0132_1032_2301_3120
Counting sequence:
1, 1, 2, 6, 20, 58, 150, 360, 824, 1830, 3986, 8572, 18276, 38730, 81710, 171792, 360176, 753358, 1572522, 3276420, 6815324, 14155314, 29359622, 60816856, 125828520, 260046198, 536870210, 1107295500, 2281700564, 4697619610, 9663675486, 19864222752, 40802188256, 83751861150, 171798690650, 352187317012, 721554504396, 1477468748418, 3023656974902, 6184752904680, 12644383717784, 25838523251014, 52776558131442, 107752139520156, 219902325553220, 448600744130538, 914793674307470, 1864771720705840, 3799912185591504, 7740561859540590, 15762598695794186, 32088147345012132, 65302194596869436, 132856189007426770, 270215977642226790, 549439154539197432, 1116892707587879816, 2269814212194726678, 4611686018427384482, 9367487224930628140, 19023204826012971444, 38622870404329369914, 78398662313265590462, 159103167635744878656, 322818021289917149120, 654859414616689078078, 1328165573307087711930, 2693224634761594531380, 5460236245818027273644, 11068046444225730964770, 22431240793630814760086, 45452777397620335176712, 92086146415958081661816, 186533476073350985935590, 377789318629571617090130, 765023370224882524613052, 1548936206381243630086436, 3135651344625444421888138, 6346860552976803167201262, 12844836833405434981246800, 25991905121714527256176304, 52588273153236369099712014, 106385472126087367374136682, 215188795891403993097692356, 435213295061266502894216220, 880097996679450039186088818, 1779538806472734145167483590, 3597763239173136423925572120, 7272897730801609115032346984, 14700537966513890764427092150, 29710560942849126597578973186, 60040091905340943332607516492, 121318123849967266940114165396, 245112127778505294430026587610, 495176015714152109959649680670, 1000255551742587262118492363872, 2020318144113740608635370724256, 4080250369484613386067513432798, 8239728901483491109728570825242, 16637914127995510894644229560660, 33592740906048079139662634932364
Generating function in Maple syntax:
-(2*x^6-6*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}-6 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 + 6*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^6-6*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}-6 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),1,1/4*2^n*n+3*2^(-1+n)-n-n^2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{2^{n} n}{4}+3 \,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) = 58
a(6) = 150
a(n+2) = -n^2-4*a(n)+4*a(n+1)+3*n+2, 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) = 58
a \! \left(6\right) = 150
a \! \left(n +2\right) = -n^{2}-4 a \! \left(n \right)+4 a \! \left(n +1\right)+3 n +2, \quad n \geq 7
Specification 1
Strategy pack name: row_placements
Tree: http://permpal.com/tree/21279/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[14,x]*F[3,x]
F[3,x] = F[1,x]+F[4,x]+F[49,x]
F[4,x] = F[14,x]*F[5,x]
F[5,x] = F[1,x]+F[34,x]+F[38,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[14,x]*F[8,x]
F[8,x] = F[28,x]+F[9,x]
F[9,x] = F[10,x]+F[18,x]
F[10,x] = F[11,x]*F[15,x]
F[11,x] = F[1,x]+F[12,x]
F[12,x] = F[13,x]
F[13,x] = F[11,x]*F[14,x]*F[15,x]
F[14,x] = x
F[15,x] = F[1,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[14,x]*F[15,x]
F[18,x] = F[19,x]*F[25,x]
F[19,x] = F[20,x]
F[20,x] = F[14,x]*F[21,x]
F[21,x] = F[1,x]+F[22,x]+F[24,x]
F[22,x] = F[23,x]
F[23,x] = F[14,x]*F[9,x]
F[24,x] = F[14,x]*F[21,x]
F[25,x] = F[1,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[14,x]*F[25,x]
F[28,x] = F[26,x]+F[29,x]
F[29,x] = F[30,x]+F[31,x]+F[32,x]
F[30,x] = 0
F[31,x] = F[14,x]*F[28,x]
F[32,x] = F[14,x]*F[33,x]
F[33,x] = F[16,x]+F[29,x]
F[34,x] = F[14,x]*F[35,x]
F[35,x] = F[1,x]+F[34,x]+F[36,x]+F[6,x]
F[36,x] = F[14,x]*F[37,x]
F[37,x] = F[25,x]+F[33,x]
F[38,x] = F[14,x]*F[39,x]
F[39,x] = F[37,x]+F[40,x]
F[40,x] = F[33,x]+F[41,x]
F[41,x] = F[42,x]+F[45,x]
F[42,x] = F[43,x]
F[43,x] = F[14,x]*F[44,x]
F[44,x] = F[16,x]+F[42,x]
F[45,x] = 2*F[30,x]+F[46,x]+F[48,x]
F[46,x] = F[14,x]*F[47,x]
F[47,x] = F[29,x]+F[45,x]
F[48,x] = F[14,x]*F[41,x]
F[49,x] = F[50,x]
F[50,x] = F[14,x]*F[51,x]
F[51,x] = F[3,x]+F[52,x]
F[52,x] = F[14,x]*F[53,x]
F[53,x] = F[36,x]+F[52,x]+F[54,x]
F[54,x] = F[1,x]+F[55,x]+F[57,x]
F[55,x] = F[56,x]
F[56,x] = F[14,x]*F[15,x]*F[25,x]
F[57,x] = F[58,x]
F[58,x] = F[14,x]*F[25,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_{14}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{49}\! \left(x \right)
F_{4}\! \left(x \right) = F_{14}\! \left(x \right) F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{34}\! \left(x \right)+F_{38}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{14}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{28}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{18}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{15}\! \left(x \right)
F_{11}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{15}\! \left(x \right)
F_{14}\! \left(x \right) = x
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right) F_{15}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{25}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{14}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{22}\! \left(x \right)+F_{24}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{14}\! \left(x \right) F_{9}\! \left(x \right)
F_{24}\! \left(x \right) = F_{14}\! \left(x \right) F_{21}\! \left(x \right)
F_{25}\! \left(x \right) = F_{1}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{14}\! \left(x \right) F_{25}\! \left(x \right)
F_{28}\! \left(x \right) = F_{26}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{31}\! \left(x \right)+F_{32}\! \left(x \right)
F_{30}\! \left(x \right) = 0
F_{31}\! \left(x \right) = F_{14}\! \left(x \right) F_{28}\! \left(x \right)
F_{32}\! \left(x \right) = F_{14}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{16}\! \left(x \right)+F_{29}\! \left(x \right)
F_{34}\! \left(x \right) = F_{14}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{1}\! \left(x \right)+F_{34}\! \left(x \right)+F_{36}\! \left(x \right)+F_{6}\! \left(x \right)
F_{36}\! \left(x \right) = F_{14}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{25}\! \left(x \right)+F_{33}\! \left(x \right)
F_{38}\! \left(x \right) = F_{14}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{37}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{33}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{45}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{14}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{16}\! \left(x \right)+F_{42}\! \left(x \right)
F_{45}\! \left(x \right) = 2 F_{30}\! \left(x \right)+F_{46}\! \left(x \right)+F_{48}\! \left(x \right)
F_{46}\! \left(x \right) = F_{14}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{29}\! \left(x \right)+F_{45}\! \left(x \right)
F_{48}\! \left(x \right) = F_{14}\! \left(x \right) F_{41}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{14}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{3}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{14}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{36}\! \left(x \right)+F_{52}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{1}\! \left(x \right)+F_{55}\! \left(x \right)+F_{57}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{14}\! \left(x \right) F_{15}\! \left(x \right) F_{25}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{14}\! \left(x \right) F_{25}\! \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_14(x)*F_3(x))
Eq(F_3(x), F_1(x) + F_4(x) + F_49(x))
Eq(F_4(x), F_14(x)*F_5(x))
Eq(F_5(x), F_1(x) + F_34(x) + F_38(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_14(x)*F_8(x))
Eq(F_8(x), F_28(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_18(x))
Eq(F_10(x), F_11(x)*F_15(x))
Eq(F_11(x), F_1(x) + F_12(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_11(x)*F_14(x)*F_15(x))
Eq(F_14(x), x)
Eq(F_15(x), F_1(x) + F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_14(x)*F_15(x))
Eq(F_18(x), F_19(x)*F_25(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_14(x)*F_21(x))
Eq(F_21(x), F_1(x) + F_22(x) + F_24(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_14(x)*F_9(x))
Eq(F_24(x), F_14(x)*F_21(x))
Eq(F_25(x), F_1(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_14(x)*F_25(x))
Eq(F_28(x), F_26(x) + F_29(x))
Eq(F_29(x), F_30(x) + F_31(x) + F_32(x))
Eq(F_30(x), 0)
Eq(F_31(x), F_14(x)*F_28(x))
Eq(F_32(x), F_14(x)*F_33(x))
Eq(F_33(x), F_16(x) + F_29(x))
Eq(F_34(x), F_14(x)*F_35(x))
Eq(F_35(x), F_1(x) + F_34(x) + F_36(x) + F_6(x))
Eq(F_36(x), F_14(x)*F_37(x))
Eq(F_37(x), F_25(x) + F_33(x))
Eq(F_38(x), F_14(x)*F_39(x))
Eq(F_39(x), F_37(x) + F_40(x))
Eq(F_40(x), F_33(x) + F_41(x))
Eq(F_41(x), F_42(x) + F_45(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_14(x)*F_44(x))
Eq(F_44(x), F_16(x) + F_42(x))
Eq(F_45(x), 2*F_30(x) + F_46(x) + F_48(x))
Eq(F_46(x), F_14(x)*F_47(x))
Eq(F_47(x), F_29(x) + F_45(x))
Eq(F_48(x), F_14(x)*F_41(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_14(x)*F_51(x))
Eq(F_51(x), F_3(x) + F_52(x))
Eq(F_52(x), F_14(x)*F_53(x))
Eq(F_53(x), F_36(x) + F_52(x) + F_54(x))
Eq(F_54(x), F_1(x) + F_55(x) + F_57(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_14(x)*F_15(x)*F_25(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_14(x)*F_25(x)*F_54(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": false, "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_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": [1, 0, 3, 2], "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, 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], "pos": [[0, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [2, 1]]}, 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