02143_02413_02431_04213_04231_04321_20143_20413_20431_21043_21403_24013_24031_24103_40213_40231_40321_42013_42031_42103
Counting sequence:
1, 1, 2, 6, 24, 100, 400, 1536, 5752, 21288, 78488, 289336, 1067576, 3942488, 14566904, 53834584, 198966456, 735348504, 2717660408, 10043577176, 37117352760, 137171480216, 506932797816, 1873428839896, 6923477837240, 25586542591768, 94558156505080, 349451103788632, 1291438818723896, 4772668261520280, 17637972184100472, 65183256566513368, 240892596681269944, 890247682282082328, 3290017821005444344, 12158658184062572376, 44933789704300049720, 166058246479175410840, 613688304662303093624, 2267959244829515778008, 8381517289918216699832, 30974909377427507761432, 114471518430914308024824, 423043321036415065788504, 1563407683644261640782392, 5777761906959990745208216, 21352416904921559884978296, 78910435394110730242470104, 291623043977470023970553016, 1077728178205420322304045592, 3982874639315972095994052344, 14719194239609659850681379160, 54396559943087949931121454904, 201029056718284701250864850584, 742927157293120947767649983864, 2745577032762522713395412646360, 10146611506703709646997596690872, 37498028224829297383403674268440, 138578491925221045319399949874168, 512133552973130494159307214496344, 1892651395155986897372402743245880, 6994521805474915009529503069018008, 25849100057452441626978378594140792, 95528471046752267799590239460139736, 353037001684675618698890661680977592, 1304690875849027228876488100722578456, 4821642698642912452147429470604364024, 17818962900501410164792665946202979160, 65852129387109653123460623503867128120, 243364497082745783557605721976283326616, 899382889992485777234967401018950944632, 3323778088043003047451528984566263180248, 12283423335579689866207704583328593720248, 45394874400264581185519011944505258382616, 167762240665179073103631216062206993677816, 619985620950135870884022185489965133706328, 2291231737611790063127322008657517654455864, 8467523597392863531197736117634079353379224, 31292756073263305900723311281777784874249336, 115646159281122385689681909211504471628059864, 427384348159143894852968689618963854355738808, 1579450473641735375545996637893564422680704536, 5837049974881558710830024160312591891865948920, 21571523120131159495015784570556606441150713176, 79720168873797472349474058359404676735729673016, 294615511842825576477686193631303320577819585176, 1088787204600855710682439405656440906449172312440, 4023744606953945364026733558525898690199308246488, 14870234140863484247955711130016482816297874918840, 54954746139193243860816392652891173361184604862232, 203091901217892194154797640309337070130316550785016, 750550648270242236781901155539331476561432426563160, 2773750564354125965047975340966403640901546641453112, 10250730195205496684992575862466240843152339376151448, 37882811412555852669127041759351868319501683341694584, 140000504665560785234679733289532875718696006353782488, 517388772790908807526621526951627608722977862275484344, 1912072694663171026299253540659793406034732501051594776, 7066295563305507256954346619445972362737225799906368760, 26114348647600537683265071062887898564726550654375845720, 96508729245600042039775501098316839833776496796851346744
Generating function in Maple syntax:
(x-1)*(3*x-1)*(2*x-1)/(14*x^3-16*x^2+7*x-1)
Generating function in latex syntax:
\frac{\left(x -1\right) \left(3 x -1\right) \left(2 x -1\right)}{14 x^{3}-16 x^{2}+7 x -1}
Generating function in sympy syntax:
(x - 1)*(2*x - 1)*(3*x - 1)/(14*x**3 - 16*x**2 + 7*x - 1)
Implicit equation for the generating function in Maple syntax:
(14*x^3-16*x^2+7*x-1)*F(x)-(x-1)*(3*x-1)*(2*x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(14 x^{3}-16 x^{2}+7 x -1\right) F \! \left(x \right)-\left(x -1\right) \left(3 x -1\right) \left(2 x -1\right) = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/1758792*(12673*((I+35/667*29^(1/2))*3^(1/2)-105/667*I*29^(1/2)-1)*2^(1/3)*(157+21*29^(1/2)*3^(1/2))^(1/3)+335008-1073*2^(2/3)*((I-224/1073*29^(1/2))*3^(1/2)-672/1073*I*29^(1/2)+1)*(157+21*29^(1/2)*3^(1/2))^(2/3))*(157/60648*((I-21/157*29^(1/2))*3^(1/2)-63/157*I*29^(1/2)+1)*2^(2/3)*(157+21*29^(1/2)*3^(1/2))^(2/3)-1/84*I*3^(1/2)*(314+42*29^(1/2)*3^(1/2))^(1/3)+1/84*(314+42*29^(1/2)*3^(1/2))^(1/3)+8/21)^(-n)+1/1758792*(-12673*2^(1/3)*((I-35/667*29^(1/2))*3^(1/2)-105/667*I*29^(1/2)+1)*(157+21*29^(1/2)*3^(1/2))^(1/3)+335008+1073*((I+224/1073*29^(1/2))*3^(1/2)-672/1073*I*29^(1/2)-1)*2^(2/3)*(157+21*29^(1/2)*3^(1/2))^(2/3))*(-157/60648*((I+21/157*29^(1/2))*3^(1/2)-63/157*I*29^(1/2)-1)*2^(2/3)*(157+21*29^(1/2)*3^(1/2))^(2/3)+1/84*I*3^(1/2)*(314+42*29^(1/2)*3^(1/2))^(1/3)+1/84*(314+42*29^(1/2)*3^(1/2))^(1/3)+8/21)^(-n)-5/6612*(2^(1/3)*(29^(1/2)*3^(1/2)-667/35)*(157+21*29^(1/2)*3^(1/2))^(1/3)-8816/35+(32/95*2^(2/3)*29^(1/2)*3^(1/2)-1073/665*2^(2/3))*(157+21*29^(1/2)*3^(1/2))^(2/3))*(1/30324*2^(2/3)*(21*29^(1/2)*3^(1/2)-157)*(157+21*29^(1/2)*3^(1/2))^(2/3)-1/42*(314+42*29^(1/2)*3^(1/2))^(1/3)+8/21)^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(12673 \left(\left(i+\frac{35 \sqrt{29}}{667}\right) \sqrt{3}-\frac{105 i \sqrt{29}}{667}-1\right) 2^{\frac{1}{3}} \left(157+21 \sqrt{29}\, \sqrt{3}\right)^{\frac{1}{3}}+335008-1073 \,2^{\frac{2}{3}} \left(\left(i-\frac{224 \sqrt{29}}{1073}\right) \sqrt{3}-\frac{672 i \sqrt{29}}{1073}+1\right) \left(157+21 \sqrt{29}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{157 \left(\left(i-\frac{21 \sqrt{29}}{157}\right) \sqrt{3}-\frac{63 i \sqrt{29}}{157}+1\right) 2^{\frac{2}{3}} \left(157+21 \sqrt{29}\, \sqrt{3}\right)^{\frac{2}{3}}}{60648}-\frac{i \sqrt{3}\, \left(314+42 \sqrt{29}\, \sqrt{3}\right)^{\frac{1}{3}}}{84}+\frac{\left(314+42 \sqrt{29}\, \sqrt{3}\right)^{\frac{1}{3}}}{84}+\frac{8}{21}\right)^{-n}}{1758792}\\+\\\frac{\left(-12673 \,2^{\frac{1}{3}} \left(\left(i-\frac{35 \sqrt{29}}{667}\right) \sqrt{3}-\frac{105 i \sqrt{29}}{667}+1\right) \left(157+21 \sqrt{29}\, \sqrt{3}\right)^{\frac{1}{3}}+335008+1073 \left(\left(i+\frac{224 \sqrt{29}}{1073}\right) \sqrt{3}-\frac{672 i \sqrt{29}}{1073}-1\right) 2^{\frac{2}{3}} \left(157+21 \sqrt{29}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{157 \left(\left(i+\frac{21 \sqrt{29}}{157}\right) \sqrt{3}-\frac{63 i \sqrt{29}}{157}-1\right) 2^{\frac{2}{3}} \left(157+21 \sqrt{29}\, \sqrt{3}\right)^{\frac{2}{3}}}{60648}+\frac{i \sqrt{3}\, \left(314+42 \sqrt{29}\, \sqrt{3}\right)^{\frac{1}{3}}}{84}+\frac{\left(314+42 \sqrt{29}\, \sqrt{3}\right)^{\frac{1}{3}}}{84}+\frac{8}{21}\right)^{-n}}{1758792}\\-\\\frac{5 \left(2^{\frac{1}{3}} \left(\sqrt{29}\, \sqrt{3}-\frac{667}{35}\right) \left(157+21 \sqrt{29}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{8816}{35}+\left(\frac{32 \,2^{\frac{2}{3}} \sqrt{29}\, \sqrt{3}}{95}-\frac{1073 \,2^{\frac{2}{3}}}{665}\right) \left(157+21 \sqrt{29}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{2^{\frac{2}{3}} \left(21 \sqrt{29}\, \sqrt{3}-157\right) \left(157+21 \sqrt{29}\, \sqrt{3}\right)^{\frac{2}{3}}}{30324}-\frac{\left(314+42 \sqrt{29}\, \sqrt{3}\right)^{\frac{1}{3}}}{42}+\frac{8}{21}\right)^{-n}}{6612} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n+3) = 14*a(n)-16*a(n+1)+7*a(n+2), n >= 4
Recurrence in latex format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a{\left(n + 3 \right)} = 14 a{\left(n \right)} - 16 a{\left(n + 1 \right)} + 7 a{\left(n + 2 \right)}, \quad n \geq 4
Specification 1
Strategy pack name: regular_insertion_encoding_left
Tree: http://permpal.com/tree/25474/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[3,x]
F[3,x] = F[16,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[2,x]+F[6,x]
F[6,x] = F[35,x]+F[7,x]+F[8,x]
F[7,x] = 0
F[8,x] = F[16,x]*F[9,x]
F[9,x] = F[10,x]+F[34,x]
F[10,x] = F[11,x]+F[2,x]
F[11,x] = F[12,x]+F[17,x]+F[7,x]
F[12,x] = F[13,x]*F[16,x]
F[13,x] = F[14,x]+F[15,x]
F[14,x] = F[2,x]
F[15,x] = F[11,x]
F[16,x] = x
F[17,x] = F[16,x]*F[18,x]
F[18,x] = F[19,x]+F[30,x]
F[19,x] = F[20,x]+F[23,x]
F[20,x] = F[21,x]
F[21,x] = F[16,x]*F[22,x]
F[22,x] = F[1,x]+F[20,x]
F[23,x] = F[24,x]+F[26,x]+F[7,x]
F[24,x] = F[16,x]*F[25,x]
F[25,x] = F[2,x]+F[23,x]
F[26,x] = F[16,x]*F[27,x]
F[27,x] = F[28,x]+F[29,x]
F[28,x] = F[20,x]
F[29,x] = F[23,x]
F[30,x] = F[31,x]
F[31,x] = F[17,x]+F[32,x]+F[7,x]
F[32,x] = F[16,x]*F[33,x]
F[33,x] = F[2,x]+F[31,x]
F[34,x] = F[6,x]
F[35,x] = F[16,x]*F[36,x]
F[36,x] = F[37,x]+F[50,x]
F[37,x] = F[38,x]+F[42,x]
F[38,x] = F[39,x]
F[39,x] = F[16,x]*F[40,x]
F[40,x] = F[22,x]+F[41,x]
F[41,x] = F[38,x]
F[42,x] = F[43,x]+F[46,x]+F[7,x]
F[43,x] = F[16,x]*F[44,x]
F[44,x] = F[25,x]+F[45,x]
F[45,x] = F[42,x]
F[46,x] = F[16,x]*F[47,x]
F[47,x] = F[48,x]+F[49,x]
F[48,x] = F[38,x]
F[49,x] = F[42,x]
F[50,x] = F[51,x]
F[51,x] = F[35,x]+F[52,x]+F[7,x]
F[52,x] = F[16,x]*F[53,x]
F[53,x] = F[33,x]+F[54,x]
F[54,x] = F[51,x]
System of equations in latex syntax:
F_{0}\! \left(x \right) = F_{1}\! \left(x \right)+F_{2}\! \left(x \right)
F_{1}\! \left(x \right) = 1
F_{2}\! \left(x \right) = F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{2}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{35}\! \left(x \right)+F_{7}\! \left(x \right)+F_{8}\! \left(x \right)
F_{7}\! \left(x \right) = 0
F_{8}\! \left(x \right) = F_{16}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{34}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{2}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{17}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{15}\! \left(x \right)
F_{14}\! \left(x \right) = F_{2}\! \left(x \right)
F_{15}\! \left(x \right) = F_{11}\! \left(x \right)
F_{16}\! \left(x \right) = x
F_{17}\! \left(x \right) = F_{16}\! \left(x \right) F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{30}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{23}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{16}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{20}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{26}\! \left(x \right)+F_{7}\! \left(x \right)
F_{24}\! \left(x \right) = F_{16}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{2}\! \left(x \right)+F_{23}\! \left(x \right)
F_{26}\! \left(x \right) = F_{16}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)
F_{28}\! \left(x \right) = F_{20}\! \left(x \right)
F_{29}\! \left(x \right) = F_{23}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{17}\! \left(x \right)+F_{32}\! \left(x \right)+F_{7}\! \left(x \right)
F_{32}\! \left(x \right) = F_{16}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{2}\! \left(x \right)+F_{31}\! \left(x \right)
F_{34}\! \left(x \right) = F_{6}\! \left(x \right)
F_{35}\! \left(x \right) = F_{16}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{50}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{42}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{16}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{22}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{38}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{46}\! \left(x \right)+F_{7}\! \left(x \right)
F_{43}\! \left(x \right) = F_{16}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{25}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{42}\! \left(x \right)
F_{46}\! \left(x \right) = F_{16}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{49}\! \left(x \right)
F_{48}\! \left(x \right) = F_{38}\! \left(x \right)
F_{49}\! \left(x \right) = F_{42}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{35}\! \left(x \right)+F_{52}\! \left(x \right)+F_{7}\! \left(x \right)
F_{52}\! \left(x \right) = F_{16}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{33}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{51}\! \left(x \right)
System of equations in sympy syntax:
Eq(F_0(x), F_1(x) + F_2(x))
Eq(F_1(x), 1)
Eq(F_2(x), F_3(x))
Eq(F_3(x), F_16(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_2(x) + F_6(x))
Eq(F_6(x), F_35(x) + F_7(x) + F_8(x))
Eq(F_7(x), 0)
Eq(F_8(x), F_16(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_34(x))
Eq(F_10(x), F_11(x) + F_2(x))
Eq(F_11(x), F_12(x) + F_17(x) + F_7(x))
Eq(F_12(x), F_13(x)*F_16(x))
Eq(F_13(x), F_14(x) + F_15(x))
Eq(F_14(x), F_2(x))
Eq(F_15(x), F_11(x))
Eq(F_16(x), x)
Eq(F_17(x), F_16(x)*F_18(x))
Eq(F_18(x), F_19(x) + F_30(x))
Eq(F_19(x), F_20(x) + F_23(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_16(x)*F_22(x))
Eq(F_22(x), F_1(x) + F_20(x))
Eq(F_23(x), F_24(x) + F_26(x) + F_7(x))
Eq(F_24(x), F_16(x)*F_25(x))
Eq(F_25(x), F_2(x) + F_23(x))
Eq(F_26(x), F_16(x)*F_27(x))
Eq(F_27(x), F_28(x) + F_29(x))
Eq(F_28(x), F_20(x))
Eq(F_29(x), F_23(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_17(x) + F_32(x) + F_7(x))
Eq(F_32(x), F_16(x)*F_33(x))
Eq(F_33(x), F_2(x) + F_31(x))
Eq(F_34(x), F_6(x))
Eq(F_35(x), F_16(x)*F_36(x))
Eq(F_36(x), F_37(x) + F_50(x))
Eq(F_37(x), F_38(x) + F_42(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_16(x)*F_40(x))
Eq(F_40(x), F_22(x) + F_41(x))
Eq(F_41(x), F_38(x))
Eq(F_42(x), F_43(x) + F_46(x) + F_7(x))
Eq(F_43(x), F_16(x)*F_44(x))
Eq(F_44(x), F_25(x) + F_45(x))
Eq(F_45(x), F_42(x))
Eq(F_46(x), F_16(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_49(x))
Eq(F_48(x), F_38(x))
Eq(F_49(x), F_42(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_35(x) + F_52(x) + F_7(x))
Eq(F_52(x), F_16(x)*F_53(x))
Eq(F_53(x), F_33(x) + F_54(x))
Eq(F_54(x), F_51(x))
Pack JSON:
{"name": "regular_insertion_encoding_left", "initial_strats": [{"class_module": "tilings.strategies.factor", "strategy_class": "FactorFactory", "interleaving": null, "unions": false, "ignore_parent": true, "workable": true, "tracked": false}, {"class_module": "tilings.strategies.requirement_insertion", "strategy_class": "CellInsertionFactory", "ignore_parent": true, "maxreqlen": 1, "extra_basis": [], "one_cell_only": true}], "inferral_strats": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}], "expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "strategy_class": "RowAndColumnPlacementFactory", "partial": false, "ignore_parent": false, "dirs": [0], "place_row": false, "place_col": true}]], "symmetries": [], "iterative": false}
Specification JSON:
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