0132_0321_1203_1320
Counting sequence:
1, 1, 2, 6, 20, 64, 196, 582, 1690, 4822, 13560, 37668, 103560, 282250, 763694, 2053914, 5496380, 14648248, 38907148, 103056270, 272358850, 718476766, 1892508672, 4978972236, 13086351120, 34368161554, 90202682966, 236625763362, 620479913540, 1626491698912, 4262484844180, 11168210590998, 29257179314410, 76634465865190, 200710642790664, 525630606493044, 1376456054595480, 3604304492976538, 9437625655438718, 24710977655025450, 64700255111962700, 169399958163419656, 443520510099224092, 1161204453087735966, 3040180810094205970, 7959518297101837294, 20838743517118238736, 54557468718252788892, 142835210750012036640, 373951330125271316002, 979025253550266256550, 2563137659849432854578, 6710414747595796530260, 17568161752033392024880, 45994183098495063806884, 120414617227032795291174, 315250136956964568598330, 825336748406981913049078, 2160762053819020194603288, 5656953376217750756797380, 14810106145284764323717800, 38773381488767982861925546, 101510071755742817860621262, 265756901800829242523909946, 695760771997325462532745820, 1821525695503994269144989784, 4768816886363723629898323756, 12484926125732053264251733422, 32685963852015677597679483490, 85572970226468438693270137342, 224032956567270509400774181920, 586525919250252736525691740908, 1535544841323602804568285357360, 4020108686181377506680544267570, 10524781382503356615910929924854, 27554235796616712624797055679554, 72137926687367554791709147886180, 188859545644416964748266790378368, 494440713041524297311921189757108, 1294462599146995706631071035328310, 3388947095884258108920269075936650, 8872378711777600647711347805575686, 23228189086596650799184312154331432, 60812188643517491499397293459206676, 159208377037383954837348236176276920, 416812942860326338567787267674425274, 1091230452336650398533210305454250526, 2856878415755078345480071192698128202, 7479404798178177241031126494250335340, 19581335985356009836316891000454475432, 51264603171197707690237384462278676540, 134212473555162309088852283365437506238, 351372817548758581304876197729595308210, 919905979201290098322175154289370474510, 2408345120277940920733017714620358475824, 6305129382083142838176756409634986170684, 16507043026882611462711291396610355465280, 43215999700406746328413763629245977070658, 113140956078061351160700423358024141412678, 296206868541303982592542618516213122449490, 775479649561062403819666185008575664400500
Generating function in Maple syntax:
(4*x^6-9*x^5+26*x^4-37*x^3+25*x^2-8*x+1)/(x^2-3*x+1)/(2*x-1)^2/(x-1)^2
Generating function in latex syntax:
\frac{4 x^{6}-9 x^{5}+26 x^{4}-37 x^{3}+25 x^{2}-8 x +1}{\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{2}}
Generating function in sympy syntax:
(4*x**6 - 9*x**5 + 26*x**4 - 37*x**3 + 25*x**2 - 8*x + 1)/((x - 1)**2*(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)^2*F(x)-4*x^6+9*x^5-26*x^4+37*x^3-25*x^2+8*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)^{2} F \! \left(x \right)-4 x^{6}+9 x^{5}-26 x^{4}+37 x^{3}-25 x^{2}+8 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,(5^(1/2)-1)*(3/2-1/2*5^(1/2))^(-n)+(-5^(1/2)-1)*(3/2+1/2*5^(1/2))^(-n)-2^(-1+n)*n-2*n+2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \left(\sqrt{5}-1\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}+\left(-\sqrt{5}-1\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}-2^{-1+n} n -2 n +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) = 20
a(5) = 64
a(6) = 196
a(n+4) = -4*a(n)+16*a(n+1)-17*a(n+2)+7*a(n+3)+2*n-4, 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) = 20
a \! \left(5\right) = 64
a \! \left(6\right) = 196
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)+2 n -4, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15429/
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[19,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[10,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[18,x]
F[15,x] = 0
F[16,x] = F[12,x]*F[17,x]
F[17,x] = F[10,x]
F[18,x] = F[12,x]*F[13,x]
F[19,x] = F[20,x]+F[29,x]
F[20,x] = F[21,x]
F[21,x] = F[12,x]*F[22,x]
F[22,x] = F[23,x]+F[9,x]
F[23,x] = F[20,x]+F[24,x]
F[24,x] = F[15,x]+F[25,x]+F[28,x]
F[25,x] = F[12,x]*F[26,x]
F[26,x] = F[17,x]+F[27,x]
F[27,x] = F[24,x]
F[28,x] = F[12,x]*F[23,x]
F[29,x] = F[15,x]+F[30,x]+F[54,x]
F[30,x] = F[12,x]*F[31,x]
F[31,x] = F[32,x]+F[39,x]
F[32,x] = F[33,x]+F[6,x]
F[33,x] = F[34,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[10,x]+F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[12,x]*F[36,x]
F[39,x] = F[29,x]+F[40,x]
F[40,x] = 2*F[15,x]+F[41,x]+F[45,x]
F[41,x] = F[12,x]*F[42,x]
F[42,x] = F[43,x]+F[44,x]
F[43,x] = F[33,x]
F[44,x] = F[40,x]
F[45,x] = F[12,x]*F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[24,x]+F[48,x]
F[48,x] = 2*F[15,x]+F[49,x]+F[53,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[51,x]+F[52,x]
F[51,x] = F[37,x]
F[52,x] = F[48,x]
F[53,x] = F[12,x]*F[47,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[56,x]+F[63,x]
F[56,x] = F[20,x]+F[57,x]
F[57,x] = F[15,x]+F[58,x]+F[62,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[60,x]+F[61,x]
F[60,x] = F[10,x]+F[37,x]
F[61,x] = F[48,x]+F[57,x]
F[62,x] = F[12,x]*F[56,x]
F[63,x] = F[24,x]+F[64,x]
F[64,x] = F[15,x]+F[49,x]+F[65,x]+F[67,x]
F[65,x] = F[12,x]*F[66,x]
F[66,x] = F[57,x]
F[67,x] = F[12,x]*F[63,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_{19}\! \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_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{10}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \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_{12}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{9}\! \left(x \right)
F_{23}\! \left(x \right) = F_{20}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{17}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{24}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{29}\! \left(x \right) = F_{15}\! \left(x \right)+F_{30}\! \left(x \right)+F_{54}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{39}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{6}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{10}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{39}\! \left(x \right) = F_{29}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{41}\! \left(x \right)+F_{45}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{44}\! \left(x \right)
F_{43}\! \left(x \right) = F_{33}\! \left(x \right)
F_{44}\! \left(x \right) = F_{40}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{24}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{49}\! \left(x \right)+F_{53}\! \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_{37}\! \left(x \right)
F_{52}\! \left(x \right) = F_{48}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \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_{63}\! \left(x \right)
F_{56}\! \left(x \right) = F_{20}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{15}\! \left(x \right)+F_{58}\! \left(x \right)+F_{62}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \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_{10}\! \left(x \right)+F_{37}\! \left(x \right)
F_{61}\! \left(x \right) = F_{48}\! \left(x \right)+F_{57}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \left(x \right)
F_{63}\! \left(x \right) = F_{24}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{15}\! \left(x \right)+F_{49}\! \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_{57}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \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_19(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_10(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_18(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_12(x)*F_17(x))
Eq(F_17(x), F_10(x))
Eq(F_18(x), F_12(x)*F_13(x))
Eq(F_19(x), F_20(x) + F_29(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_12(x)*F_22(x))
Eq(F_22(x), F_23(x) + F_9(x))
Eq(F_23(x), F_20(x) + F_24(x))
Eq(F_24(x), F_15(x) + F_25(x) + F_28(x))
Eq(F_25(x), F_12(x)*F_26(x))
Eq(F_26(x), F_17(x) + F_27(x))
Eq(F_27(x), F_24(x))
Eq(F_28(x), F_12(x)*F_23(x))
Eq(F_29(x), F_15(x) + F_30(x) + F_54(x))
Eq(F_30(x), F_12(x)*F_31(x))
Eq(F_31(x), F_32(x) + F_39(x))
Eq(F_32(x), F_33(x) + F_6(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_10(x) + F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_12(x)*F_36(x))
Eq(F_39(x), F_29(x) + F_40(x))
Eq(F_40(x), 2*F_15(x) + F_41(x) + F_45(x))
Eq(F_41(x), F_12(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_44(x))
Eq(F_43(x), F_33(x))
Eq(F_44(x), F_40(x))
Eq(F_45(x), F_12(x)*F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_24(x) + F_48(x))
Eq(F_48(x), 2*F_15(x) + F_49(x) + F_53(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_37(x))
Eq(F_52(x), F_48(x))
Eq(F_53(x), F_12(x)*F_47(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_63(x))
Eq(F_56(x), F_20(x) + F_57(x))
Eq(F_57(x), F_15(x) + F_58(x) + F_62(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_61(x))
Eq(F_60(x), F_10(x) + F_37(x))
Eq(F_61(x), F_48(x) + F_57(x))
Eq(F_62(x), F_12(x)*F_56(x))
Eq(F_63(x), F_24(x) + F_64(x))
Eq(F_64(x), F_15(x) + F_49(x) + F_65(x) + F_67(x))
Eq(F_65(x), F_12(x)*F_66(x))
Eq(F_66(x), F_57(x))
Eq(F_67(x), F_12(x)*F_63(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:
{"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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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