0132_0231_1320_2013_3120
Counting sequence:
1, 1, 2, 6, 19, 53, 134, 316, 709, 1535, 3240, 6718, 13759, 27945, 56442, 113584, 228041, 457155, 915612, 1832786, 3667427, 7337037, 14676622, 29356196, 58715789, 117435463, 234875344, 469755686, 939516999, 1879040305, 3758087650, 7516183128, 15032374929, 30064759435, 60129529412, 120259070394, 240518153451, 481036320725, 962072656502, 1924145329356, 3848290676437, 7696581372047, 15393162764792, 30786325551886, 61572651127759, 123145302281273, 246290604590154, 492581209209856, 985162418451289, 1970324836936275, 3940649673908460, 7881299347855138, 15762598695750899, 31525197391544925, 63050394783135582, 126100789566319604, 252201579132690461, 504403158265435095, 1008806316530927392, 2017612633061915126, 4035225266123893847, 8070450532247854657, 16140901064495779762, 32281802128991633576, 64563604257983344929, 129127208515966771483, 258254417031933628564, 516508834063867346826, 1033017668127734787579, 2066035336255469673445, 4132070672510939449670, 8264141345021879006748, 16528282690043758125669, 33056565380087516368415, 66113130760175032858952, 132226261520350065845214, 264452523040700131823071, 528905046081400263784265, 1057810092162800527712282, 2115620184325601055574096, 4231240368651202111303657, 8462480737302404222768867, 16924961474604808445705532, 33849922949209616891585266, 67699845898419233783351299, 135399691796838467566890093, 270799383593676935133974574, 541598767187353870268150596, 1083197534374707740536509869, 2166395068749415481073235815, 4332790137498830962146695280, 8665580274997661924293621958, 17331160549995323848587483239, 34662321099990647697175213905, 69324642199981295394350683522, 138649284399962590788701631224, 277298568799925181577403535281, 554597137599850363154807352235, 1109194275199700726309614995172, 2218388550399401452619230290266, 4436777100798802905238460889867
Generating function in Maple syntax:
-(2*x^6-2*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{2 x^{6}-2 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:
(-2*x**6 + 2*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)+2*x^6-2*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)+2 x^{6}-2 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),1,1+7*2^(-1+n)-37/6*n+1/2*n^2-1/3*n^3)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 1+7 \,2^{-1+n}-\frac{37 n}{6}+\frac{n^{2}}{2}-\frac{n^{3}}{3} & \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) = 53
a(6) = 134
a(n+1) = 2*a(n)+1/6*(2*n-3)*(n^2-3*n+14), n >= 7
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) = 53
a \! \left(6\right) = 134
a \! \left(n +1\right) = 2 a \! \left(n \right)+\frac{\left(2 n -3\right) \left(n^{2}-3 n +14\right)}{6}, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19144/
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[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[19,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[7,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[11,x]
F[18,x] = F[14,x]
F[19,x] = F[20,x]+F[29,x]
F[20,x] = F[21,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[6,x]
F[23,x] = F[20,x]+F[24,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[28,x]
F[27,x] = F[7,x]
F[28,x] = F[24,x]
F[29,x] = F[30,x]+F[31,x]+F[47,x]
F[30,x] = 0
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[38,x]
F[33,x] = F[34,x]+F[7,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]
F[37,x] = F[7,x]
F[38,x] = F[29,x]+F[39,x]
F[39,x] = 2*F[30,x]+F[40,x]+F[44,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[43,x]
F[42,x] = F[34,x]
F[43,x] = F[39,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[24,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[64,x]
F[49,x] = F[20,x]+F[50,x]
F[50,x] = F[30,x]+F[51,x]+F[63,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[53,x]+F[56,x]
F[53,x] = F[11,x]+F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[37,x]*F[4,x]
F[56,x] = F[50,x]+F[57,x]
F[57,x] = 2*F[30,x]+F[58,x]+F[62,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[60,x]+F[61,x]
F[60,x] = F[54,x]
F[61,x] = F[57,x]
F[62,x] = F[4,x]*F[46,x]
F[63,x] = F[4,x]*F[49,x]
F[64,x] = F[24,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_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{19}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \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_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{29}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{6}\! \left(x \right)
F_{23}\! \left(x \right) = F_{20}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)
F_{27}\! \left(x \right) = F_{7}\! \left(x \right)
F_{28}\! \left(x \right) = F_{24}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{31}\! \left(x \right)+F_{47}\! \left(x \right)
F_{30}\! \left(x \right) = 0
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{38}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{7}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{7}\! \left(x \right)
F_{38}\! \left(x \right) = F_{29}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = 2 F_{30}\! \left(x \right)+F_{40}\! \left(x \right)+F_{44}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{43}\! \left(x \right)
F_{42}\! \left(x \right) = F_{34}\! \left(x \right)
F_{43}\! \left(x \right) = F_{39}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{24}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{64}\! \left(x \right)
F_{49}\! \left(x \right) = F_{20}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{30}\! \left(x \right)+F_{51}\! \left(x \right)+F_{63}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{56}\! \left(x \right)
F_{53}\! \left(x \right) = F_{11}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{56}\! \left(x \right) = F_{50}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = 2 F_{30}\! \left(x \right)+F_{58}\! \left(x \right)+F_{62}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{61}\! \left(x \right)
F_{60}\! \left(x \right) = F_{54}\! \left(x \right)
F_{61}\! \left(x \right) = F_{57}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{64}\! \left(x \right) = F_{24}\! \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_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_19(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_7(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_11(x))
Eq(F_18(x), F_14(x))
Eq(F_19(x), F_20(x) + F_29(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_6(x))
Eq(F_23(x), F_20(x) + F_24(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_28(x))
Eq(F_27(x), F_7(x))
Eq(F_28(x), F_24(x))
Eq(F_29(x), F_30(x) + F_31(x) + F_47(x))
Eq(F_30(x), 0)
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_38(x))
Eq(F_33(x), F_34(x) + F_7(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_7(x))
Eq(F_38(x), F_29(x) + F_39(x))
Eq(F_39(x), 2*F_30(x) + F_40(x) + F_44(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_43(x))
Eq(F_42(x), F_34(x))
Eq(F_43(x), F_39(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_24(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_64(x))
Eq(F_49(x), F_20(x) + F_50(x))
Eq(F_50(x), F_30(x) + F_51(x) + F_63(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_56(x))
Eq(F_53(x), F_11(x) + F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_37(x)*F_4(x))
Eq(F_56(x), F_50(x) + F_57(x))
Eq(F_57(x), 2*F_30(x) + F_58(x) + F_62(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_61(x))
Eq(F_60(x), F_54(x))
Eq(F_61(x), F_57(x))
Eq(F_62(x), F_4(x)*F_46(x))
Eq(F_63(x), F_4(x)*F_49(x))
Eq(F_64(x), F_24(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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