0132_0321_1320_2301_3021
Counting sequence:
1, 1, 2, 6, 19, 53, 133, 318, 756, 1828, 4525, 11435, 29327, 75916, 197574, 515698, 1348103, 3526833, 9230201, 24161098, 63249880, 165584720, 433499777, 1134909351, 2971222179, 7778750168, 20365020298, 53316301598, 139583874171, 365435309293, 956722040685, 2504730798230, 6557470337852, 17167680197436, 44945570234709, 117669030484963, 308061521196343, 806515533077988, 2111485078009166, 5527939700918538, 14472334024712815, 37889062373183465, 99194853094798177, 259695496911168546, 679891637638661664, 1779979416004767208, 4660046610375587113, 12200160415121937503, 31940434634990164811, 83621143489848492208, 218922995834555242770, 573147844013817162550, 1500520536206896166627, 3928413764606871254181, 10284720757613717507669, 26925748508234281175278, 70492524767089125919108, 184551825793033096477268, 483162952612010163401981, 1264937032042997393611803, 3311648143516982017310175, 8670007398507948658188860, 22698374052006863957119702, 59425114757512643213026466, 155576970220531065681808599, 407305795904080553832240673, 1066340417491710595814746953, 2791715456571051233611825658, 7308805952221443105020547176, 19134702400093278081449624448, 50095301248058391139328125905, 131151201344081895336534543895, 343358302784187294870275287027, 898923707008479989274291088776, 2353412818241252672952597740954, 6161314747715278029583501885518, 16130531424904581415797907656523, 42230279526998466217810220814173, 110560307156090817237632754505021, 289450641941273985495088042408518, 757791618667731139247631372416460, 1983924214061919432247806074524780, 5193981023518027157495786850829477, 13598018856492162040239554477622611, 35600075545958458963222876581684359, 93202207781383214849429075267063188, 244006547798191185585064349219124318, 638817435613190341905763972389914938, 1672445759041379840132227567950211391, 4378519841510949178490918731460295513, 11463113765491467695340528626430236465, 30010821454963453907530667147829959890, 78569350599398894027251472817059173552, 205697230343233228174223751303347075096, 538522340430300790495419781092981549689, 1409869790947669143312035591975597055183, 3691087032412706639440686994833809079963, 9663391306290450775010025392525829631328, 25299086886458645685589389182743679242786, 66233869353085486281758142155705207507558, 173402521172797813159685037284371942671795
Generating function in Maple syntax:
(x^7-x^6-6*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{x^{7}-x^{6}-6 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:
(x**7 - x**6 - 6*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)+x^7-x^6-6*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)+x^{7}-x^{6}-6 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),1,1/30*(-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)+2/3*n^3-4*n^2+25/3*n-6)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \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{2 n^{3}}{3}-4 n^{2}+\\\frac{25 n}{3}-6 & \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) = 133
a(7) = 318
a(n+2) = -a(n)+3*a(n+1)-1/3*(-1+n)*(2*n^2-4*n-9), 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) = 53
a \! \left(6\right) = 133
a \! \left(7\right) = 318
a \! \left(n +2\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)-\frac{\left(-1+n \right) \left(2 n^{2}-4 n -9\right)}{3}, \quad n \geq 8
Specification 1
Strategy pack name: row_and_col_placements
Tree: http://permpal.com/tree/21762/
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[4,x]
F[3,x] = x
F[4,x] = F[1,x]+F[34,x]+F[5,x]
F[5,x] = F[3,x]*F[6,x]
F[6,x] = F[19,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[17,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[3,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[3,x]*F[8,x]
F[17,x] = F[18,x]
F[18,x] = F[10,x]^2*F[15,x]*F[3,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[3,x]
F[21,x] = F[22,x]+F[33,x]
F[22,x] = F[13,x]*F[23,x]
F[23,x] = F[24,x]+F[27,x]
F[24,x] = F[1,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[24,x]*F[3,x]
F[27,x] = F[11,x]+F[28,x]
F[28,x] = F[29,x]+F[30,x]+F[32,x]
F[29,x] = 0
F[30,x] = F[3,x]*F[31,x]
F[31,x] = F[25,x]+F[28,x]
F[32,x] = F[27,x]*F[3,x]
F[33,x] = F[10,x]*F[11,x]*F[24,x]
F[34,x] = F[3,x]*F[35,x]
F[35,x] = F[36,x]+F[4,x]
F[36,x] = F[37,x]
F[37,x] = F[3,x]*F[38,x]
F[38,x] = F[39,x]+F[40,x]+F[58,x]
F[39,x] = F[24,x]+F[25,x]
F[40,x] = F[41,x]
F[41,x] = F[3,x]*F[42,x]
F[42,x] = F[43,x]+F[53,x]
F[43,x] = F[44,x]+F[52,x]
F[44,x] = F[24,x]*F[45,x]
F[45,x] = F[46,x]+F[47,x]
F[46,x] = F[1,x]+F[3,x]
F[47,x] = F[11,x]+F[48,x]
F[48,x] = F[29,x]+F[49,x]+F[51,x]
F[49,x] = F[3,x]*F[50,x]
F[50,x] = F[3,x]+F[48,x]
F[51,x] = F[11,x]*F[3,x]
F[52,x] = F[10,x]*F[25,x]
F[53,x] = F[24,x]*F[54,x]
F[54,x] = F[3,x]+F[55,x]
F[55,x] = F[29,x]+F[56,x]+F[57,x]
F[56,x] = F[3,x]*F[54,x]
F[57,x] = F[11,x]*F[3,x]
F[58,x] = F[59,x]
F[59,x] = F[10,x]^2*F[24,x]*F[3,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_{4}\! \left(x \right)
F_{3}\! \left(x \right) = x
F_{4}\! \left(x \right) = F_{1}\! \left(x \right)+F_{34}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{3}\! \left(x \right) F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{19}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{17}\! \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_{3}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{3}\! \left(x \right) F_{8}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{10} \left(x \right)^{2} F_{15}\! \left(x \right) F_{3}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{3}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{33}\! \left(x \right)
F_{22}\! \left(x \right) = F_{13}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{27}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{24}\! \left(x \right) F_{3}\! \left(x \right)
F_{27}\! \left(x \right) = F_{11}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)+F_{32}\! \left(x \right)
F_{29}\! \left(x \right) = 0
F_{30}\! \left(x \right) = F_{3}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{32}\! \left(x \right) = F_{27}\! \left(x \right) F_{3}\! \left(x \right)
F_{33}\! \left(x \right) = F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{24}\! \left(x \right)
F_{34}\! \left(x \right) = F_{3}\! \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_{3}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{40}\! \left(x \right)+F_{58}\! \left(x \right)
F_{39}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{3}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{53}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{52}\! \left(x \right)
F_{44}\! \left(x \right) = F_{24}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{47}\! \left(x \right)
F_{46}\! \left(x \right) = F_{1}\! \left(x \right)+F_{3}\! \left(x \right)
F_{47}\! \left(x \right) = F_{11}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{29}\! \left(x \right)+F_{49}\! \left(x \right)+F_{51}\! \left(x \right)
F_{49}\! \left(x \right) = F_{3}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{3}\! \left(x \right)+F_{48}\! \left(x \right)
F_{51}\! \left(x \right) = F_{11}\! \left(x \right) F_{3}\! \left(x \right)
F_{52}\! \left(x \right) = F_{10}\! \left(x \right) F_{25}\! \left(x \right)
F_{53}\! \left(x \right) = F_{24}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{3}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{29}\! \left(x \right)+F_{56}\! \left(x \right)+F_{57}\! \left(x \right)
F_{56}\! \left(x \right) = F_{3}\! \left(x \right) F_{54}\! \left(x \right)
F_{57}\! \left(x \right) = F_{11}\! \left(x \right) F_{3}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{10} \left(x \right)^{2} F_{24}\! \left(x \right) F_{3}\! \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)*F_4(x))
Eq(F_3(x), x)
Eq(F_4(x), F_1(x) + F_34(x) + F_5(x))
Eq(F_5(x), F_3(x)*F_6(x))
Eq(F_6(x), F_19(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_17(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_3(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_3(x)*F_8(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_10(x)**2*F_15(x)*F_3(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x)*F_3(x))
Eq(F_21(x), F_22(x) + F_33(x))
Eq(F_22(x), F_13(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_27(x))
Eq(F_24(x), F_1(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_24(x)*F_3(x))
Eq(F_27(x), F_11(x) + F_28(x))
Eq(F_28(x), F_29(x) + F_30(x) + F_32(x))
Eq(F_29(x), 0)
Eq(F_30(x), F_3(x)*F_31(x))
Eq(F_31(x), F_25(x) + F_28(x))
Eq(F_32(x), F_27(x)*F_3(x))
Eq(F_33(x), F_10(x)*F_11(x)*F_24(x))
Eq(F_34(x), F_3(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_3(x)*F_38(x))
Eq(F_38(x), F_39(x) + F_40(x) + F_58(x))
Eq(F_39(x), F_24(x) + F_25(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_3(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_53(x))
Eq(F_43(x), F_44(x) + F_52(x))
Eq(F_44(x), F_24(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_47(x))
Eq(F_46(x), F_1(x) + F_3(x))
Eq(F_47(x), F_11(x) + F_48(x))
Eq(F_48(x), F_29(x) + F_49(x) + F_51(x))
Eq(F_49(x), F_3(x)*F_50(x))
Eq(F_50(x), F_3(x) + F_48(x))
Eq(F_51(x), F_11(x)*F_3(x))
Eq(F_52(x), F_10(x)*F_25(x))
Eq(F_53(x), F_24(x)*F_54(x))
Eq(F_54(x), F_3(x) + F_55(x))
Eq(F_55(x), F_29(x) + F_56(x) + F_57(x))
Eq(F_56(x), F_3(x)*F_54(x))
Eq(F_57(x), F_11(x)*F_3(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_10(x)**2*F_24(x)*F_3(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": true, "place_row": true, "strategy_class": "RowAndColumnPlacementFactory"}]], "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": "row_and_col_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, 3, 2, 1], "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], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], 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