0132_0213_1320_2301_3021
Counting sequence:
1, 1, 2, 6, 19, 52, 127, 300, 716, 1753, 4399, 11239, 29039, 75511, 197024, 514972, 1347167, 3525650, 9228731, 24159298, 63247704, 165582119, 433496699, 1134905741, 2971217979, 7778745317, 20365014732, 53316295250, 139583866971, 365435301168, 956722031559, 2504730788024, 6557470326484, 17167680184821, 44945570220759, 117669030469587, 308061521179447, 806515533059475, 2111485077988936, 5527939700896488, 14472334024688839, 37889062373157454, 99194853094770019, 259695496911138126, 679891637638628864, 1779979416004731907, 4660046610375549187, 12200160415121896825, 31940434634990121251, 83621143489848445633, 218922995834555193044, 573147844013817109534, 1500520536206896110179, 3928413764606871194156, 10284720757613717443919, 26925748508234281107652, 70492524767089125847452, 184551825793033096401425, 483162952612010163321791, 1264937032042997393527103, 3311648143516982017220799, 8670007398507948658094639, 22698374052006863957020464, 59425114757512643212922036, 155576970220531065681698799, 407305795904080553832125322, 1066340417491710595814625867, 2791715456571051233611698650, 7308805952221443105020414056, 19134702400093278081449485023, 50095301248058391139327979979, 131151201344081895336534391269, 343358302784187294870275127499, 898923707008479989274290922141, 2353412818241252672952597567004, 6161314747715278029583501704042, 16130531424904581415797907467307, 42230279526998466217810220617000, 110560307156090817237632754299671, 289450641941273985495088042194768, 757791618667731139247631372194084, 1983924214061919432247806074293549, 5193981023518027157495786850589159, 13598018856492162040239554477372971, 35600075545958458963222876581425159, 93202207781383214849429075266794187, 244006547798191185585064349218845272, 638817435613190341905763972389625600, 1672445759041379840132227567949911511, 4378519841510949178490918731459984838, 11463113765491467695340528626429914739, 30010821454963453907530667147829626854, 78569350599398894027251472817058828944, 205697230343233228174223751303346718651, 538522340430300790495419781092981181139, 1409869790947669143312035591975596674257, 3691087032412706639440686994833808686387, 9663391306290450775010025392525829224825, 25299086886458645685589389182743678823076, 66233869353085486281758142155705207074358, 173402521172797813159685037284371942224819
Generating function in Maple syntax:
-(2*x^8-6*x^7+7*x^5-8*x^4+15*x^3-14*x^2+6*x-1)/(x^2-3*x+1)/(x-1)^4
Generating function in latex syntax:
-\frac{2 x^{8}-6 x^{7}+7 x^{5}-8 x^{4}+15 x^{3}-14 x^{2}+6 x -1}{\left(x^{2}-3 x +1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(-2*x**8 + 6*x**7 - 7*x**5 + 8*x**4 - 15*x**3 + 14*x**2 - 6*x + 1)/((x - 1)**4*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x-1)^4*F(x)+2*x^8-6*x^7+7*x^5-8*x^4+15*x^3-14*x^2+6*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x -1\right)^{4} F \! \left(x \right)+2 x^{8}-6 x^{7}+7 x^{5}-8 x^{4}+15 x^{3}-14 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/30*(-3*5^(1/2)+15)*(3/2-1/2*5^(1/2))^(-n)+1/30*(3*5^(1/2)+15)*(3/2+1/2*5^(1/2))^(-n)+1/6*n^3+3/2*n^2-35/3*n+18)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(-3 \sqrt{5}+15\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{30}+\frac{\left(3 \sqrt{5}+15\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{30}+\frac{n^{3}}{6}+\frac{3 n^{2}}{2}-\\\frac{35 n}{3}+18 & \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) = 52
a(6) = 127
a(7) = 300
a(8) = 716
a(n+2) = -1/6*n^3-2*n^2-a(n)+3*a(n+1)+55/6*n-4, n >= 9
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) = 52
a \! \left(6\right) = 127
a \! \left(7\right) = 300
a \! \left(8\right) = 716
a \! \left(n +2\right) = -\frac{n^{3}}{6}-2 n^{2}-a \! \left(n \right)+3 a \! \left(n +1\right)+\frac{55 n}{6}-4, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21638/
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[21,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[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[8,x]
F[13,x] = F[14,x]+F[6,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[8,x]
F[16,x] = F[17,x]+F[20,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[8,x]
F[19,x] = F[1,x]+F[17,x]
F[20,x] = F[15,x]
F[21,x] = F[22,x]
F[22,x] = F[23,x]*F[8,x]
F[23,x] = F[24,x]+F[56,x]
F[24,x] = F[25,x]+F[46,x]
F[25,x] = F[26,x]+F[27,x]
F[26,x] = F[19,x]*F[5,x]
F[27,x] = F[10,x]*F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[8,x]
F[30,x] = F[28,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[33,x]*F[40,x]*F[8,x]
F[33,x] = F[10,x]+F[34,x]
F[34,x] = F[17,x]+F[35,x]
F[35,x] = F[36,x]+F[37,x]+F[39,x]
F[36,x] = 0
F[37,x] = F[38,x]*F[8,x]
F[38,x] = F[11,x]+F[35,x]
F[39,x] = F[34,x]*F[8,x]
F[40,x] = F[1,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[43,x]*F[8,x]
F[43,x] = F[40,x]+F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[19,x]*F[43,x]*F[8,x]
F[46,x] = F[47,x]
F[47,x] = F[48,x]*F[8,x]
F[48,x] = F[49,x]+F[53,x]
F[49,x] = F[10,x]*F[50,x]
F[50,x] = F[51,x]+F[52,x]
F[51,x] = F[33,x]*F[40,x]
F[52,x] = F[19,x]^2*F[8,x]
F[53,x] = F[19,x]^2*F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[11,x]*F[8,x]
F[56,x] = F[57,x]
F[57,x] = F[19,x]*F[58,x]
F[58,x] = F[59,x]+F[63,x]
F[59,x] = F[10,x]*F[60,x]
F[60,x] = F[28,x]+F[61,x]
F[61,x] = F[17,x]*F[62,x]
F[62,x] = F[1,x]+F[8,x]
F[63,x] = F[17,x]*F[54,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_{21}\! \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_{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_{8}\! \left(x \right)
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_{16}\! \left(x \right) F_{8}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{20}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{8}\! \left(x \right)
F_{19}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{20}\! \left(x \right) = F_{15}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{8}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{56}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{46}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)
F_{26}\! \left(x \right) = F_{19}\! \left(x \right) F_{5}\! \left(x \right)
F_{27}\! \left(x \right) = F_{10}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{8}\! \left(x \right)
F_{30}\! \left(x \right) = F_{28}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{40}\! \left(x \right) F_{8}\! \left(x \right)
F_{33}\! \left(x \right) = F_{10}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{17}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{37}\! \left(x \right)+F_{39}\! \left(x \right)
F_{36}\! \left(x \right) = 0
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{8}\! \left(x \right)
F_{38}\! \left(x \right) = F_{11}\! \left(x \right)+F_{35}\! \left(x \right)
F_{39}\! \left(x \right) = F_{34}\! \left(x \right) F_{8}\! \left(x \right)
F_{40}\! \left(x \right) = F_{1}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right) F_{8}\! \left(x \right)
F_{43}\! \left(x \right) = F_{40}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{19}\! \left(x \right) F_{43}\! \left(x \right) F_{8}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right) F_{8}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{53}\! \left(x \right)
F_{49}\! \left(x \right) = F_{10}\! \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_{33}\! \left(x \right) F_{40}\! \left(x \right)
F_{52}\! \left(x \right) = F_{19} \left(x \right)^{2} F_{8}\! \left(x \right)
F_{53}\! \left(x \right) = F_{19} \left(x \right)^{2} F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{11}\! \left(x \right) F_{8}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{19}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{63}\! \left(x \right)
F_{59}\! \left(x \right) = F_{10}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{28}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{17}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{1}\! \left(x \right)+F_{8}\! \left(x \right)
F_{63}\! \left(x \right) = F_{17}\! \left(x \right) F_{54}\! \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_21(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_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_8(x))
Eq(F_13(x), F_14(x) + F_6(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_8(x))
Eq(F_16(x), F_17(x) + F_20(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x)*F_8(x))
Eq(F_19(x), F_1(x) + F_17(x))
Eq(F_20(x), F_15(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_23(x)*F_8(x))
Eq(F_23(x), F_24(x) + F_56(x))
Eq(F_24(x), F_25(x) + F_46(x))
Eq(F_25(x), F_26(x) + F_27(x))
Eq(F_26(x), F_19(x)*F_5(x))
Eq(F_27(x), F_10(x)*F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_8(x))
Eq(F_30(x), F_28(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_33(x)*F_40(x)*F_8(x))
Eq(F_33(x), F_10(x) + F_34(x))
Eq(F_34(x), F_17(x) + F_35(x))
Eq(F_35(x), F_36(x) + F_37(x) + F_39(x))
Eq(F_36(x), 0)
Eq(F_37(x), F_38(x)*F_8(x))
Eq(F_38(x), F_11(x) + F_35(x))
Eq(F_39(x), F_34(x)*F_8(x))
Eq(F_40(x), F_1(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_43(x)*F_8(x))
Eq(F_43(x), F_40(x) + F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_19(x)*F_43(x)*F_8(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_48(x)*F_8(x))
Eq(F_48(x), F_49(x) + F_53(x))
Eq(F_49(x), F_10(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_52(x))
Eq(F_51(x), F_33(x)*F_40(x))
Eq(F_52(x), F_19(x)**2*F_8(x))
Eq(F_53(x), F_19(x)**2*F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_11(x)*F_8(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_19(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_63(x))
Eq(F_59(x), F_10(x)*F_60(x))
Eq(F_60(x), F_28(x) + F_61(x))
Eq(F_61(x), F_17(x)*F_62(x))
Eq(F_62(x), F_1(x) + F_8(x))
Eq(F_63(x), F_17(x)*F_54(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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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