0132_0213_2130_2301_3120
Counting sequence:
1, 1, 2, 6, 19, 49, 103, 192, 335, 567, 955, 1630, 2851, 5133, 9503, 18012, 34759, 67939, 133939, 265530, 528251, 1053177, 2102455, 4200376, 8395519, 16785039, 33563243, 67118742, 134228755, 268447717, 536884495, 1073756820, 2147500151, 4294985403, 8589954403, 17179890802, 34359761899, 68719502289, 137438981159, 274877936880, 549755846191, 1099511662567, 2199023292955, 4398046551246, 8796093065219, 17592186090429, 35184372137983, 70368744230092, 140737488411175, 281474976770067, 562949953484435, 1125899906909610, 2251799813756251, 4503599627445673, 9007199254820503, 18014398509565992, 36028797019052639, 72057594038021439, 144115188075954379, 288230376151815430, 576460752303532531, 1152921504606961557, 2305843009213814255, 4611686018427514116, 9223372036854908119, 18446744073709690219, 36893488147419248323, 73786976294838358242, 147573952589676571595, 295147905179352991617, 590295810358705824775, 1180591620717411484000, 2361183241434822795151, 4722366482869645409943, 9444732965739290631803, 18889465931478581067582, 37778931862957161930979, 75557863725914323649389, 151115727451828647077599, 302231454903657293925180, 604462909807314587611271, 1208925819614629174974147, 2417851639229258349690355, 4835703278458516699112986, 9671406556917033397948219, 19342813113834066795608409, 38685626227668133590918263, 77371252455336267181527192, 154742504910672534362734015, 309485009821345068725136367, 618970019642690137449929515, 1237940039285380274899503990, 2475880078570760549798640851, 4951760157141521099596902213, 9903520314283042199193412303, 19807040628566084398386419572, 39614081257132168796772420919, 79228162514264337593544410139, 158456325028528675187088374819, 316912650057057350374176290130, 633825300114114700748352106411
Generating function in Maple syntax:
-(6*x^7-5*x^6-6*x^5+4*x^4-8*x^3+10*x^2-5*x+1)/(2*x-1)/(x-1)^4
Generating function in latex syntax:
-\frac{6 x^{7}-5 x^{6}-6 x^{5}+4 x^{4}-8 x^{3}+10 x^{2}-5 x +1}{\left(2 x -1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(-6*x**7 + 5*x**6 + 6*x**5 - 4*x**4 + 8*x**3 - 10*x**2 + 5*x - 1)/((x - 1)**4*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^4*F(x)+6*x^7-5*x^6-6*x^5+4*x^4-8*x^3+10*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+6 x^{7}-5 x^{6}-6 x^{5}+4 x^{4}-8 x^{3}+10 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,23-13*n+1/2*n^3+1/2*n^2+2^(-1+n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 23-13 n +\frac{n^{3}}{2}+\frac{n^{2}}{2}+2^{-1+n} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 49
a(6) = 103
a(7) = 192
a(n+1) = -1/2*n^3+n^2+2*a(n)+31/2*n-35, 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) = 19
a \! \left(5\right) = 49
a \! \left(6\right) = 103
a \! \left(7\right) = 192
a \! \left(n +1\right) = -\frac{n^{3}}{2}+n^{2}+2 a \! \left(n \right)+\frac{31 n}{2}-35, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18571/
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[12,x]*F[4,x]
F[4,x] = F[25,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[18,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[9,x]
F[17,x] = F[14,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[21,x]+F[24,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[1,x]+F[21,x]
F[24,x] = F[19,x]
F[25,x] = F[26,x]+F[31,x]
F[26,x] = F[27,x]
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[29,x]+F[30,x]
F[29,x] = F[1,x]+F[14,x]
F[30,x] = F[26,x]
F[31,x] = F[32,x]+F[33,x]+F[61,x]
F[32,x] = 0
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[35,x]+F[60,x]
F[35,x] = F[36,x]+F[6,x]
F[36,x] = F[32,x]+F[37,x]+F[43,x]
F[37,x] = F[12,x]*F[38,x]
F[38,x] = F[39,x]+F[42,x]
F[39,x] = F[10,x]+F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[12,x]*F[39,x]
F[42,x] = F[18,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[20,x]+F[45,x]
F[45,x] = F[46,x]+F[53,x]
F[46,x] = F[32,x]+F[47,x]+F[49,x]
F[47,x] = F[12,x]*F[48,x]
F[48,x] = F[10,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[51,x]+F[52,x]
F[51,x] = F[21,x]
F[52,x] = F[46,x]
F[53,x] = F[54,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[56,x]+F[59,x]
F[56,x] = F[57,x]
F[57,x] = F[12,x]*F[58,x]
F[58,x] = F[21,x]+F[56,x]
F[59,x] = F[54,x]
F[60,x] = F[31,x]
F[61,x] = F[12,x]*F[62,x]
F[62,x] = F[63,x]+F[68,x]
F[63,x] = F[21,x]+F[64,x]
F[64,x] = F[32,x]+F[65,x]+F[67,x]
F[65,x] = F[12,x]*F[66,x]
F[66,x] = F[10,x]+F[64,x]
F[67,x] = F[12,x]*F[63,x]
F[68,x] = F[69,x]+F[75,x]
F[69,x] = F[32,x]+F[70,x]+F[72,x]
F[70,x] = F[12,x]*F[71,x]
F[71,x] = F[14,x]
F[72,x] = F[12,x]*F[73,x]
F[73,x] = F[51,x]+F[74,x]
F[74,x] = F[69,x]
F[75,x] = 2*F[32,x]+F[76,x]+F[78,x]
F[76,x] = F[12,x]*F[77,x]
F[77,x] = F[18,x]
F[78,x] = F[12,x]*F[79,x]
F[79,x] = F[56,x]+F[80,x]
F[80,x] = F[78,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{25}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{9}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{24}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{1}\! \left(x \right)+F_{21}\! \left(x \right)
F_{24}\! \left(x \right) = F_{19}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{31}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)
F_{29}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{30}\! \left(x \right) = F_{26}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)+F_{61}\! \left(x \right)
F_{32}\! \left(x \right) = 0
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{60}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{6}\! \left(x \right)
F_{36}\! \left(x \right) = F_{32}\! \left(x \right)+F_{37}\! \left(x \right)+F_{43}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{42}\! \left(x \right)
F_{39}\! \left(x \right) = F_{10}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{42}\! \left(x \right) = F_{18}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{20}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{53}\! \left(x \right)
F_{46}\! \left(x \right) = F_{32}\! \left(x \right)+F_{47}\! \left(x \right)+F_{49}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{10}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{52}\! \left(x \right)
F_{51}\! \left(x \right) = F_{21}\! \left(x \right)
F_{52}\! \left(x \right) = F_{46}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{59}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{21}\! \left(x \right)+F_{56}\! \left(x \right)
F_{59}\! \left(x \right) = F_{54}\! \left(x \right)
F_{60}\! \left(x \right) = F_{31}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{68}\! \left(x \right)
F_{63}\! \left(x \right) = F_{21}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{32}\! \left(x \right)+F_{65}\! \left(x \right)+F_{67}\! \left(x \right)
F_{65}\! \left(x \right) = F_{12}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{10}\! \left(x \right)+F_{64}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)+F_{75}\! \left(x \right)
F_{69}\! \left(x \right) = F_{32}\! \left(x \right)+F_{70}\! \left(x \right)+F_{72}\! \left(x \right)
F_{70}\! \left(x \right) = F_{12}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{14}\! \left(x \right)
F_{72}\! \left(x \right) = F_{12}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{51}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{69}\! \left(x \right)
F_{75}\! \left(x \right) = 2 F_{32}\! \left(x \right)+F_{76}\! \left(x \right)+F_{78}\! \left(x \right)
F_{76}\! \left(x \right) = F_{12}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{18}\! \left(x \right)
F_{78}\! \left(x \right) = F_{12}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{56}\! \left(x \right)+F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{78}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_25(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_18(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_9(x))
Eq(F_17(x), F_14(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_21(x) + F_24(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_12(x)*F_23(x))
Eq(F_23(x), F_1(x) + F_21(x))
Eq(F_24(x), F_19(x))
Eq(F_25(x), F_26(x) + F_31(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_29(x) + F_30(x))
Eq(F_29(x), F_1(x) + F_14(x))
Eq(F_30(x), F_26(x))
Eq(F_31(x), F_32(x) + F_33(x) + F_61(x))
Eq(F_32(x), 0)
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_60(x))
Eq(F_35(x), F_36(x) + F_6(x))
Eq(F_36(x), F_32(x) + F_37(x) + F_43(x))
Eq(F_37(x), F_12(x)*F_38(x))
Eq(F_38(x), F_39(x) + F_42(x))
Eq(F_39(x), F_10(x) + F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_12(x)*F_39(x))
Eq(F_42(x), F_18(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_20(x) + F_45(x))
Eq(F_45(x), F_46(x) + F_53(x))
Eq(F_46(x), F_32(x) + F_47(x) + F_49(x))
Eq(F_47(x), F_12(x)*F_48(x))
Eq(F_48(x), F_10(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_52(x))
Eq(F_51(x), F_21(x))
Eq(F_52(x), F_46(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_59(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_12(x)*F_58(x))
Eq(F_58(x), F_21(x) + F_56(x))
Eq(F_59(x), F_54(x))
Eq(F_60(x), F_31(x))
Eq(F_61(x), F_12(x)*F_62(x))
Eq(F_62(x), F_63(x) + F_68(x))
Eq(F_63(x), F_21(x) + F_64(x))
Eq(F_64(x), F_32(x) + F_65(x) + F_67(x))
Eq(F_65(x), F_12(x)*F_66(x))
Eq(F_66(x), F_10(x) + F_64(x))
Eq(F_67(x), F_12(x)*F_63(x))
Eq(F_68(x), F_69(x) + F_75(x))
Eq(F_69(x), F_32(x) + F_70(x) + F_72(x))
Eq(F_70(x), F_12(x)*F_71(x))
Eq(F_71(x), F_14(x))
Eq(F_72(x), F_12(x)*F_73(x))
Eq(F_73(x), F_51(x) + F_74(x))
Eq(F_74(x), F_69(x))
Eq(F_75(x), 2*F_32(x) + F_76(x) + F_78(x))
Eq(F_76(x), F_12(x)*F_77(x))
Eq(F_77(x), F_18(x))
Eq(F_78(x), F_12(x)*F_79(x))
Eq(F_79(x), F_56(x) + F_80(x))
Eq(F_80(x), F_78(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "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": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
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