0213_0321_1302_3120
Counting sequence:
1, 1, 2, 6, 20, 62, 178, 480, 1232, 3042, 7286, 17036, 39076, 88254, 196826, 434424, 950552, 2064698, 4456798, 9568644, 20447660, 43516374, 92275202, 195035696, 411042400, 864027282, 1811940038, 3791651580, 7918846772, 16508781422, 34359739306, 71403832296, 148176372776, 307090162794, 635655161006, 1314259993844, 2714419332412, 5600637355398, 11544872093138, 23776938952224, 48928267437680, 100605313943234, 206708186023702, 424411488323436, 870813209200580, 1785606883510302, 3659174697240698, 7494271254923480, 15340386230733112, 31384459903240602, 64176294690032126, 131167339147168356, 267964177828547276, 547187354725518134, 1116892707587885986, 2278821411449474064, 4647714815446355072, 9475573615987526898, 19311435202164690278, 39343446344708656604, 80128044570175868500, 163138392901868850894, 332041393326771933002, 675612001699612331976, 1374282433491361599560, 2794681727166997074122, 5681597174702541902158, 11547661790142179316180, 23464258461758549660252, 47666386686465481380582, 96808512898827726885746, 196568504849448982025216, 399039967802485020562576, 809885851812144154154274, 1643383536038636534371766, 3333990736905969520875084, 6762428803469331946018532, 13713752266253449700579198, 27805293851136471018248218, 56366166339532085270681784, 114243489953582457009740120, 231509294456201486956239354, 469063218010476119786003102, 950215694217098531319061316, 1924609904826489646132239340, 3897576842437564459252718742, 7891867750444299252481924418, 15977163632026939172916829680, 32341183526330559681739628192, 65456079577214482035291201362, 132459584203535689414206300166, 268014018505284829515660402876, 542217737206996560405816418676, 1096814874806846923560624071214, 2218388550399401452619230618346, 4486294702370218116234426196904, 9071624607883266654460782322792, 18341319622052194152905424512298, 37078780056675709993778568766958, 74949841738494063363492577027764, 151484246727273413478856033052540
Generating function in Maple syntax:
-(2*x^6-2*x^5+7*x^4-14*x^3+14*x^2-6*x+1)/(2*x-1)^2/(x-1)^3
Generating function in latex syntax:
-\frac{2 x^{6}-2 x^{5}+7 x^{4}-14 x^{3}+14 x^{2}-6 x +1}{\left(2 x -1\right)^{2} \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-2*x**6 + 2*x**5 - 7*x**4 + 14*x**3 - 14*x**2 + 6*x - 1)/((x - 1)**3*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^3*F(x)+2*x^6-2*x^5+7*x^4-14*x^3+14*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)+2 x^{6}-2 x^{5}+7 x^{4}-14 x^{3}+14 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,8+5/4*2^n*n-11*2^(-1+n)+n+n^2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 8+\frac{5 \,2^{n} n}{4}-11 \,2^{-1+n}+n +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) = 62
a(6) = 178
a(n+2) = n^2-4*a(n)+4*a(n+1)-3*n+6, 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) = 62
a \! \left(6\right) = 178
a \! \left(n +2\right) = n^{2}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-3 n +6, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21369/
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[8,x]
F[4,x] = F[34,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[8,x]*F[9,x]
F[8,x] = x
F[9,x] = F[10,x]+F[18,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[8,x]
F[13,x] = F[10,x]+F[14,x]
F[14,x] = F[15,x]+F[8,x]
F[15,x] = F[16,x]
F[16,x] = F[17,x]*F[8,x]
F[17,x] = F[14,x]
F[18,x] = F[19,x]+F[22,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[8,x]
F[21,x] = F[1,x]+F[19,x]
F[22,x] = F[23,x]+F[24,x]+F[26,x]
F[23,x] = 0
F[24,x] = F[25,x]*F[8,x]
F[25,x] = F[11,x]+F[22,x]
F[26,x] = F[27,x]*F[8,x]
F[27,x] = F[18,x]+F[28,x]
F[28,x] = F[29,x]+F[31,x]
F[29,x] = F[30,x]
F[30,x] = F[19,x]*F[8,x]
F[31,x] = F[32,x]
F[32,x] = F[33,x]*F[8,x]
F[33,x] = F[28,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[8,x]
F[36,x] = F[37,x]+F[47,x]
F[37,x] = F[38,x]+F[4,x]
F[38,x] = F[39,x]
F[39,x] = F[40,x]*F[44,x]
F[40,x] = F[41,x]+F[6,x]
F[41,x] = F[1,x]+F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[21,x]*F[41,x]*F[8,x]
F[44,x] = F[45,x]
F[45,x] = F[21,x]*F[46,x]*F[8,x]
F[46,x] = F[1,x]+F[44,x]
F[47,x] = F[48,x]
F[48,x] = F[49,x]*F[8,x]
F[49,x] = F[50,x]+F[64,x]
F[50,x] = F[51,x]+F[61,x]
F[51,x] = F[46,x]*F[52,x]
F[52,x] = F[53,x]+F[56,x]
F[53,x] = F[41,x]*F[54,x]
F[54,x] = F[1,x]+F[55,x]
F[55,x] = F[54,x]*F[8,x]
F[56,x] = F[57,x]+F[58,x]
F[57,x] = F[42,x]*F[54,x]
F[58,x] = F[59,x]
F[59,x] = F[21,x]*F[42,x]*F[60,x]*F[8,x]
F[60,x] = F[55,x]
F[61,x] = F[54,x]*F[62,x]
F[62,x] = F[63,x]
F[63,x] = F[42,x]*F[44,x]*F[54,x]*F[8,x]
F[64,x] = F[44,x]*F[65,x]
F[65,x] = F[52,x]+F[57,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_{8}\! \left(x \right)
F_{4}\! \left(x \right) = F_{34}\! \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_{8}\! \left(x \right) F_{9}\! \left(x \right)
F_{8}\! \left(x \right) = x
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{18}\! \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_{13}\! \left(x \right) F_{8}\! \left(x \right)
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_{8}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{8}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{22}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{8}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{26}\! \left(x \right)
F_{23}\! \left(x \right) = 0
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{8}\! \left(x \right)
F_{25}\! \left(x \right) = F_{11}\! \left(x \right)+F_{22}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{8}\! \left(x \right)
F_{27}\! \left(x \right) = F_{18}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{31}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{19}\! \left(x \right) F_{8}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{8}\! \left(x \right)
F_{33}\! \left(x \right) = F_{28}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{8}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{47}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right) F_{44}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{6}\! \left(x \right)
F_{41}\! \left(x \right) = F_{1}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{21}\! \left(x \right) F_{41}\! \left(x \right) F_{8}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{21}\! \left(x \right) F_{46}\! \left(x \right) F_{8}\! \left(x \right)
F_{46}\! \left(x \right) = F_{1}\! \left(x \right)+F_{44}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right) F_{8}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{64}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{61}\! \left(x \right)
F_{51}\! \left(x \right) = F_{46}\! \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_{41}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{1}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{54}\! \left(x \right) F_{8}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{58}\! \left(x \right)
F_{57}\! \left(x \right) = F_{42}\! \left(x \right) F_{54}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{21}\! \left(x \right) F_{42}\! \left(x \right) F_{60}\! \left(x \right) F_{8}\! \left(x \right)
F_{60}\! \left(x \right) = F_{55}\! \left(x \right)
F_{61}\! \left(x \right) = F_{54}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{42}\! \left(x \right) F_{44}\! \left(x \right) F_{54}\! \left(x \right) F_{8}\! \left(x \right)
F_{64}\! \left(x \right) = F_{44}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{52}\! \left(x \right)+F_{57}\! \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_8(x))
Eq(F_4(x), F_34(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_8(x)*F_9(x))
Eq(F_8(x), x)
Eq(F_9(x), F_10(x) + F_18(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_8(x))
Eq(F_13(x), F_10(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_8(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_17(x)*F_8(x))
Eq(F_17(x), F_14(x))
Eq(F_18(x), F_19(x) + F_22(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x)*F_8(x))
Eq(F_21(x), F_1(x) + F_19(x))
Eq(F_22(x), F_23(x) + F_24(x) + F_26(x))
Eq(F_23(x), 0)
Eq(F_24(x), F_25(x)*F_8(x))
Eq(F_25(x), F_11(x) + F_22(x))
Eq(F_26(x), F_27(x)*F_8(x))
Eq(F_27(x), F_18(x) + F_28(x))
Eq(F_28(x), F_29(x) + F_31(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_19(x)*F_8(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_33(x)*F_8(x))
Eq(F_33(x), F_28(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_8(x))
Eq(F_36(x), F_37(x) + F_47(x))
Eq(F_37(x), F_38(x) + F_4(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_40(x)*F_44(x))
Eq(F_40(x), F_41(x) + F_6(x))
Eq(F_41(x), F_1(x) + F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_21(x)*F_41(x)*F_8(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_21(x)*F_46(x)*F_8(x))
Eq(F_46(x), F_1(x) + F_44(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_49(x)*F_8(x))
Eq(F_49(x), F_50(x) + F_64(x))
Eq(F_50(x), F_51(x) + F_61(x))
Eq(F_51(x), F_46(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_56(x))
Eq(F_53(x), F_41(x)*F_54(x))
Eq(F_54(x), F_1(x) + F_55(x))
Eq(F_55(x), F_54(x)*F_8(x))
Eq(F_56(x), F_57(x) + F_58(x))
Eq(F_57(x), F_42(x)*F_54(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_21(x)*F_42(x)*F_60(x)*F_8(x))
Eq(F_60(x), F_55(x))
Eq(F_61(x), F_54(x)*F_62(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_42(x)*F_44(x)*F_54(x)*F_8(x))
Eq(F_64(x), F_44(x)*F_65(x))
Eq(F_65(x), F_52(x) + F_57(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": 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, 2, 1, 3], "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, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 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, 2, 1, 3], "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, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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