0132_0231_1320_3021
Counting sequence:
1, 1, 2, 6, 20, 65, 206, 651, 2085, 6818, 22791, 77729, 269656, 948742, 3376701, 12132856, 43941634, 160215519, 587540644, 2165472370, 8016582789, 29794498975, 111126455206, 415801091478, 1560323361337, 5870780701507, 22142921691638, 83704534575828, 317077118563037, 1203421300685781, 4575613372320576, 17426400698494295, 66473091616486145, 253934317351249118, 971394614364921123, 3720776968087103465, 14269315703119210180, 54786658824945709318, 210582190039321542413, 810249448810894131921, 3120636481801624122824, 12030224127781433313398, 46418454683561277578225, 179256861039406680292959, 692806645464551455706128, 2679684147228355055504208, 10372314658836381504649691, 40176716866875705338992391, 155728084388150097996520226, 604005691155381065591497284, 2344149521633660336675375887, 9103094615374904717366451645, 35370579710355552234069895622, 137510277530324352823295825438, 534883277668186000548122909441, 2081638354836215055399089962839, 8105240741274550464016071434356, 31574271437217909305823719525088, 123055305331436995802850647173631, 479799845950409798139788652139637, 1871570287269111400409002067445506, 7303521220085741084566121738258286, 28512329846798420455190895421624897, 111352968305613445458702694637567376, 435045076651881096058063556582678906, 1700297908544202027056054201415266903, 6647666947063847595865508642338632860, 25999341011831476998478692703426475538, 101718482828138407344796790764679854909, 398086672039956938805735516193312762767, 1558445423350109202725760685866735574550, 6102923032886320405185276068003134154834, 23906325192990672806246311350116063825921, 93672657483086872539046444423912466124615, 367142403448278659207345284898030521172914, 1439377382970963214483719526194849166632384, 5644559944021809764291180925715891928284577, 22141027965134612140618588544684815654138577, 86871006229552933361314630425191950148505692, 340924741279486753603554188489017634263375548, 1338277276635493343035885099096806145085559387, 5254543907019612886931741124892986282019916753, 20635870767281676177863768432896605011788743444, 81060080723700628410800177422343922255621864386, 318482383324443397285856724140148148644192471331, 1251571661728358001046743241855733302311378866803, 4919441536444075789250029521194887391167206599910, 19340317008638185467100769530253788713129508731748, 76049624255462613005635539636741836764919131014649, 299098559799819801886376043052409693997103913008957, 1176558338534175315116885890542200875952337056651660, 4629058802097829070779421729012032097809093870188354, 18215884653366076278767457403049614837301826854878431, 71694276124519332281074672158909448041537323388261275, 282223848446357270421602168442951920476421750604449454, 1111159384119251586302903392338035239840488067529355154, 4375532302492940995578306100013059916785277606517837597, 17232799276084316718012195349247220843770983817963186927, 67881232837697562790708652539925040725752783105047314282, 267430644614738057562408639032812063728834734259186048728, 1053752997085066568154083162729748749289773305175856118017
Generating function in Maple syntax:
1/2*((-3*x^4+9*x^3-10*x^2+5*x-1)*(1-4*x)^(1/2)+7*x^4-11*x^3+10*x^2-5*x+1)/x/(2*x-1)/(-1+x)^3
Generating function in latex syntax:
\frac{\left(-3 x^{4}+9 x^{3}-10 x^{2}+5 x -1\right) \sqrt{1-4 x}+7 x^{4}-11 x^{3}+10 x^{2}-5 x +1}{2 x \left(2 x -1\right) \left(-1+x \right)^{3}}
Generating function in sympy syntax:
(7*x**4/2 - 11*x**3/2 + 5*x**2 - 5*x/2 + sqrt(1 - 4*x)*(-3*x**4 + 9*x**3 - 10*x**2 + 5*x - 1)/2 + 1/2)/(x*(x - 1)**3*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
x*(2*x-1)^2*(-1+x)^6*F(x)^2-(2*x-1)*(7*x^4-11*x^3+10*x^2-5*x+1)*(-1+x)^3*F(x)+9*x^8-44*x^7+116*x^6-180*x^5+176*x^4-111*x^3+44*x^2-10*x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(2 x -1\right)^{2} \left(-1+x \right)^{6} F \! \left(x \right)^{2}-\left(2 x -1\right) \left(7 x^{4}-11 x^{3}+10 x^{2}-5 x +1\right) \left(-1+x \right)^{3} F \! \left(x \right)+9 x^{8}-44 x^{7}+116 x^{6}-180 x^{5}+176 x^{4}-111 x^{3}+44 x^{2}-10 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 65
a(6) = 206
a(7) = 651
a(n+5) = 12*(2*n+1)/(6+n)*a(n)-6*(11*n+15)/(6+n)*a(1+n)+(71*n+174)/(6+n)*a(n+2)-2*(19*n+69)/(6+n)*a(n+3)+2*(5*n+24)/(6+n)*a(n+4)-(3*n-4)*(n-3)/(6+n), n >= 8
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 20
a \! \left(5\right) = 65
a \! \left(6\right) = 206
a \! \left(7\right) = 651
a \! \left(n +5\right) = \frac{12 \left(2 n +1\right) a \! \left(n \right)}{6+n}-\frac{6 \left(11 n +15\right) a \! \left(1+n \right)}{6+n}+\frac{\left(71 n +174\right) a \! \left(n +2\right)}{6+n}-\frac{2 \left(19 n +69\right) a \! \left(n +3\right)}{6+n}+\frac{2 \left(5 n +24\right) a \! \left(n +4\right)}{6+n}-\frac{\left(3 n -4\right) \left(n -3\right)}{6+n}, \quad n \geq 8
Specification 1
Strategy pack name: row_placements
Tree: http://permpal.com/tree/21245/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[13,x]*F[3,x]
F[3,x] = F[1,x]+F[18,x]+F[4,x]
F[4,x] = F[13,x]*F[5,x]
F[5,x] = F[1,x]+F[11,x]+F[15,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[13,x]*F[5,x]*F[8,x]
F[8,x] = F[1,x]+F[9,x]
F[9,x] = F[10,x]*F[13,x]
F[10,x] = F[1,x]+F[11,x]+F[14,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[13,x]*F[8,x]
F[13,x] = x
F[14,x] = F[10,x]*F[13,x]
F[15,x] = F[13,x]*F[16,x]
F[16,x] = F[1,x]+F[11,x]+F[15,x]+F[17,x]
F[17,x] = F[13,x]*F[16,x]
F[18,x] = F[13,x]*F[19,x]
F[19,x] = F[1,x]+F[18,x]+F[20,x]+F[41,x]
F[20,x] = F[13,x]*F[21,x]
F[21,x] = F[1,x]+F[15,x]+F[22,x]+F[24,x]+F[27,x]
F[22,x] = F[23,x]
F[23,x] = F[13,x]*F[21,x]*F[8,x]
F[24,x] = F[25,x]
F[25,x] = F[13,x]*F[26,x]*F[8,x]
F[26,x] = F[1,x]+F[14,x]+F[24,x]+F[27,x]
F[27,x] = F[13,x]*F[28,x]
F[28,x] = F[29,x]+F[32,x]
F[29,x] = F[1,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[13,x]*F[29,x]
F[32,x] = F[33,x]+F[36,x]
F[33,x] = F[34,x]
F[34,x] = F[13,x]*F[35,x]
F[35,x] = F[1,x]+F[34,x]
F[36,x] = F[37,x]+F[38,x]+F[40,x]
F[37,x] = 0
F[38,x] = F[13,x]*F[39,x]
F[39,x] = F[30,x]+F[36,x]
F[40,x] = F[13,x]*F[32,x]
F[41,x] = F[42,x]
F[42,x] = F[13,x]*F[43,x]
F[43,x] = F[44,x]+F[54,x]
F[44,x] = F[45,x]+F[48,x]
F[45,x] = F[1,x]+F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[13,x]*F[28,x]
F[48,x] = F[33,x]+F[49,x]
F[49,x] = F[37,x]+F[50,x]+F[52,x]
F[50,x] = F[13,x]*F[51,x]
F[51,x] = F[46,x]+F[49,x]
F[52,x] = F[13,x]*F[53,x]
F[53,x] = F[32,x]
F[54,x] = F[55,x]
F[55,x] = F[13,x]*F[35,x]*F[56,x]
F[56,x] = F[45,x]+F[46,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_{13}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{13}\! \left(x \right) F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)+F_{15}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{13}\! \left(x \right) F_{5}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{1}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{13}\! \left(x \right) F_{8}\! \left(x \right)
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{10}\! \left(x \right) F_{13}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)+F_{15}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right)
F_{18}\! \left(x \right) = F_{13}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)+F_{20}\! \left(x \right)+F_{41}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{24}\! \left(x \right)+F_{27}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{13}\! \left(x \right) F_{21}\! \left(x \right) F_{8}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{13}\! \left(x \right) F_{26}\! \left(x \right) F_{8}\! \left(x \right)
F_{26}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)+F_{24}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{13}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{32}\! \left(x \right)
F_{29}\! \left(x \right) = F_{1}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{13}\! \left(x \right) F_{29}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{36}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{13}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{1}\! \left(x \right)+F_{34}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{38}\! \left(x \right)+F_{40}\! \left(x \right)
F_{37}\! \left(x \right) = 0
F_{38}\! \left(x \right) = F_{13}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{30}\! \left(x \right)+F_{36}\! \left(x \right)
F_{40}\! \left(x \right) = F_{13}\! \left(x \right) F_{32}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{13}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{54}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{48}\! \left(x \right)
F_{45}\! \left(x \right) = F_{1}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{13}\! \left(x \right) F_{28}\! \left(x \right)
F_{48}\! \left(x \right) = F_{33}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{37}\! \left(x \right)+F_{50}\! \left(x \right)+F_{52}\! \left(x \right)
F_{50}\! \left(x \right) = F_{13}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{46}\! \left(x \right)+F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{13}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{32}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{13}\! \left(x \right) F_{35}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{45}\! \left(x \right)+F_{46}\! \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_13(x)*F_3(x))
Eq(F_3(x), F_1(x) + F_18(x) + F_4(x))
Eq(F_4(x), F_13(x)*F_5(x))
Eq(F_5(x), F_1(x) + F_11(x) + F_15(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_13(x)*F_5(x)*F_8(x))
Eq(F_8(x), F_1(x) + F_9(x))
Eq(F_9(x), F_10(x)*F_13(x))
Eq(F_10(x), F_1(x) + F_11(x) + F_14(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_13(x)*F_8(x))
Eq(F_13(x), x)
Eq(F_14(x), F_10(x)*F_13(x))
Eq(F_15(x), F_13(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_11(x) + F_15(x) + F_17(x))
Eq(F_17(x), F_13(x)*F_16(x))
Eq(F_18(x), F_13(x)*F_19(x))
Eq(F_19(x), F_1(x) + F_18(x) + F_20(x) + F_41(x))
Eq(F_20(x), F_13(x)*F_21(x))
Eq(F_21(x), F_1(x) + F_15(x) + F_22(x) + F_24(x) + F_27(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_13(x)*F_21(x)*F_8(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_13(x)*F_26(x)*F_8(x))
Eq(F_26(x), F_1(x) + F_14(x) + F_24(x) + F_27(x))
Eq(F_27(x), F_13(x)*F_28(x))
Eq(F_28(x), F_29(x) + F_32(x))
Eq(F_29(x), F_1(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_13(x)*F_29(x))
Eq(F_32(x), F_33(x) + F_36(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_13(x)*F_35(x))
Eq(F_35(x), F_1(x) + F_34(x))
Eq(F_36(x), F_37(x) + F_38(x) + F_40(x))
Eq(F_37(x), 0)
Eq(F_38(x), F_13(x)*F_39(x))
Eq(F_39(x), F_30(x) + F_36(x))
Eq(F_40(x), F_13(x)*F_32(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_13(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_54(x))
Eq(F_44(x), F_45(x) + F_48(x))
Eq(F_45(x), F_1(x) + F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_13(x)*F_28(x))
Eq(F_48(x), F_33(x) + F_49(x))
Eq(F_49(x), F_37(x) + F_50(x) + F_52(x))
Eq(F_50(x), F_13(x)*F_51(x))
Eq(F_51(x), F_46(x) + F_49(x))
Eq(F_52(x), F_13(x)*F_53(x))
Eq(F_53(x), F_32(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_13(x)*F_35(x)*F_56(x))
Eq(F_56(x), F_45(x) + F_46(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": false, "place_row": true, "strategy_class": "RowAndColumnPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "row_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, 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