------- ---------------------------------------- Generating functions and Growth rates for the Enumerating Sequences for Como\ sitions Avoiding Each of the Possible subsets of the set of compositions of, 5, into exactly, 1, parts By Shalosh B. Ekhad The subsets of the set of compositions of, 5, with exactly, 1, parts, that yield the 1, -th largest growth, that is, 1.9275619754829253043, are , {[5]} Theorem Number, 1, : 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, 0.003, seconds to generate. ---------------------------------------------------------- ---------------------------------------- Generating functions and Growth rates for the Enumerating Sequences for Como\ sitions Avoiding Each of the Possible subsets of the set of compositions of, 5, into exactly, 2, parts By Shalosh B. Ekhad The subsets of the set of compositions of, 5, with exactly, 2, parts, that yield the 1, -th largest growth, that is, 1.7142853291425735181, are , {[1, 4], [2, 3], [3, 2], [4, 1]} Theorem Number, 1, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, {[1, 4], [2, 3], [3, 2], [4, 1]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 10 8 5 3 \ n x - x - 2 x + x - x + 1 ) a(n) x = - ------------------------------- / 6 4 2 ----- (-1 + x) (x - x - x - x + 1) n = 0 and in Maple format -(x^10-x^8-2*x^5+x^3-x+1)/(-1+x)/(x^6-x^4-x^2-x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 12, 20, 34, 58, 98, 167, 286, 490, 839, 1437, 2463, 4222, 7237, 12405, 21265, 36454, 62492, 107128, 183647, 314823, 539696, 925192, 1586042, 2718928, 4661018, 7990314] The limit of a(n+1)/a(n) as n goes to infinity is 1.71428532914 a(n) is asymptotic to .758700778796*1.71428532914^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 5, with exactly, 2, parts, that yield the 2, -th largest growth, that is, 1.7548776662466927601, are , {[1, 4], [2, 3], [4, 1]}, {[1, 4], [3, 2], [4, 1]} Theorem Number, 2, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, {[1, 4], [2, 3], [4, 1]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 7 6 5 4 3 2 \ n x - x + 2 x - x + x - 2 x + 2 x - 1 ) a(n) x = - ----------------------------------------- / 3 2 ----- (-1 + x) (x - x + 2 x - 1) n = 0 and in Maple format -(x^7-x^6+2*x^5-x^4+x^3-2*x^2+2*x-1)/(-1+x)/(x^3-x^2+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 13, 22, 38, 66, 115, 201, 352, 617, 1082, 1898, 3330, 5843, 10253, 17992, 31573, 55406, 97230, 170626, 299427, 525457, 922112, 1618193, 2839730, 4983378, 8745218, 15346787] 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 subsets of the set of compositions of, 5, with exactly, 2, parts, that yield the 3, -th largest growth, that is, 1.8124036192680426608, are , {[2, 3], [3, 2]} Theorem Number, 3, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, {[2, 3], [3, 2]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 4 3 \ n x + x + 1 ) a(n) x = - -------------------- / 5 4 2 ----- x + x + x + x - 1 n = 0 and in Maple format -(x^4+x^3+1)/(x^5+x^4+x^2+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 14, 25, 45, 82, 149, 270, 489, 886, 1606, 2911, 5276, 9562, 17330, 31409, 56926, 103173, 186991, 338903, 614229, 1113231, 2017624, 3656749, 6627505, 12011714, 21770074, 39456161] The limit of a(n+1)/a(n) as n goes to infinity is 1.81240361927 a(n) is asymptotic to .705451568911*1.81240361927^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 5, with exactly, 2, parts, 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 \ subcompositions, the members of the set, {[1, 4], [4, 1]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 3 2 4 2 \ n (x + x - 1) (x + x - x + 1) ) a(n) x = - ------------------------------- / 3 2 ----- (-1 + x) (x + x + x - 1) n = 0 and in Maple format -(x^3+x^2-1)*(x^4+x^2-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, 14, 25, 45, 82, 150, 275, 505, 928, 1706, 3137, 5769, 10610, 19514, 35891, 66013, 121416, 223318, 410745, 755477, 1389538, 2555758, 4700771, 8646065, 15902592, 29249426, 53798081] 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 ---------------------------------------------------------------------------- The subsets of the set of compositions of, 5, with exactly, 2, parts, 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 ---------------------------------------------------------------------------- This ends this article, that took, 0.101, seconds to generate. ---------------------------------------------------------- ---------------------------------------- Generating functions and Growth rates for the Enumerating Sequences for Como\ sitions Avoiding Each of the Possible subsets of the set of compositions of, 5, into exactly, 3, parts By Shalosh B. Ekhad The subsets of the set of compositions of, 5, with exactly, 3, parts, that yield the 1, -th largest growth, that is, 1.3802775690976141157, are , {[1, 1, 3], [1, 2, 2], [2, 1, 2], [3, 1, 1]}, {[1, 1, 3], [2, 1, 2], [2, 2, 1], [3, 1, 1]} Theorem Number, 1, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, {[1, 1, 3], [1, 2, 2], [2, 1, 2], [3, 1, 1]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 11 10 9 7 6 5 2 \ n x + x - x - x - x - x + 2 x - 2 x + 1 ) a(n) x = - ---------------------------------------------- / 4 2 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format -(x^11+x^10-x^9-x^7-x^6-x^5+2*x^2-2*x+1)/(x^4+x-1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 12, 16, 20, 26, 33, 42, 54, 71, 94, 125, 167, 225, 305, 415, 566, 774, 1061, 1457, 2003, 2756, 3795, 5229, 7208, 9939, 13708, 18910] The limit of a(n+1)/a(n) as n goes to infinity is 1.38027756910 a(n) is asymptotic to 1.19389291458*1.38027756910^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 5, with exactly, 3, parts, that yield the 2, -th largest growth, that is, 1.4655712318767680267, are , {[1, 1, 3], [2, 1, 2], [3, 1, 1]} Theorem Number, 2, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, {[1, 1, 3], [2, 1, 2], [3, 1, 1]} 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 8 7 6 2 2 x + x - x - 2 x + x - 2 x - 2 x + 2 x - 2 x + 1 - ------------------------------------------------------------ 2 3 2 (x + 1) (x - x + 1) (x + x - 1) (-1 + x) and in Maple format -(2*x^13+x^11-x^10-2*x^9+x^8-2*x^7-2*x^6+2*x^2-2*x+1)/(x+1)/(x^2-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, 13, 19, 24, 32, 43, 59, 81, 113, 159, 227, 326, 471, 682, 991, 1443, 2106, 3077, 4500, 6584, 9638, 14113, 20672, 30284, 44371, 65015, 95270] The limit of a(n+1)/a(n) as n goes to infinity is 1.46557123188 a(n) is asymptotic to .996775270563*1.46557123188^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 5, with exactly, 3, parts, that yield the 3, -th largest growth, that is, 1.6180339887498948482, are , {[1, 1, 3], [3, 1, 1]} Theorem Number, 3, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, {[1, 1, 3], [3, 1, 1]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 10 9 7 6 5 3 2 \ n x + 2 x - 2 x - 2 x - x + x + x - 2 x + 1 ) a(n) x = - ------------------------------------------------- / 2 2 ----- (x + x - 1) (-1 + x) n = 0 and in Maple format -(x^10+2*x^9-2*x^7-2*x^6-x^5+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, 14, 22, 32, 46, 68, 103, 159, 249, 394, 628, 1006, 1617, 2605, 4203, 6788, 10970, 17736, 28683, 46395, 75053, 121422, 196448, 317842, 514261, 832073, 1346303] 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 ---------------------------------------------------------------------------- The subsets of the set of compositions of, 5, with exactly, 3, parts, that yield the 4, -th largest growth, that is, 1.7548776662466927601, are , {[2, 1, 2]} Theorem Number, 4, : 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 ---------------------------------------------------------------------------- This ends this article, that took, 2.701, seconds to generate. ---------------------------------------------------------- ---------------------------------------- Generating functions and Growth rates for the Enumerating Sequences for Como\ sitions Avoiding Each of the Possible subsets of the set of compositions of, 5, into exactly, 4, parts By Shalosh B. Ekhad The subsets of the set of compositions of, 5, with exactly, 4, parts, that yield the 1, -th largest growth, that is, 1, are , {[1, 1, 1, 2], [2, 1, 1, 1]} Theorem Number, 1, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, {[1, 1, 1, 2], [2, 1, 1, 1]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 8 7 6 4 2 \ n x - x - x + x + 2 x - 2 x + 1 ) a(n) x = - ---------------------------------- / 3 ----- (-1 + x) n = 0 and in Maple format -(x^8-x^7-x^6+x^4+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, 8, 14, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465] ---------------------------------------------------------------------------- This ends this article, that took, 0.348, seconds to generate. ---------------------------------------------------------- ---------------------------------------- Generating functions and Growth rates for the Enumerating Sequences for Como\ sitions Avoiding Each of the Possible subsets of the set of compositions of, 5, into exactly, 5, parts By Shalosh B. Ekhad The subsets of the set of compositions of, 5, with exactly, 5, parts, that yield the 1, -th largest growth, that is, 1, are , {[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] 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] ---------------------------------------------------------------------------- This ends this article, that took, 0.003, seconds to generate. ---------------------------------------------------------- ---------------------------------------- Generating functions and Growth rates for the Enumerating Sequences for Como\ sitions Avoiding Each of the Possible subsets of the set of compositions of, 6, into exactly, 1, parts By Shalosh B. Ekhad The subsets of the set of compositions of, 6, with exactly, 1, parts, that yield the 1, -th largest growth, that is, 1.9659482366454853372, are , {[6]} Theorem Number, 1, : 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, 0.003, seconds to generate. ---------------------------------------------------------- ---------------------------------------- Generating functions and Growth rates for the Enumerating Sequences for Como\ sitions Avoiding Each of the Possible subsets of the set of compositions of, 6, into exactly, 2, parts By Shalosh B. Ekhad The subsets of the set of compositions of, 6, with exactly, 2, parts, that yield the 1, -th largest growth, that is, 1.8446944862958596627, are , {[1, 5], [2, 4], [3, 3], [4, 2], [5, 1]} Theorem Number, 1, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, {[1, 5], [2, 4], [3, 3], [4, 2], [5, 1]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 12 7 6 4 2 \ n (x - x - 2 x + x - x + 1) (x + 1) (x - x + 1) ) a(n) x = - --------------------------------------------------- / 10 7 5 4 2 ----- (-1 + x) (x + x - 2 x - x - x - x + 1) n = 0 and in Maple format -(x^12-x^7-2*x^6+x^4-x+1)*(x+1)*(x^2-x+1)/(-1+x)/(x^10+x^7-2*x^5-x^4-x^2-x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 27, 49, 90, 166, 306, 562, 1035, 1909, 3522, 6497, 11983, 22102, 40771, 75211, 138742, 255935, 472118, 870912, 1606568, 2963629, 5466989, 10084920, 18603593, 34317947, 63306131] The limit of a(n+1)/a(n) as n goes to infinity is 1.84469448630 a(n) is asymptotic to .666365137719*1.84469448630^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 2, parts, that yield the 2, -th largest growth, that is, 1.8600730504341370742, are , {[1, 5], [2, 4], [3, 3], [5, 1]}, {[1, 5], [3, 3], [4, 2], [5, 1]} Theorem Number, 2, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, {[1, 5], [2, 4], [3, 3], [5, 1]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 7 6 4 2 \ n (2 x + x - x + x - 1) (x + 1) (x - x + 1) ) a(n) x = - --------------------------------------------- / 5 4 2 ----- (-1 + x) (2 x + x + x + x - 1) n = 0 and in Maple format -(2*x^7+x^6-x^4+x-1)*(x+1)*(x^2-x+1)/(-1+x)/(2*x^5+x^4+x^2+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 28, 51, 94, 175, 325, 603, 1120, 2082, 3873, 7204, 13399, 24921, 46353, 86220, 160376, 298311, 554878, 1032111, 1919801, 3570971, 6642268, 12355102, 22981389, 42747060, 79512655] The limit of a(n+1)/a(n) as n goes to infinity is 1.86007305043 a(n) is asymptotic to .652433481979*1.86007305043^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 2, parts, that yield the 3, -th largest growth, that is, 1.8679418846712806945, are , {[1, 5], [2, 4], [4, 2], [5, 1]} Theorem Number, 3, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, {[1, 5], [2, 4], [4, 2], [5, 1]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 13 12 10 7 6 4 \ n x + x - x - x - 2 x + x - x + 1 ) a(n) x = - ----------------------------------------- / 8 7 5 3 2 ----- (-1 + x) (x + x - x - x - x - x + 1) n = 0 and in Maple format -(x^13+x^12-x^10-x^7-2*x^6+x^4-x+1)/(-1+x)/(x^8+x^7-x^5-x^3-x^2-x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 28, 51, 95, 177, 330, 615, 1147, 2142, 4001, 7473, 13958, 26071, 48698, 90965, 169917, 317394, 592872, 1107449, 2068650, 3864118, 7217947, 13482704, 25184906, 47043940, 87875346] The limit of a(n+1)/a(n) as n goes to infinity is 1.86794188467 a(n) is asymptotic to .635281535027*1.86794188467^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 2, parts, that yield the 4, -th largest growth, that is, 1.8788717116981317107, are , {[2, 4], [3, 3], [4, 2]} Theorem Number, 4, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, {[2, 4], [3, 3], [4, 2]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 9 6 3 \ n x + x - x + x - 1 ) a(n) x = - ------------------------------------------ / 2 8 7 4 2 ----- (x - x + 1) (x + x + x - 2 x - x + 1) n = 0 and in Maple format -(x^9+x^6-x^3+x-1)/(x^2-x+1)/(x^8+x^7+x^4-2*x^2-x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 29, 54, 101, 190, 358, 673, 1264, 2374, 4460, 8380, 15746, 29585, 55586, 104438, 196225, 368682, 692707, 1301508, 2445366, 4594528, 8632528, 16219413, 30474197, 57257107, 107578758] The limit of a(n+1)/a(n) as n goes to infinity is 1.87887171170 a(n) is asymptotic to .652848921262*1.87887171170^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 2, parts, that yield the 5, -th largest growth, that is, 1.8825804521685165608, are , {[1, 5], [3, 3], [5, 1]} Theorem Number, 5, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, {[1, 5], [3, 3], [5, 1]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 11 10 9 7 6 3 \ n x + 2 x + x + x + x - x + x - 1 ) a(n) x = - --------------------------------------- / 6 5 4 2 ----- (-1 + x) (x + 2 x + x + x + x - 1) n = 0 and in Maple format -(x^11+2*x^10+x^9+x^7+x^6-x^3+x-1)/(-1+x)/(x^6+2*x^5+x^4+x^2+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 29, 53, 99, 186, 350, 658, 1237, 2327, 4380, 8246, 15524, 29224, 55014, 103566, 194971, 367050, 691002, 1300865, 2448979, 4610397, 8679444, 16339755, 30760905, 57909875, 109019992] The limit of a(n+1)/a(n) as n goes to infinity is 1.88258045217 a(n) is asymptotic to .623590941750*1.88258045217^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 2, parts, that yield the 6, -th largest growth, that is, 1.8885188454844147017, are , {[1, 5], [2, 4], [5, 1]}, {[1, 5], [4, 2], [5, 1]} Theorem Number, 6, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, {[1, 5], [2, 4], [5, 1]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 10 8 7 6 4 \ n x + x + x + x - x + x - 1 ) a(n) x = - ------------------------------- / 5 3 2 ----- (-1 + x) (x + x + x + x - 1) n = 0 and in Maple format -(x^10+x^8+x^7+x^6-x^4+x-1)/(-1+x)/(x^5+x^3+x^2+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 29, 54, 101, 190, 358, 675, 1274, 2405, 4541, 8575, 16193, 30580, 57750, 109061, 205963, 388964, 734565, 1387239, 2619826, 4947590, 9343616, 17645594, 33324036, 62933069, 118850286] The limit of a(n+1)/a(n) as n goes to infinity is 1.88851884548 a(n) is asymptotic to .618529670710*1.88851884548^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 2, parts, that yield the 7, -th largest growth, that is, 1.8981723275022933773, are , {[2, 4], [4, 2]} Theorem Number, 7, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, {[2, 4], [4, 2]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 7 6 4 \ n x + x - x + x - 1 ) a(n) x = - --------------------------- / 8 7 5 4 ----- x + x - x + x - 2 x + 1 n = 0 and in Maple format -(x^7+x^6-x^4+x-1)/(x^8+x^7-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, 30, 56, 106, 201, 382, 726, 1378, 2615, 4963, 9420, 17881, 33942, 64428, 122295, 232136, 440633, 836397, 1587626, 3013588, 5720309, 10858131, 20610602, 39122473, 74261195, 140960545] The limit of a(n+1)/a(n) as n goes to infinity is 1.89817232750 a(n) is asymptotic to .629547396142*1.89817232750^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 2, parts, that yield the 8, -th largest growth, that is, 1.9051661677540189096, are , {[2, 4], [3, 3]}, {[3, 3], [4, 2]} Theorem Number, 8, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, {[2, 4], [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 = -------------------------------- / 2 4 2 ----- (x - x + 1) (x - 2 x - x + 1) n = 0 and in Maple format (x^3-x+1)/(x^2-x+1)/(x^4-2*x^2-x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 30, 57, 108, 206, 393, 749, 1427, 2718, 5178, 9865, 18795, 35808, 68220, 129970, 247614, 471746, 898755, 1712278, 3262174, 6214983, 11840575, 22558263, 42977240, 81878784, 155992689] The limit of a(n+1)/a(n) as n goes to infinity is 1.90516616775 a(n) is asymptotic to .623904940296*1.90516616775^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 2, parts, that yield the 9, -th largest growth, that is, 1.9275619754829253043, are , {[1, 5], [5, 1]} Theorem Number, 9, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, {[1, 5], [5, 1]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 9 8 7 6 5 \ n x + x + x + x - x + x - 1 ) a(n) x = - ------------------------------- / 4 3 2 ----- (-1 + x) (x + x + x + x - 1) n = 0 and in Maple format -(x^9+x^8+x^7+x^6-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, 30, 57, 109, 209, 402, 774, 1491, 2873, 5537, 10672, 20570, 39649, 76425, 147313, 283954, 547338, 1055027, 2033629, 3919945, 7555936, 14564534, 28074041, 54114453, 104308961, 201061986] 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 ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 2, parts, that yield the 10, -th largest growth, that is, 1.9331849818995204468, are , {[2, 4]}, {[4, 2]} Theorem Number, 10, : 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 subsets of the set of compositions of, 6, with exactly, 2, parts, that yield the 11, -th largest growth, that is, 1.9417130342786384772, are , {[3, 3]} Theorem Number, 11, : 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 ---------------------------------------------------------------------------- This ends this article, that took, 0.290, seconds to generate. ---------------------------------------------------------- ---------------------------------------- Generating functions and Growth rates for the Enumerating Sequences for Como\ sitions Avoiding Each of the Possible subsets of the set of compositions of, 6, into exactly, 3, parts By Shalosh B. Ekhad The subsets of the set of compositions of, 6, with exactly, 3, parts, that yield the 1, -th largest growth, that is, 1.6221667292817237350, are , {[1, 1, 4], [1, 2, 3], [1, 4, 1], [2, 1, 3], [2, 2, 2], [3, 1, 2], [3, 2, 1], [4, 1, 1] } Theorem Number, 1, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, {[1, 1, 4], [1, 2, 3], [1, 4, 1], [2, 1, 3], [2, 2, 2], [3, 1, 2], [3, 2, 1], [4, 1, 1]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 19 18 17 16 15 14 13 12 ) a(n) x = - (x - 2 x + 4 x - 3 x + 2 x - 3 x + x - 4 x / ----- n = 0 11 10 9 8 7 6 5 4 3 2 - x + 3 x - x + x + x + 3 x + x - 2 x + x - 2 x + 2 x - 1) / 10 8 5 3 2 / ((x + x - 2 x - x - x + 1) (-1 + x) ) / and in Maple format -(x^19-2*x^18+4*x^17-3*x^16+2*x^15-3*x^14+x^13-4*x^12-x^11+3*x^10-x^9+x^8+x^7+3 *x^6+x^5-2*x^4+x^3-2*x^2+2*x-1)/(x^10+x^8-2*x^5-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, 24, 35, 53, 84, 134, 211, 335, 533, 859, 1389, 2247, 3635, 5881, 9528, 15448, 25054, 40632, 65893, 106869, 173344, 281187, 456126, 739895, 1200205, 1946905] The limit of a(n+1)/a(n) as n goes to infinity is 1.62216672928 a(n) is asymptotic to .969338238307*1.62216672928^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 3, parts, that yield the 2, -th largest growth, that is, 1.6439001594857552252, are , { [1, 1, 4], [1, 2, 3], [1, 3, 2], [1, 4, 1], [2, 1, 3], [2, 2, 2], [4, 1, 1] }, { [1, 1, 4], [1, 4, 1], [2, 2, 2], [2, 3, 1], [3, 1, 2], [3, 2, 1], [4, 1, 1] } Theorem Number, 2, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, { [1, 1, 4], [1, 2, 3], [1, 3, 2], [1, 4, 1], [2, 1, 3], [2, 2, 2], [4, 1, 1] } 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 10 9 8 7 6 5 4 / (2 x + 2 x + x - 3 x - 5 x - 3 x + x + 2 x - x + 1) / ( / 5 3 2 (2 x + x + x - 1) (-1 + x) ) and in Maple format -(x^2-x+1)*(2*x^11+2*x^10+x^9-3*x^8-5*x^7-3*x^6+x^5+2*x^4-x+1)/(2*x^5+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, 25, 38, 60, 95, 153, 246, 395, 643, 1051, 1721, 2822, 4626, 7593, 12474, 20496, 33684, 55358, 90985, 149559, 245848, 404137, 664345, 1092093, 1795275, 2951240] The limit of a(n+1)/a(n) as n goes to infinity is 1.64390015949 a(n) is asymptotic to .985682559739*1.64390015949^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 3, parts, that yield the 3, -th largest growth, that is, 1.6479148203514353171, are , {[1, 1, 4], [1, 3, 2], [2, 1, 3], [2, 2, 2], [3, 1, 2], [4, 1, 1]}, {[1, 1, 4], [2, 1, 3], [2, 2, 2], [2, 3, 1], [3, 1, 2], [4, 1, 1]} Theorem Number, 3, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, {[1, 1, 4], [1, 3, 2], [2, 1, 3], [2, 2, 2], [3, 1, 2], [4, 1, 1]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 19 18 17 16 15 14 13 12 ) a(n) x = - (x + x + x - x - 2 x - 4 x - 2 x - x / ----- n = 0 11 10 9 7 6 5 4 3 2 / + 4 x - 2 x + 4 x + 3 x + x + x - 2 x + x - 2 x + 2 x - 1) / / 10 8 7 5 3 2 ((x + x - x - 2 x - x - x + 1) (-1 + x) ) and in Maple format -(x^19+x^18+x^17-x^16-2*x^15-4*x^14-2*x^13-x^12+4*x^11-2*x^10+4*x^9+3*x^7+x^6+x ^5-2*x^4+x^3-2*x^2+2*x-1)/(x^10+x^8-x^7-2*x^5-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, 26, 40, 62, 96, 155, 249, 400, 648, 1056, 1733, 2849, 4686, 7708, 12683, 20885, 34405, 56690, 93410, 153911, 253607, 417898, 688648, 1134828, 1870089, 3081723] The limit of a(n+1)/a(n) as n goes to infinity is 1.64791482035 a(n) is asymptotic to .956631396571*1.64791482035^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 3, parts, that yield the 4, -th largest growth, that is, 1.6582126420503060110, are , { [1, 1, 4], [1, 2, 3], [1, 3, 2], [1, 4, 1], [2, 1, 3], [3, 1, 2], [3, 2, 1] }, { [1, 2, 3], [1, 4, 1], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1], [4, 1, 1] } Theorem Number, 4, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, { [1, 1, 4], [1, 2, 3], [1, 3, 2], [1, 4, 1], [2, 1, 3], [3, 1, 2], [3, 2, 1] } We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 13 12 11 10 9 8 7 6 ) a(n) x = - (x + 1) (x + x - x - 2 x - x + x - 2 x - 2 x / ----- n = 0 5 4 3 2 / - x + 2 x - x + 2 x - 2 x + 1) / ( / 7 6 5 4 3 2 2 (x + 2 x + 2 x + x + x + x - 1) (-1 + x) ) and in Maple format -(x+1)*(x^13+x^12-x^11-2*x^10-x^9+x^8-2*x^7-2*x^6-x^5+2*x^4-x^3+2*x^2-2*x+1)/(x ^7+2*x^6+2*x^5+x^4+x^3+x^2-1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 25, 38, 60, 98, 160, 260, 423, 693, 1143, 1890, 3127, 5174, 8566, 14192, 23524, 38999, 64656, 107196, 177735, 294707, 488675, 810314, 1343654, 2228040, 3694541] The limit of a(n+1)/a(n) as n goes to infinity is 1.65821264205 a(n) is asymptotic to .951372232410*1.65821264205^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 3, parts, that yield the 5, -th largest growth, that is, 1.6663019373129334689, are , {[1, 1, 4], [2, 2, 2], [3, 1, 2], [3, 2, 1]}, {[1, 2, 3], [2, 1, 3], [2, 2, 2], [4, 1, 1]} Theorem Number, 5, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, {[1, 1, 4], [2, 2, 2], [3, 1, 2], [3, 2, 1]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 2 ) a(n) x = - (x + 1) / ----- n = 0 14 12 11 10 9 8 7 5 4 3 2 (x - 2 x + x + x - x - x - x - 2 x + x + x + x - 2 x + 1) / 2 5 4 3 2 2 / ((x - x + 1) (x + x + 2 x + x - 1) (-1 + x) ) / and in Maple format -(x^2+1)*(x^14-2*x^12+x^11+x^10-x^9-x^8-x^7-2*x^5+x^4+x^3+x^2-2*x+1)/(x^2-x+1)/ (x^5+x^4+2*x^3+x^2-1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 28, 46, 74, 119, 196, 325, 538, 892, 1478, 2454, 4083, 6797, 11320, 18853, 31402, 52314, 87159, 145224, 241977, 403192, 671825, 1119446, 1865320, 3108174, 5179140] The limit of a(n+1)/a(n) as n goes to infinity is 1.66630193731 a(n) is asymptotic to 1.15250905943*1.66630193731^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 3, parts, that yield the 6, -th largest growth, that is, 1.6775217478410968373, are , { [1, 1, 4], [1, 2, 3], [1, 3, 2], [1, 4, 1], [2, 3, 1], [3, 1, 2], [4, 1, 1] }, { [1, 1, 4], [1, 3, 2], [1, 4, 1], [2, 1, 3], [2, 3, 1], [3, 2, 1], [4, 1, 1] } Theorem Number, 6, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, { [1, 1, 4], [1, 2, 3], [1, 3, 2], [1, 4, 1], [2, 3, 1], [3, 1, 2], [4, 1, 1] } 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 12 11 10 9 8 7 6 5 4 3 2 x + 3 x - 3 x - 8 x - x + 7 x + 5 x - x - 2 x - x + x + x - 1 / 7 5 2 2 ) / ((x - x - x - x + 1) (-1 + x) ) / and in Maple format -(x^2-x+1)*(x^12+3*x^11-3*x^10-8*x^9-x^8+7*x^7+5*x^6-x^5-2*x^4-x^3+x^2+x-1)/(x^ 7-x^5-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, 25, 38, 61, 99, 161, 263, 434, 721, 1203, 2010, 3362, 5629, 9432, 15812, 26514, 44465, 74577, 125090, 209827, 351975, 590430, 990442, 1661470, 2787134, 4675460] The limit of a(n+1)/a(n) as n goes to infinity is 1.67752174784 a(n) is asymptotic to .850700702888*1.67752174784^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 3, parts, that yield the 7, -th largest growth, that is, 1.6790045840971653763, are , {[1, 1, 4], [1, 2, 3], [1, 4, 1], [2, 1, 3], [2, 3, 1], [3, 1, 2]}, {[1, 3, 2], [1, 4, 1], [2, 1, 3], [3, 1, 2], [3, 2, 1], [4, 1, 1]} Theorem Number, 7, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, {[1, 1, 4], [1, 2, 3], [1, 4, 1], [2, 1, 3], [2, 3, 1], [3, 1, 2]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 13 12 11 10 9 8 7 6 ) a(n) x = - (x + 1) (x - x - x - x - 2 x + 2 x - 3 x - x / ----- n = 0 5 4 3 2 / - x + 2 x - x + 2 x - 2 x + 1) / ( / 8 7 6 5 4 3 2 2 (x + 2 x + 2 x + 2 x + x + x + x - 1) (-1 + x) ) and in Maple format -(x+1)*(x^13-x^12-x^11-x^10-2*x^9+2*x^8-3*x^7-x^6-x^5+2*x^4-x^3+2*x^2-2*x+1)/(x ^8+2*x^7+2*x^6+2*x^5+x^4+x^3+x^2-1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 26, 41, 66, 109, 181, 301, 500, 832, 1389, 2325, 3898, 6538, 10968, 18403, 30885, 51844, 87036, 146124, 245330, 411893, 691551, 1161099, 1949476, 3273166, 5495645] The limit of a(n+1)/a(n) as n goes to infinity is 1.67900458410 a(n) is asymptotic to .973777177703*1.67900458410^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 3, parts, that yield the 8, -th largest growth, that is, 1.6797202028955816256, are , {[1, 1, 4], [2, 1, 3], [2, 2, 2], [3, 1, 2], [4, 1, 1]} Theorem Number, 8, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, {[1, 1, 4], [2, 1, 3], [2, 2, 2], [3, 1, 2], [4, 1, 1]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 20 19 17 16 15 14 12 ) a(n) x = - (2 x + 3 x - 2 x - 6 x - 4 x - 3 x + 3 x / ----- n = 0 11 10 9 8 7 5 4 3 2 / + 3 x - x + 4 x + 2 x + 3 x + x - 2 x + x - 2 x + 2 x - 1) / / 11 10 7 5 3 2 ((x + x - 2 x - 2 x - x - x + 1) (-1 + x) ) and in Maple format -(2*x^20+3*x^19-2*x^17-6*x^16-4*x^15-3*x^14+3*x^12+3*x^11-x^10+4*x^9+2*x^8+3*x^ 7+x^5-2*x^4+x^3-2*x^2+2*x-1)/(x^11+x^10-2*x^7-2*x^5-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, 27, 44, 69, 107, 174, 285, 470, 778, 1291, 2155, 3606, 6048, 10152, 17039, 28604, 48020, 80636, 135431, 227476, 382095, 641797, 1078010, 1810715, 3041450, 5108771] The limit of a(n+1)/a(n) as n goes to infinity is 1.67972020290 a(n) is asymptotic to .893731389973*1.67972020290^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 3, parts, that yield the 9, -th largest growth, that is, 1.6800446669757394336, are , {[1, 1, 4], [1, 3, 2], [2, 1, 3], [2, 2, 2], [4, 1, 1]}, {[1, 1, 4], [2, 2, 2], [2, 3, 1], [3, 1, 2], [4, 1, 1]} Theorem Number, 9, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, {[1, 1, 4], [1, 3, 2], [2, 1, 3], [2, 2, 2], [4, 1, 1]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 13 12 11 10 9 8 6 5 4 ) a(n) x = - (x + 3 x + x - 3 x - 3 x + x - 2 x - 2 x + x / ----- n = 0 3 2 2 / 6 5 3 2 + x + x - 2 x + 1) (x + 1) / ((x + 2 x + x + x - 1) (-1 + x) ) / and in Maple format -(x^13+3*x^12+x^11-3*x^10-3*x^9+x^8-2*x^6-2*x^5+x^4+x^3+x^2-2*x+1)*(x^2+1)/(x^6 +2*x^5+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, 27, 42, 67, 107, 174, 286, 469, 776, 1295, 2167, 3633, 6092, 10216, 17147, 28796, 48369, 81253, 136489, 229280, 385180, 647107, 1087162, 1826469, 3068517, 5155207] The limit of a(n+1)/a(n) as n goes to infinity is 1.68004466698 a(n) is asymptotic to .896641901858*1.68004466698^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 3, parts, that yield the 10, -th largest growth, that is, 1.6851371101652493043, are , {[1, 2, 3], [1, 3, 2], [1, 4, 1], [2, 2, 2], [3, 2, 1]}, {[1, 2, 3], [1, 4, 1], [2, 2, 2], [2, 3, 1], [3, 2, 1]} Theorem Number, 10, : Let a(n) be the number of compositions of n avoiding as\ subcompositions, the members of the set, {[1, 2, 3], [1, 3, 2], [1, 4, 1], [2, 2, 2], [3, 2, 1]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 2 10 8 7 6 5 4 \ n (x - x + 1) (x - 2 x - 3 x - x + 2 x + x - x + 1) ) a(n) x = - --------------------------------------------------------- / 5 4 3 2 ----- (x + x + x + x - 1) (-1 + x) n = 0 and in Maple format -(x^2-x+1)*(x^10-2*x^8-3*x^7-x^6+2*x^5+x^4-x+1)/(x^5+x^4+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, 27, 45, 77, 131, 220, 367, 616, 1038, 1748, 2941, 4950, 8338, 14049, 23670, 39879, 67194, 113227, 190799, 321514, 541784, 912972, 1538478, 2592539, 4368771, 7361965] The limit of a(n+1)/a(n) as n goes to infinity is 1.68513711017 a(n) is asymptotic to 1.16932607374*1.68513711017^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 3, parts, that yield the 11, -th largest growth, that is, 1.6895452189413031152, are , { [1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 2, 2], [2, 3, 1], [3, 1, 2], [3, 2, 1] } Theorem Number, 11, : Let a(n) be the number of compositions of n avoiding as\ subcompositions, the members of the set, { [1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 2, 2], [2, 3, 1], [3, 1, 2], [3, 2, 1] } We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 16 15 14 13 12 11 10 9 ) a(n) x = - (x - 2 x + x - 2 x + x - 3 x + 4 x - 3 x / ----- n = 0 8 7 6 5 4 3 2 / + 3 x - 4 x + 2 x + 3 x - 3 x + 3 x - 4 x + 3 x - 1) / ( / 10 8 6 5 4 3 2 2 (x + x + x - 2 x + x - x + x - 2 x + 1) (-1 + x) ) and in Maple format -(x^16-2*x^15+x^14-2*x^13+x^12-3*x^11+4*x^10-3*x^9+3*x^8-4*x^7+2*x^6+3*x^5-3*x^ 4+3*x^3-4*x^2+3*x-1)/(x^10+x^8+x^6-2*x^5+x^4-x^3+x^2-2*x+1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 25, 40, 65, 108, 182, 305, 510, 854, 1434, 2416, 4078, 6886, 11627, 19632, 33153, 55999, 94604, 159833, 270039, 456229, 770796, 1302269, 2200224, 3717371, 6280665] The limit of a(n+1)/a(n) as n goes to infinity is 1.68954521894 a(n) is asymptotic to .922378056075*1.68954521894^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 3, parts, that yield the 12, -th largest growth, that is, 1.7049027760416460701, are , {[1, 2, 3], [1, 4, 1], [2, 1, 3]}, {[1, 4, 1], [3, 1, 2], [3, 2, 1]} Theorem Number, 12, : Let a(n) be the number of compositions of n avoiding as\ subcompositions, the members of the set, {[1, 2, 3], [1, 4, 1], [2, 1, 3]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 2 6 5 4 3 2 \ n (x + 1) (x + x - 2 x - 2 x + x + x - 1) ) a(n) x = - --------------------------------------------- / 5 2 ----- (-1 + x) (x + x + x - 1) n = 0 and in Maple format -(x^2+1)*(x^6+x^5-2*x^4-2*x^3+x^2+x-1)/(-1+x)/(x^5+x^2+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 29, 50, 85, 145, 248, 424, 724, 1235, 2106, 3591, 6123, 10440, 17800, 30348, 51741, 88214, 150397, 256413, 437160, 745316, 1270692, 2166407, 3693514, 6297083, 10735915] The limit of a(n+1)/a(n) as n goes to infinity is 1.70490277604 a(n) is asymptotic to 1.20185082919*1.70490277604^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 3, parts, that yield the 13, -th largest growth, that is, 1.7079480740587477835, are , {[1, 1, 4], [1, 2, 3], [3, 1, 2], [4, 1, 1]}, {[1, 1, 4], [2, 1, 3], [3, 2, 1], [4, 1, 1]} Theorem Number, 13, : Let a(n) be the number of compositions of n avoiding as\ subcompositions, the members of the set, {[1, 1, 4], [1, 2, 3], [3, 1, 2], [4, 1, 1]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 17 16 14 13 12 11 10 9 ) a(n) x = - (2 x + 2 x - 5 x - 4 x + x + x - 3 x + 2 x / ----- n = 0 8 7 6 3 2 / + 3 x + x + 2 x - x - x + 2 x - 1) / ( / 8 7 6 5 2 2 (x + x - x - x - x - x + 1) (-1 + x) ) and in Maple format -(2*x^17+2*x^16-5*x^14-4*x^13+x^12+x^11-3*x^10+2*x^9+3*x^8+x^7+2*x^6-x^3-x^2+2* x-1)/(x^8+x^7-x^6-x^5-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, 28, 46, 73, 116, 191, 318, 532, 896, 1517, 2578, 4393, 7491, 12779, 21808, 37228, 63567, 108556, 185393, 316622, 540750, 923548, 1577352, 2694020, 4601229, 7858636] The limit of a(n+1)/a(n) as n goes to infinity is 1.70794807406 a(n) is asymptotic to .833882899690*1.70794807406^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 3, parts, that yield the 14, -th largest growth, that is, 1.7090250246706445950, are , {[1, 1, 4], [2, 1, 3], [2, 2, 2], [4, 1, 1]}, {[1, 1, 4], [2, 2, 2], [3, 1, 2], [4, 1, 1]} Theorem Number, 14, : Let a(n) be the number of compositions of n avoiding as\ subcompositions, the members of the set, {[1, 1, 4], [2, 1, 3], [2, 2, 2], [4, 1, 1]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 17 16 15 14 13 12 11 ) a(n) x = - (2 x + 3 x + 2 x + 3 x - x - 3 x - 3 x / ----- n = 0 10 9 8 7 5 4 3 2 / - 2 x - 3 x - 3 x - 2 x - x + 2 x - x + 2 x - 2 x + 1) / ( / 8 7 6 5 3 2 (x + x + x + 2 x + x + x - 1) (-1 + x) ) and in Maple format -(2*x^17+3*x^16+2*x^15+3*x^14-x^13-3*x^12-3*x^11-2*x^10-3*x^9-3*x^8-2*x^7-x^5+2 *x^4-x^3+2*x^2-2*x+1)/(x^8+x^7+x^6+2*x^5+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, 28, 46, 74, 118, 194, 324, 543, 916, 1552, 2638, 4497, 7674, 13102, 22377, 38223, 65304, 111589, 190692, 325884, 556927, 951778, 1626587, 2779851, 4750813, 8119244] The limit of a(n+1)/a(n) as n goes to infinity is 1.70902502467 a(n) is asymptotic to .845397851407*1.70902502467^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 3, parts, that yield the 15, -th largest growth, that is, 1.7124772620832746605, are , {[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]} Theorem Number, 15, : Let a(n) be the number of compositions of n avoiding as\ subcompositions, the members of the set, {[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 14 13 12 11 10 9 8 7 ) a(n) x = - (x - x - 2 x - x + 2 x + 2 x - 2 x - 4 x / ----- n = 0 6 4 2 / 8 7 5 3 2 + 4 x + x - 3 x + 3 x - 1) / ((x + x - x + x - 2 x + 1) (-1 + x) / ) and in Maple format -(x^14-x^13-2*x^12-x^11+2*x^10+2*x^9-2*x^8-4*x^7+4*x^6+x^4-3*x^2+3*x-1)/(x^8+x^ 7-x^5+x^3-2*x+1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 26, 42, 69, 116, 197, 335, 569, 967, 1647, 2812, 4809, 8230, 14087, 24113, 41278, 70671, 121008, 207213, 354839, 607642, 1040554, 1781899, 3051434, 5225489, 8948516] The limit of a(n+1)/a(n) as n goes to infinity is 1.71247726208 a(n) is asymptotic to .877010781717*1.71247726208^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 3, parts, that yield the 16, -th largest growth, that is, 1.7142853291425735181, are , {[1, 1, 4], [1, 4, 1], [2, 1, 3], [3, 1, 2], [4, 1, 1]} Theorem Number, 16, : Let a(n) be the number of compositions of n avoiding as\ subcompositions, the members of the set, {[1, 1, 4], [1, 4, 1], [2, 1, 3], [3, 1, 2], [4, 1, 1]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 26 24 23 22 21 20 18 17 16 ) a(n) x = - (x - x - x - x + x + x - 3 x - 4 x + 4 x / ----- n = 0 15 14 13 12 11 10 9 8 7 6 + 5 x + 3 x + 2 x - 3 x - 2 x + 2 x + 2 x - x - 3 x - 2 x 3 2 / 6 4 2 12 8 7 4 + x + x - 2 x + 1) / ((x - x - x - x + 1) (x + x + x - x - 1) / 2 (-1 + x) ) and in Maple format -(x^26-x^24-x^23-x^22+x^21+x^20-3*x^18-4*x^17+4*x^16+5*x^15+3*x^14+2*x^13-3*x^ 12-2*x^11+2*x^10+2*x^9-x^8-3*x^7-2*x^6+x^3+x^2-2*x+1)/(x^6-x^4-x^2-x+1)/(x^12+x ^8+x^7-x^4-1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 27, 44, 70, 113, 190, 321, 546, 930, 1583, 2702, 4617, 7903, 13542, 23209, 39783, 68184, 116863, 200309, 343359, 588607, 1009041, 1729791, 2965351, 5083418, 8714374] The limit of a(n+1)/a(n) as n goes to infinity is 1.71428532914 a(n) is asymptotic to .827448287658*1.71428532914^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 3, parts, that yield the 17, -th largest growth, that is, 1.7178301450652747076, are , {[1, 3, 2], [1, 4, 1], [2, 2, 2]}, {[1, 4, 1], [2, 2, 2], [2, 3, 1]} Theorem Number, 17, : Let a(n) be the number of compositions of n avoiding as\ subcompositions, the members of the set, {[1, 3, 2], [1, 4, 1], [2, 2, 2]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 2 7 6 3 2 \ n (x + 1) (x - x + x + x - 2 x + 1) ) a(n) x = - -------------------------------------- / 6 5 4 3 2 ----- (x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format -(x^2+1)*(x^7-x^6+x^3+x^2-2*x+1)/(x^6+x^5+x^4+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, 29, 51, 89, 156, 272, 471, 812, 1398, 2406, 4139, 7116, 12229, 21012, 36101, 62023, 106553, 183047, 314450, 540179, 927945, 1594062, 2738337, 4704006, 8080692, 13881267] The limit of a(n+1)/a(n) as n goes to infinity is 1.71783014507 a(n) is asymptotic to 1.23886226563*1.71783014507^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 3, parts, that yield the 18, -th largest growth, that is, 1.7220838057390422450, are , {[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 2, 2], [2, 3, 1], [3, 2, 1]}, {[1, 2, 3], [1, 3, 2], [2, 2, 2], [2, 3, 1], [3, 1, 2], [3, 2, 1]} Theorem Number, 18, : Let a(n) be the number of compositions of n avoiding as\ subcompositions, the members of the set, {[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 2, 2], [2, 3, 1], [3, 2, 1]} 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 ) a(n) x = - (x - 4 x + 5 x - 3 x + 3 x - 3 x + x + 3 x / ----- n = 0 4 3 2 / 2 4 3 2 - 3 x + 3 x - 4 x + 3 x - 1) / ((x - x + 1) (x - x - x - x + 1) / 2 (-1 + x) ) and in Maple format -(x^12-4*x^11+5*x^10-3*x^9+3*x^8-3*x^7+x^6+3*x^5-3*x^4+3*x^3-4*x^2+3*x-1)/(x^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, 26, 43, 72, 122, 208, 354, 604, 1034, 1774, 3049, 5244, 9022, 15527, 26728, 46017, 79235, 136438, 234945, 404581, 696707, 1199773, 2066095, 3557973, 6127111, 10551380] The limit of a(n+1)/a(n) as n goes to infinity is 1.72208380574 a(n) is asymptotic to .874338607439*1.72208380574^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 3, parts, that yield the 19, -th largest growth, that is, 1.7302248027051109170, are , {[1, 3, 2], [2, 1, 3], [2, 2, 2], [3, 1, 2]}, {[2, 1, 3], [2, 2, 2], [2, 3, 1], [3, 1, 2]} Theorem Number, 19, : Let a(n) be the number of compositions of n avoiding as\ subcompositions, the members of the set, {[1, 3, 2], [2, 1, 3], [2, 2, 2], [3, 1, 2]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 9 6 5 4 2 \ n x - x - x - 2 x - x + x - 1 ) a(n) x = - -------------------------------------------------- / 9 8 7 6 5 3 ----- (-1 + x) (x + x + 2 x + x + 2 x + x + x - 1) n = 0 and in Maple format -(x^9-x^6-x^5-2*x^4-x^2+x-1)/(-1+x)/(x^9+x^8+2*x^7+x^6+2*x^5+x^3+x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 28, 48, 82, 140, 243, 423, 735, 1274, 2203, 3808, 6585, 11393, 19717, 34122, 59044, 102158, 176748, 305804, 529106, 915480, 1584002, 2740693, 4742015, 8204735, 14196012] The limit of a(n+1)/a(n) as n goes to infinity is 1.73022480271 a(n) is asymptotic to 1.02115226399*1.73022480271^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 3, parts, that yield the 20, -th largest growth, that is, 1.7437001659048548749, are , {[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 2, 1]}, {[1, 2, 3], [1, 3, 2], [2, 3, 1], [3, 1, 2], [3, 2, 1]} Theorem Number, 20, : Let a(n) be the number of compositions of n avoiding as\ subcompositions, the members of the set, {[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 2, 1]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 12 11 10 8 6 3 2 \ n x - x - 2 x + x - 3 x + x + x - 2 x + 1 ) a(n) x = - ------------------------------------------------- / 6 5 2 2 ----- (x + x + x + x - 1) (-1 + x) n = 0 and in Maple format -(x^12-x^11-2*x^10+x^8-3*x^6+x^3+x^2-2*x+1)/(x^6+x^5+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, 27, 45, 76, 130, 224, 387, 670, 1162, 2019, 3513, 6118, 10660, 18579, 32386, 56460, 98437, 171632, 299262, 521810, 909866, 1586518, 2766395, 4823746, 8411149, 14666504] The limit of a(n+1)/a(n) as n goes to infinity is 1.74370016590 a(n) is asymptotic to .835933152793*1.74370016590^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 3, parts, that yield the 21, -th largest growth, that is, 1.7484141350538300125, are , {[1, 2, 3], [2, 1, 3], [3, 1, 2]}, {[2, 1, 3], [3, 1, 2], [3, 2, 1]} Theorem Number, 21, : Let a(n) be the number of compositions of n avoiding as\ subcompositions, the members of the set, {[1, 2, 3], [2, 1, 3], [3, 1, 2]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 10 8 7 6 3 2 \ n x - x - x - x + x + x - 2 x + 1 ) a(n) x = - -------------------------------------- / 9 8 5 3 ----- (-1 + x) (x + x - x + x - 2 x + 1) n = 0 and in Maple format -(x^10-x^8-x^7-x^6+x^3+x^2-2*x+1)/(-1+x)/(x^9+x^8-x^5+x^3-2*x+1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 29, 51, 87, 149, 259, 453, 795, 1393, 2436, 4255, 7431, 12984, 22697, 39686, 69394, 121333, 212134, 370879, 648424, 1133692, 1982156, 3465638, 6059382, 10594304, 18523194] The limit of a(n+1)/a(n) as n goes to infinity is 1.74841413505 a(n) is asymptotic to .973609572775*1.74841413505^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 3, parts, that yield the 22, -th largest growth, that is, 1.7548776662466927601, are , {[1, 1, 4], [1, 4, 1], [2, 1, 3], [4, 1, 1]}, {[1, 1, 4], [1, 4, 1], [3, 1, 2], [4, 1, 1]} Theorem Number, 22, : Let a(n) be the number of compositions of n avoiding as\ subcompositions, the members of the set, {[1, 1, 4], [1, 4, 1], [2, 1, 3], [4, 1, 1]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = / ----- n = 0 2 13 12 11 10 8 7 6 5 2 (x + 1) (x - x + x - x - x + x + x - 3 x + 3 x - 3 x + 1) - ----------------------------------------------------------------------- 3 2 4 2 (x - x + 2 x - 1) (x + 1) (-1 + x) and in Maple format -(x^2+1)*(x^13-x^12+x^11-x^10-x^8+x^7+x^6-3*x^5+3*x^2-3*x+1)/(x^3-x^2+2*x-1)/(x ^4+1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 28, 47, 78, 131, 224, 387, 672, 1172, 2048, 3584, 6279, 11007, 19304, 33864, 59414, 104251, 182933, 321009, 563315, 988531, 1734733, 3044226, 5342225, 9374932, 16451838] The limit of a(n+1)/a(n) as n goes to infinity is 1.75487766625 a(n) is asymptotic to .774119870096*1.75487766625^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 3, parts, that yield the 23, -th largest growth, that is, 1.7610793284505662547, are , {[1, 4, 1], [2, 2, 2]} Theorem Number, 23, : Let a(n) be the number of compositions of n avoiding as\ subcompositions, the members of the set, {[1, 4, 1], [2, 2, 2]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 4 3 2 6 5 3 \ n (x - x + x - x + 1) (x + x + x - x + 1) ) a(n) x = - --------------------------------------------- / 7 6 5 4 3 2 ----- (x + 2 x + x + x + x + x - 1) (-1 + x) n = 0 and in Maple format -(x^4-x^3+x^2-x+1)*(x^6+x^5+x^3-x+1)/(x^7+2*x^6+x^5+x^4+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, 30, 54, 96, 171, 305, 542, 959, 1692, 2982, 5255, 9261, 16318, 28745, 50627, 89161, 157024, 276541, 487024, 857700, 1510484, 2660084, 4684623, 8250005, 14528933, 25586622] The limit of a(n+1)/a(n) as n goes to infinity is 1.76107932845 a(n) is asymptotic to 1.08304254925*1.76107932845^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 3, parts, that yield the 24, -th largest growth, that is, 1.7764132269071608870, are , {[1, 3, 2], [2, 1, 3], [2, 2, 2]}, {[2, 2, 2], [2, 3, 1], [3, 1, 2]} Theorem Number, 24, : Let a(n) be the number of compositions of n avoiding as\ subcompositions, the members of the set, {[1, 3, 2], [2, 1, 3], [2, 2, 2]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 5 4 3 2 \ n x - 2 x + x - 2 x + 2 x - 1 ) a(n) x = - ---------------------------------------- / 5 4 3 2 ----- (-1 + x) (2 x - x + x - x + 2 x - 1) n = 0 and in Maple format -(x^5-2*x^4+x^3-2*x^2+2*x-1)/(-1+x)/(2*x^5-x^4+x^3-x^2+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 29, 51, 90, 159, 283, 505, 899, 1598, 2838, 5039, 8950, 15900, 28248, 50184, 89149, 158363, 281314, 499727, 887723, 1576969, 2801355, 4976366, 8840078, 15703623, 27896118] The limit of a(n+1)/a(n) as n goes to infinity is 1.77641322691 a(n) is asymptotic to .910382551314*1.77641322691^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 3, parts, that yield the 25, -th largest growth, that is, 1.7845989333686468028, are , {[1, 3, 2], [3, 1, 2]}, {[2, 1, 3], [2, 3, 1]} Theorem Number, 25, : Let a(n) be the number of compositions of n avoiding as\ subcompositions, the members of the set, {[1, 3, 2], [3, 1, 2]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 6 5 4 3 \ n x + x + x + x - x + 1 ) a(n) x = - ------------------------- / 5 3 ----- x - x + 2 x - 1 n = 0 and in Maple format -(x^6+x^5+x^4+x^3-x+1)/(x^5-x^3+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 30, 54, 96, 170, 302, 538, 960, 1714, 3060, 5462, 9748, 17396, 31044, 55400, 98866, 176436, 314868, 561914, 1002792, 1789582, 3193686, 5699448, 10171228, 18151562, 32393258] The limit of a(n+1)/a(n) as n goes to infinity is 1.78459893337 a(n) is asymptotic to .920949390152*1.78459893337^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 3, parts, that yield the 26, -th largest growth, that is, 1.7898182262908775163, are , {[2, 1, 3], [2, 2, 2], [3, 1, 2]} Theorem Number, 26, : Let a(n) be the number of compositions of n avoiding as\ subcompositions, the members of the set, {[2, 1, 3], [2, 2, 2], [3, 1, 2]} 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 + 2 x + x + 2 x + x - x + 1 ) a(n) x = - ---------------------------------------------- / 9 7 6 5 4 3 2 ----- x + 2 x - x + 2 x - x + x - x + 2 x - 1 n = 0 and in Maple format -(x^8+x^7+2*x^6+x^5+2*x^4+x^2-x+1)/(x^9+2*x^7-x^6+2*x^5-x^4+x^3-x^2+2*x-1) The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 29, 52, 92, 162, 288, 514, 921, 1652, 2961, 5304, 9494, 16989, 30400, 54401, 97362, 174261, 311905, 558270, 999218, 1788427, 3200952, 5729098, 10254011, 18352787, 32848146] The limit of a(n+1)/a(n) as n goes to infinity is 1.78981822629 a(n) is asymptotic to .855546509811*1.78981822629^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 3, parts, that yield the 27, -th largest growth, that is, 1.7927428383414338089, are , {[1, 2, 3], [3, 1, 2]}, {[2, 1, 3], [3, 2, 1]} Theorem Number, 27, : Let a(n) be the number of compositions of n avoiding as\ subcompositions, the members of the set, {[1, 2, 3], [3, 1, 2]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 13 12 10 9 8 6 3 2 \ n x + x - x + x + x + x - x - x + 2 x - 1 ) a(n) x = -------------------------------------------------- / 7 6 5 2 2 ----- (x + 2 x + x + x + x - 1) (-1 + x) n = 0 and in Maple format (x^13+x^12-x^10+x^9+x^8+x^6-x^3-x^2+2*x-1)/(x^7+2*x^6+x^5+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, 30, 54, 95, 167, 297, 532, 955, 1713, 3069, 5495, 9841, 17633, 31607, 56665, 101590, 182124, 326490, 585291, 1049253, 1881028, 3372204, 6045511, 10838058, 19429839, 34832664] The limit of a(n+1)/a(n) as n goes to infinity is 1.79274283834 a(n) is asymptotic to .863865960060*1.79274283834^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 3, parts, that yield the 28, -th largest growth, that is, 1.7958388326162244753, are , {[1, 2, 3], [1, 3, 2], [2, 2, 2], [2, 3, 1], [3, 2, 1]} Theorem Number, 28, : Let a(n) be the number of compositions of n avoiding as\ subcompositions, the members of the set, {[1, 2, 3], [1, 3, 2], [2, 2, 2], [2, 3, 1], [3, 2, 1]} 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 - 3 x + 2 x - x + 3 x - 3 x + 2 x + x - 2 x / ----- n = 0 3 2 / 6 5 3 2 2 + 3 x - 4 x + 3 x - 1) / ((x - x - x + x - 2 x + 1) (-1 + x) ) / and in Maple format -(x^12-3*x^11+2*x^10-x^9+3*x^8-3*x^7+2*x^6+x^5-2*x^4+3*x^3-4*x^2+3*x-1)/(x^6-x^ 5-x^3+x^2-2*x+1)/(-1+x)^2 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 27, 47, 83, 147, 260, 461, 822, 1471, 2636, 4726, 8477, 15213, 27311, 49037, 88051, 158111, 283927, 509874, 915639, 1644325, 2952924, 5302956, 9523237, 17102183, 30712746] The limit of a(n+1)/a(n) as n goes to infinity is 1.79583883262 a(n) is asymptotic to .723265064566*1.79583883262^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 3, parts, that yield the 29, -th largest growth, that is, 1.8124036192680426608, are , {[1, 2, 3], [1, 3, 2], [2, 3, 1], [3, 2, 1]} Theorem Number, 29, : Let a(n) be the number of compositions of n avoiding as\ subcompositions, the members of the set, {[1, 2, 3], [1, 3, 2], [2, 3, 1], [3, 2, 1]} 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 x - x - 2 x + x - x - 2 x + x - x + x + x - 2 x + 1 - --------------------------------------------------------------- 5 4 2 2 (x + x + x + x - 1) (-1 + x) and in Maple format -(x^11-x^10-2*x^9+x^8-x^7-2*x^6+x^5-x^4+x^3+x^2-2*x+1)/(x^5+x^4+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, 28, 49, 87, 155, 277, 497, 895, 1616, 2922, 5288, 9575, 17344, 31424, 56942, 103190, 187009, 338922, 614249, 1113252, 2017646, 3656772, 6627529, 12011739, 21770100, 39456188] The limit of a(n+1)/a(n) as n goes to infinity is 1.81240361927 a(n) is asymptotic to .705451568911*1.81240361927^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 3, parts, that yield the 30, -th largest growth, that is, 1.8271232214769008001, are , {[2, 1, 3], [2, 2, 2]}, {[2, 2, 2], [3, 1, 2]} Theorem Number, 30, : Let a(n) be the number of compositions of n avoiding as\ subcompositions, the members of the set, {[2, 1, 3], [2, 2, 2]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 6 5 4 3 2 \ n x - x + 2 x - x + 2 x - 2 x + 1 ) a(n) x = - ----------------------------------------------- / 7 6 5 4 3 2 ----- x - 2 x + 3 x - 2 x + 2 x - 3 x + 3 x - 1 n = 0 and in Maple format -(x^6-x^5+2*x^4-x^3+2*x^2-2*x+1)/(x^7-2*x^6+3*x^5-2*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, 30, 55, 100, 181, 329, 600, 1096, 2004, 3664, 6697, 12238, 22360, 40852, 74638, 136369, 249161, 455249, 831800, 1519807, 2776880, 5073704, 9270280, 16937937, 30947689, 56545234] The limit of a(n+1)/a(n) as n goes to infinity is 1.82712322148 a(n) is asymptotic to .793160204557*1.82712322148^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 3, parts, that yield the 31, -th largest growth, that is, 1.8392867552141611326, are , {[1, 1, 4], [1, 4, 1], [4, 1, 1]} Theorem Number, 31, : Let a(n) be the number of compositions of n avoiding as\ subcompositions, the members of the set, {[1, 1, 4], [1, 4, 1], [4, 1, 1]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 2 9 8 6 5 4 3 \ n (x + 1) (x + x - 3 x - 2 x + x + 2 x - 2 x + 1) ) a(n) x = - ------------------------------------------------------ / 3 2 2 ----- (x + x + x - 1) (-1 + x) n = 0 and in Maple format -(x^2+1)*(x^9+x^8-3*x^6-2*x^5+x^4+2*x^3-2*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, 29, 51, 89, 158, 284, 515, 939, 1718, 3150, 5783, 10625, 19530, 35908, 66031, 121435, 223338, 410766, 755499, 1389561, 2555782, 4700796, 8646091, 15902619, 29249454, 53798110] 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 ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 3, parts, that yield the 32, -th largest growth, that is, 1.8667603991738620930, are , {[2, 1, 3]}, {[3, 1, 2]} Theorem Number, 32, : 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 subsets of the set of compositions of, 6, with exactly, 3, parts, that yield the 33, -th largest growth, that is, 1.9087907387871591034, are , {[2, 2, 2]} Theorem Number, 33, : 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 ---------------------------------------------------------------------------- This ends this article, that took, 1628.368, seconds to generate. ---------------------------------------------------------- ---------------------------------------- Generating functions and Growth rates for the Enumerating Sequences for Como\ sitions Avoiding Each of the Possible subsets of the set of compositions of, 6, into exactly, 4, parts By Shalosh B. Ekhad The subsets of the set of compositions of, 6, with exactly, 4, parts, that yield the 1, -th largest growth, that is, 1.3247179572447460260, are , {[1, 1, 1, 3], [1, 1, 2, 2], [1, 3, 1, 1], [2, 1, 1, 2], [2, 1, 2, 1], [3, 1, 1, 1]}, { [1, 1, 1, 3], [1, 1, 3, 1], [1, 2, 1, 2], [2, 1, 1, 2], [2, 2, 1, 1], [3, 1, 1, 1]} Theorem Number, 1, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, {[1, 1, 1, 3], [1, 1, 2, 2], [1, 3, 1, 1], [2, 1, 1, 2], [2, 1, 2, 1], [3, 1, 1, 1]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 16 15 14 13 12 11 10 9 ) a(n) x = - (x - 2 x + x + 4 x - 7 x + x + 3 x + x / ----- n = 0 8 7 6 4 3 2 / 2 - 5 x + 3 x + 2 x - x + 2 x - 4 x + 3 x - 1) / ((x - x + 1) / 3 2 3 (x + x - 1) (-1 + x) ) and in Maple format -(x^16-2*x^15+x^14+4*x^13-7*x^12+x^11+3*x^10+x^9-5*x^8+3*x^7+2*x^6-x^4+2*x^3-4* x^2+3*x-1)/(x^2-x+1)/(x^3+x^2-1)/(-1+x)^3 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 26, 35, 45, 58, 75, 94, 117, 144, 176, 217, 268, 332, 412, 512, 640, 805, 1019, 1297, 1658, 2129, 2746, 3557, 4625, 6032, 7887] The limit of a(n+1)/a(n) as n goes to infinity is 1.32471795724 a(n) is asymptotic to 1.61564391382*1.32471795724^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 4, parts, that yield the 2, -th largest growth, that is, 1.3802775690976141157, are , {[1, 1, 1, 3], [1, 1, 3, 1], [1, 2, 1, 2], [1, 2, 2, 1], [3, 1, 1, 1]}, {[1, 1, 1, 3], [1, 2, 2, 1], [1, 3, 1, 1], [2, 1, 2, 1], [3, 1, 1, 1]} Theorem Number, 2, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, {[1, 1, 1, 3], [1, 1, 3, 1], [1, 2, 1, 2], [1, 2, 2, 1], [3, 1, 1, 1]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n ) a(n) x = - / ----- n = 0 10 9 7 6 5 4 3 2 (2 x - 3 x + 4 x - 7 x + 4 x + 2 x - 4 x + 5 x - 3 x + 1) 3 2 / 4 3 (x + x - 1) / ((x + x - 1) (-1 + x) ) / and in Maple format -(2*x^10-3*x^9+4*x^7-7*x^6+4*x^5+2*x^4-4*x^3+5*x^2-3*x+1)*(x^3+x^2-1)/(x^4+x-1) /(-1+x)^3 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 27, 38, 50, 66, 86, 110, 140, 179, 230, 296, 382, 496, 649, 855, 1133, 1510, 2024, 2727, 3690, 5011, 6826, 9323, 12761, 17497, 24024] The limit of a(n+1)/a(n) as n goes to infinity is 1.38027756910 a(n) is asymptotic to 1.49162659540*1.38027756910^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 4, parts, that yield the 3, -th largest growth, that is, 1.3970352753465915477, are , {[1, 1, 1, 3], [1, 2, 1, 2], [2, 1, 1, 2], [3, 1, 1, 1]}, {[1, 1, 1, 3], [2, 1, 1, 2], [2, 1, 2, 1], [3, 1, 1, 1]} Theorem Number, 3, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, {[1, 1, 1, 3], [1, 2, 1, 2], [2, 1, 1, 2], [3, 1, 1, 1]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 18 17 16 15 14 13 12 11 ) a(n) x = - (x + 2 x - x - 2 x - 2 x - 3 x + 3 x + 2 x / ----- n = 0 10 8 7 4 3 2 / 7 5 - x + x + 3 x - x + 2 x - 4 x + 3 x - 1) / ((x + x + x - 1) / 3 (-1 + x) ) and in Maple format -(x^18+2*x^17-x^16-2*x^15-2*x^14-3*x^13+3*x^12+2*x^11-x^10+x^8+3*x^7-x^4+2*x^3-\ 4*x^2+3*x-1)/(x^7+x^5+x-1)/(-1+x)^3 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 28, 44, 61, 79, 101, 129, 164, 207, 261, 331, 425, 553, 727, 963, 1283, 1720, 2322, 3157, 4319, 5936, 8185, 11313, 15668, 21740, 30215] The limit of a(n+1)/a(n) as n goes to infinity is 1.39703527535 a(n) is asymptotic to 1.30895667560*1.39703527535^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 4, parts, that yield the 4, -th largest growth, that is, 1.4036021248742166433, are , {[1, 1, 2, 2], [1, 1, 3, 1], [1, 2, 2, 1], [2, 1, 1, 2]}, {[1, 2, 2, 1], [1, 3, 1, 1], [2, 1, 1, 2], [2, 2, 1, 1]} Theorem Number, 4, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, {[1, 1, 2, 2], [1, 1, 3, 1], [1, 2, 2, 1], [2, 1, 1, 2]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 2 7 6 5 4 3 2 \ n (x + 1) (x - x - 2 x + 2 x + x + x - 2 x + 1) ) a(n) x = - ---------------------------------------------------- / 5 4 3 2 ----- (x - x + x + x - 1) (-1 + x) n = 0 and in Maple format -(x^2+1)*(x^7-x^6-2*x^5+2*x^4+x^3+x^2-2*x+1)/(x^5-x^4+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, 28, 45, 69, 103, 152, 222, 321, 461, 658, 935, 1325, 1873, 2643, 3725, 5244, 7377, 10372, 14576, 20478, 28763, 40392, 56716, 79629, 111790, 156933] The limit of a(n+1)/a(n) as n goes to infinity is 1.40360212487 a(n) is asymptotic to 6.00587058931*1.40360212487^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 4, parts, that yield the 5, -th largest growth, that is, 1.4655712318767680267, are , {[1, 1, 1, 3], [1, 2, 1, 2], [2, 1, 2, 1], [3, 1, 1, 1]} Theorem Number, 5, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, {[1, 1, 1, 3], [1, 2, 1, 2], [2, 1, 2, 1], [3, 1, 1, 1]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- \ n 16 14 13 12 11 10 9 8 ) a(n) x = - (2 x - x - x - 4 x + 4 x - 3 x + x + 3 x / ----- n = 0 7 6 5 3 2 / 2 - x + 5 x - 3 x + 2 x - 4 x + 3 x - 1) / ((x + 1) (x - x + 1) / 3 3 (x + x - 1) (-1 + x) ) and in Maple format -(2*x^16-x^14-x^13-4*x^12+4*x^11-3*x^10+x^9+3*x^8-x^7+5*x^6-3*x^5+2*x^3-4*x^2+3 *x-1)/(x+1)/(x^2-x+1)/(x^3+x-1)/(-1+x)^3 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 28, 43, 60, 79, 104, 136, 179, 237, 318, 431, 591, 818, 1144, 1614, 2296, 3287, 4731, 6837, 9914, 14413, 20997, 30635, 44749, 65421, 95705] The limit of a(n+1)/a(n) as n goes to infinity is 1.46557123188 a(n) is asymptotic to .996775270563*1.46557123188^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 4, parts, that yield the 6, -th largest growth, that is, 1.5289463545197057618, are , {[1, 1, 2, 2], [1, 2, 1, 2], [1, 2, 2, 1], [2, 1, 1, 2]}, {[1, 1, 2, 2], [1, 2, 1, 2], [2, 1, 1, 2], [2, 1, 2, 1]}, {[1, 2, 1, 2], [2, 1, 1, 2], [2, 1, 2, 1], [2, 2, 1, 1]}, {[1, 2, 2, 1], [2, 1, 1, 2], [2, 1, 2, 1], [2, 2, 1, 1]} Theorem Number, 6, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, {[1, 1, 2, 2], [1, 2, 1, 2], [1, 2, 2, 1], [2, 1, 1, 2]} 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 - 3 x + 6 x - 7 x + 4 x - 1 ) a(n) x = - ------------------------------------------------- / 5 2 3 ----- (x - x + 2 x - 1) (-1 + x) n = 0 and in Maple format -(x^10-x^7+x^6+x^5-3*x^4+6*x^3-7*x^2+4*x-1)/(x^5-x^2+2*x-1)/(-1+x)^3 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 28, 45, 69, 104, 157, 238, 361, 546, 823, 1239, 1868, 2825, 4286, 6518, 9926, 15127, 23065, 35187, 53710, 82026, 125319, 191509, 292699, 447392, 683883] The limit of a(n+1)/a(n) as n goes to infinity is 1.52894635452 a(n) is asymptotic to 2.00901953273*1.52894635452^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 4, parts, that yield the 7, -th largest growth, that is, 1.6180339887498948482, are , {[1, 1, 2, 2], [1, 2, 1, 2], [1, 2, 2, 1]}, {[1, 1, 2, 2], [1, 2, 1, 2], [2, 1, 2, 1]}, {[1, 2, 1, 2], [2, 1, 2, 1], [2, 2, 1, 1]}, {[1, 2, 2, 1], [2, 1, 2, 1], [2, 2, 1, 1]} Theorem Number, 7, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, {[1, 1, 2, 2], [1, 2, 1, 2], [1, 2, 2, 1]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 9 7 6 5 4 3 2 \ n x - 2 x + 4 x - 2 x - 2 x + 6 x - 7 x + 4 x - 1 ) a(n) x = - ------------------------------------------------------ / 2 2 3 ----- (x - x + 1) (x + x - 1) (-1 + x) n = 0 and in Maple format -(x^9-2*x^7+4*x^6-2*x^5-2*x^4+6*x^3-7*x^2+4*x-1)/(x^2-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, 29, 49, 80, 129, 207, 331, 528, 842, 1345, 2154, 3458, 5562, 8959, 14447, 23318, 37663, 60865, 98397, 159114, 257344, 416271, 673408, 1089452, 1762612, 2851793] The limit of a(n+1)/a(n) as n goes to infinity is 1.61803398875 a(n) is asymptotic to 1.53262379212*1.61803398875^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 4, parts, that yield the 8, -th largest growth, that is, 1.6530424890094669421, are , {[1, 2, 1, 2], [2, 1, 1, 2], [2, 1, 2, 1]} Theorem Number, 8, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, {[1, 2, 1, 2], [2, 1, 1, 2], [2, 1, 2, 1]} 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 + x + x - 2 x + x - x + 4 x - 9 x / ----- n = 0 3 2 / 7 6 5 3 2 + 13 x - 11 x + 5 x - 1) / ((x - x + x + x - 3 x + 3 x - 1) / 3 (-1 + x) ) and in Maple format -(x^12-x^11+x^10+x^9-2*x^8+x^7-x^6+4*x^5-9*x^4+13*x^3-11*x^2+5*x-1)/(x^7-x^6+x^ 5+x^3-3*x^2+3*x-1)/(-1+x)^3 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 29, 49, 79, 125, 198, 317, 514, 841, 1382, 2273, 3736, 6137, 10083, 16583, 27313, 45051, 74392, 122926, 203184, 335858, 555132, 917506, 1516393, 2506252, 4142472] The limit of a(n+1)/a(n) as n goes to infinity is 1.65304248901 a(n) is asymptotic to 1.17141943318*1.65304248901^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 4, parts, that yield the 9, -th largest growth, that is, 1.6736485462998415616, are , {[1, 1, 2, 2], [1, 2, 2, 1], [2, 1, 1, 2]}, {[1, 2, 2, 1], [2, 1, 1, 2], [2, 2, 1, 1]} Theorem Number, 9, : Let a(n) be the number of compositions of n avoiding as \ subcompositions, the members of the set, {[1, 1, 2, 2], [1, 2, 2, 1], [2, 1, 1, 2]} We have the following rational function for the (ordinary) generating functi\ on of a(n) infinity ----- 10 9 7 6 5 4 3 2 \ n x - x + x - 2 x + 5 x - 6 x + 7 x - 7 x + 4 x - 1 ) a(n) x = - ---------------------------------------------------------- / 5 4 3 2 3 ----- (x - x + x - x + 2 x - 1) (-1 + x) n = 0 and in Maple format -(x^10-x^9+x^7-2*x^6+5*x^5-6*x^4+7*x^3-7*x^2+4*x-1)/(x^5-x^4+x^3-x^2+2*x-1)/(-1 +x)^3 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 29, 49, 80, 129, 208, 337, 549, 899, 1479, 2443, 4049, 6729, 11206, 18690, 31206, 52143, 87173, 145789, 243879, 408034, 682758, 1142535, 1912025, 3199865, 5355240] The limit of a(n+1)/a(n) as n goes to infinity is 1.67364854630 a(n) is asymptotic to 1.04428813147*1.67364854630^n ---------------------------------------------------------------------------- The subsets of the set of compositions of, 6, with exactly, 4, parts, that yield the 10, -th largest growth, that is, 1.7548776662466927601, are , {[1, 2, 1, 2], [2, 1, 2, 1]} Theorem Number, 10, : Let a(n) be the number of compositions of n avoiding as\ subcompositions, the members of the set, {[1, 2, 1, 2], [2, 1, 2, 1]} 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 4 3 2 x - x + x + 2 x - 6 x + 6 x - 8 x + 13 x - 11 x + 5 x - 1 - -------------------------------------------------------------------- 3 3 2 3 (x - x + 1) (x - x + 2 x - 1) (-1 + x) and in Maple format -(x^11-x^10+x^9+2*x^8-6*x^7+6*x^6-8*x^4+13*x^3-11*x^2+5*x-1)/(x^3-x+1)/(x^3-x^2 +2*x-1)/(-1+x)^3 The first, 31, terms of a(n), starting at n=0, are [1, 1, 2, 4, 8, 16, 30, 53, 91, 155, 265, 456, 789, 1370, 2384, 4155, 7252, 12675, 22181, 38855, 68111, 119448, 209532, 367606, 644986, 1131727, 1985870, 3484765, 6115133, 10731110, 18831593] 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 ---------------------------------------------------------------------------- This ends this article, that took, 8979.745, seconds to generate. ---------------------------------------------------------- ---------------------------------------- Generating functions and Growth rates for the Enumerating Sequences for Como\ sitions Avoiding Each of the Possible subsets of the set of compositions of, 6, into exactly, 5, parts By Shalosh B. Ekhad The subsets of the set of compositions of, 6, with exactly, 5, parts, that yield the 1, -th largest growth, that is, 1, 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]} 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] ---------------------------------------------------------------------------- This ends this article, that took, 127.046, seconds to generate. ---------------------------------------------------------- ---------------------------------------- Generating functions and Growth rates for the Enumerating Sequences for Como\ sitions Avoiding Each of the Possible subsets of the set of compositions of, 6, into exactly, 6, parts By Shalosh B. Ekhad The subsets of the set of compositions of, 6, with exactly, 6, parts, that yield the 1, -th largest growth, that is, 1, are , {[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, 1] 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] ---------------------------------------------------------------------------- This ends this article, that took, 0.005, seconds to generate. ---------------------------------------------------------- ----------------------------------------