0132_0213_1203_1320_3021
Counting sequence:
1, 1, 2, 6, 19, 56, 154, 404, 1030, 2585, 6440, 16011, 39855, 99523, 249575, 628812, 1591909, 4048730, 10341424, 26516942, 68226532, 176065691, 455514302, 1181040261, 3067683789, 7980068341, 20784441149, 54188706054, 141395801395, 369193392140, 964506653830, 2520836908856, 6590756315890, 17236399653773, 45087304133060, 117961088236527, 308662816590939, 807752483629975, 2114027698616579, 5533162381115952, 14483054263046257, 37911052605698606, 99239933071493452, 259787855887854866, 680080753638588784, 1780366444097690351, 4660838258747527994, 12201778896237961929, 31943741965966448985, 83627898889289477545, 218936788108413989945, 573175991511488148426, 1500577957102145055775, 3928530858197182797248, 10284959448393968048770, 26926234896994037088092, 70493515559007147321694, 184553843405666158346945, 483167059894870325175584, 1264945390723905793127187, 3311665149109174968171207, 8670041986153086863459371, 22698444380218644974639327, 59425257719779214461898196, 155577260756750226607085629, 407306386199890912537723586, 1066341616530075386935425064, 2791717891541268963272453654, 7308810895948854859180186300, 19134712435122054179445500675, 50095321613263848514672897910, 131151242664788620445929942221, 343358386606192365806477598885, 898923877013673372581518540573, 2353413162974005922436698089205, 6161315446625517494290993250382, 16130532841614526276691471492299, 42230282398197287802554510457140, 110560312974046324133035657482814, 289450653728300726737722495486272, 757791642544016076636557572543882, 1983924262418952216832973062439861, 5193981121441018546280750001685052, 13598019054755996457038739129077751, 35600075947321831075279762583763699, 93202208593781365630459880669228671, 244006549442330300260960026819124859, 638817438940154197485223461180898104, 1672445765772678803746482812713771465, 4378519855128289610630101755350219030, 11463113793035633569440239742935292100, 30010821510670755675372779518290075866, 78569350712051437602221077832878987720, 205697230571014195402733721884785425623, 538522340890814485109581243355455237490, 1409869791878600052854641558699737930897, 3691087034294375499154465012680477342129, 9663391310093402575694713597015938671185, 25299086894143777449473029929317441830769, 66233869368614206134553952324039821160082, 173402521204172165515333714971415346372407
Generating function in Maple syntax:
(x^7-10*x^6+33*x^5-62*x^4+62*x^3-33*x^2+9*x-1)/(x^2-3*x+1)/(2*x-1)^2/(x-1)^3
Generating function in latex syntax:
\frac{x^{7}-10 x^{6}+33 x^{5}-62 x^{4}+62 x^{3}-33 x^{2}+9 x -1}{\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{3}}
Generating function in sympy syntax:
(x**7 - 10*x**6 + 33*x**5 - 62*x**4 + 62*x**3 - 33*x**2 + 9*x - 1)/((x - 1)**3*(2*x - 1)**2*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(2*x-1)^2*(x-1)^3*F(x)-x^7+10*x^6-33*x^5+62*x^4-62*x^3+33*x^2-9*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)-x^{7}+10 x^{6}-33 x^{5}+62 x^{4}-62 x^{3}+33 x^{2}-9 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/10*(5-5^(1/2))*(3/2-1/2*5^(1/2))^(-n)+1/10*(5+5^(1/2))*(3/2+1/2*5^(1/2))^(-n)-1/2*(n-1)*(n-2^(n-1)))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(5-\sqrt{5}\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{10}+\frac{\left(5+\sqrt{5}\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{10}-\frac{\left(n -1\right) \left(n -2^{n -1}\right)}{2} & \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) = 56
a(6) = 154
a(7) = 404
a(n+4) = -4*a(n)+16*a(n+1)-17*a(n+2)+7*a(n+3)+1/2*(n+1)*(n-4), 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) = 56
a \! \left(6\right) = 154
a \! \left(7\right) = 404
a \! \left(n +4\right) = -4 a \! \left(n \right)+16 a \! \left(n +1\right)-17 a \! \left(n +2\right)+7 a \! \left(n +3\right)+\frac{\left(n +1\right) \left(n -4\right)}{2}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18398/
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[16,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[6,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[13,x]
F[16,x] = F[17,x]+F[28,x]
F[17,x] = F[18,x]
F[18,x] = F[12,x]*F[19,x]
F[19,x] = F[20,x]+F[9,x]
F[20,x] = F[17,x]+F[21,x]
F[21,x] = F[22,x]+F[23,x]+F[27,x]
F[22,x] = 0
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[25,x]+F[26,x]
F[25,x] = F[10,x]
F[26,x] = F[21,x]
F[27,x] = F[12,x]*F[20,x]
F[28,x] = F[22,x]+F[29,x]+F[57,x]
F[29,x] = F[12,x]*F[30,x]
F[30,x] = F[31,x]+F[38,x]
F[31,x] = F[32,x]+F[6,x]
F[32,x] = F[33,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[10,x]+F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[12,x]*F[35,x]
F[38,x] = F[28,x]+F[39,x]
F[39,x] = 2*F[22,x]+F[40,x]+F[44,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[42,x]+F[43,x]
F[42,x] = F[32,x]
F[43,x] = F[39,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[47,x]+F[51,x]
F[47,x] = F[48,x]
F[48,x] = F[12,x]*F[49,x]
F[49,x] = F[25,x]+F[50,x]
F[50,x] = F[47,x]
F[51,x] = 2*F[22,x]+F[52,x]+F[56,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[54,x]+F[55,x]
F[54,x] = F[36,x]
F[55,x] = F[51,x]
F[56,x] = F[12,x]*F[46,x]
F[57,x] = F[12,x]*F[58,x]
F[58,x] = F[59,x]+F[70,x]
F[59,x] = F[17,x]+F[60,x]
F[60,x] = F[22,x]+F[61,x]+F[69,x]
F[61,x] = F[12,x]*F[62,x]
F[62,x] = F[63,x]+F[64,x]
F[63,x] = F[10,x]+F[36,x]
F[64,x] = F[60,x]+F[65,x]
F[65,x] = 2*F[22,x]+F[56,x]+F[66,x]
F[66,x] = F[12,x]*F[67,x]
F[67,x] = F[54,x]+F[68,x]
F[68,x] = F[65,x]
F[69,x] = F[12,x]*F[59,x]
F[70,x] = F[71,x]+F[72,x]
F[71,x] = F[48,x]
F[72,x] = 2*F[22,x]+F[52,x]+F[73,x]
F[73,x] = F[12,x]*F[70,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_{16}\! \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_{6}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{28}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{9}\! \left(x \right)
F_{20}\! \left(x \right) = F_{17}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{27}\! \left(x \right)
F_{22}\! \left(x \right) = 0
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)
F_{25}\! \left(x \right) = F_{10}\! \left(x \right)
F_{26}\! \left(x \right) = F_{21}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{28}\! \left(x \right) = F_{22}\! \left(x \right)+F_{29}\! \left(x \right)+F_{57}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{38}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{6}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
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_{35}\! \left(x \right) = F_{10}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{38}\! \left(x \right) = F_{28}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{40}\! \left(x \right)+F_{44}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \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_{32}\! \left(x \right)
F_{43}\! \left(x \right) = F_{39}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{51}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{25}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{47}\! \left(x \right)
F_{51}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{52}\! \left(x \right)+F_{56}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{55}\! \left(x \right)
F_{54}\! \left(x \right) = F_{36}\! \left(x \right)
F_{55}\! \left(x \right) = F_{51}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{70}\! \left(x \right)
F_{59}\! \left(x \right) = F_{17}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{22}\! \left(x \right)+F_{61}\! \left(x \right)+F_{69}\! \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_{64}\! \left(x \right)
F_{63}\! \left(x \right) = F_{10}\! \left(x \right)+F_{36}\! \left(x \right)
F_{64}\! \left(x \right) = F_{60}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{56}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{54}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{65}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)+F_{72}\! \left(x \right)
F_{71}\! \left(x \right) = F_{48}\! \left(x \right)
F_{72}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{52}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \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_16(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_6(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_13(x))
Eq(F_16(x), F_17(x) + F_28(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_12(x)*F_19(x))
Eq(F_19(x), F_20(x) + F_9(x))
Eq(F_20(x), F_17(x) + F_21(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_27(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_25(x) + F_26(x))
Eq(F_25(x), F_10(x))
Eq(F_26(x), F_21(x))
Eq(F_27(x), F_12(x)*F_20(x))
Eq(F_28(x), F_22(x) + F_29(x) + F_57(x))
Eq(F_29(x), F_12(x)*F_30(x))
Eq(F_30(x), F_31(x) + F_38(x))
Eq(F_31(x), F_32(x) + F_6(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_10(x) + F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_12(x)*F_35(x))
Eq(F_38(x), F_28(x) + F_39(x))
Eq(F_39(x), 2*F_22(x) + F_40(x) + F_44(x))
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_43(x))
Eq(F_42(x), F_32(x))
Eq(F_43(x), F_39(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_47(x) + F_51(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_12(x)*F_49(x))
Eq(F_49(x), F_25(x) + F_50(x))
Eq(F_50(x), F_47(x))
Eq(F_51(x), 2*F_22(x) + F_52(x) + F_56(x))
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_55(x))
Eq(F_54(x), F_36(x))
Eq(F_55(x), F_51(x))
Eq(F_56(x), F_12(x)*F_46(x))
Eq(F_57(x), F_12(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_70(x))
Eq(F_59(x), F_17(x) + F_60(x))
Eq(F_60(x), F_22(x) + F_61(x) + F_69(x))
Eq(F_61(x), F_12(x)*F_62(x))
Eq(F_62(x), F_63(x) + F_64(x))
Eq(F_63(x), F_10(x) + F_36(x))
Eq(F_64(x), F_60(x) + F_65(x))
Eq(F_65(x), 2*F_22(x) + F_56(x) + F_66(x))
Eq(F_66(x), F_12(x)*F_67(x))
Eq(F_67(x), F_54(x) + F_68(x))
Eq(F_68(x), F_65(x))
Eq(F_69(x), F_12(x)*F_59(x))
Eq(F_70(x), F_71(x) + F_72(x))
Eq(F_71(x), F_48(x))
Eq(F_72(x), 2*F_22(x) + F_52(x) + F_73(x))
Eq(F_73(x), F_12(x)*F_70(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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