0213_1230_1302_2031

Counting sequence:
1, 1, 2, 6, 20, 65, 206, 641, 1964, 5936, 17726, 52375, 153316, 445121, 1282936, 3673832, 10459790, 29626245, 83522980, 234480885, 655771460, 1827643970, 5077584312, 14065857373, 38861775920, 107107256211, 294535425536, 808262750404, 2213759677924, 6052449823995, 16519956732626, 45020744823017, 122514386043486, 332945178294962, 903663282420936, 2449755118358315, 6633639598831430, 17944143061679689, 48491032164335400, 130915821445969922, 353132045977842180, 951737149527179051, 2563013481829554812, 6896939815388764071, 18545993259579620120, 49836457742544802410, 133832590731491541306, 359176190116732166831, 963377582281437907074, 2582500870759987991911, 6919109802850203707506, 18528316265578566219900, 49591498089068223802876, 132670087600284638370061, 354765810889197593323336, 948246844033616406219327, 2533491449119018019507790, 6766170456047945925014660, 18063360568858154331017140, 48205056547251959769045635, 128597058441924925297261440, 342941079894598876422911795, 914246648180813924324207122, 2436507240745061619345884638, 6491384236982726670354730670, 17289275047784697240359517731, 46035244804114121489373936036, 122541298035771665648753143769, 326104492894843017245031112734, 867595178396490281643260025720, 2307634902482327280433819719306, 6136349763661987711684737707217, 16313585072110551578840236872846, 43359886195703179737330824843927, 115220365812841272369560179415636, 306108655636671569174673953336710, 813073754133083653793387659665030, 2159209882198691721924979022251329, 5732880171841074273695075180286610, 15218307605047369524916567731472447, 40390352554676494850833840323453110, 107178810436196993404361106268536126, 284355967676472311804270922207426322, 754292740136506856269990529151679961, 2000523402109070144715298343294728120, 5304877489178271266081137027853733005, 14064908502036182019026393388811133006, 37284647505708434857755240584599626446, 98822635838971676926569021008346213984, 261889272489706568456509850476237888861, 693929632342274187710638190659456603286, 1838446999299418461621783566359109881325, 4869969120738647201171542669277887350136, 12898586443417533847013595423206158109576, 34158611136868920368051745735820183631386, 90448584694034168829491727860907043858947, 239468337581902722714403666538934660376140, 633928679775776792769637707595265350096125, 1677953275162190373489966428049527451124100, 4440865644181958737708216535969445882215360, 11751811648994046878098562575733758951802170

Generating function in Maple syntax:
(x^7-5*x^6+12*x^5-23*x^4+33*x^3-24*x^2+8*x-1)/(x^3-2*x^2+3*x-1)/(x^2-3*x+1)^2

Generating function in latex syntax:
\frac{x^{7}-5 x^{6}+12 x^{5}-23 x^{4}+33 x^{3}-24 x^{2}+8 x -1}{\left(x^{3}-2 x^{2}+3 x -1\right) \left(x^{2}-3 x +1\right)^{2}}

Generating function in sympy syntax:
(x**7 - 5*x**6 + 12*x**5 - 23*x**4 + 33*x**3 - 24*x**2 + 8*x - 1)/((x**2 - 3*x + 1)**2*(x**3 - 2*x**2 + 3*x - 1))

Implicit equation for the generating function in Maple syntax:
(x^3-2*x^2+3*x-1)*(x^2-3*x+1)^2*F(x)-x^7+5*x^6-12*x^5+23*x^4-33*x^3+24*x^2-8*x+1 = 0

Implicit equation for the generating function in latex syntax:
\left(x^{3}-2 x^{2}+3 x -1\right) \left(x^{2}-3 x +1\right)^{2} F \! \left(x \right)-x^{7}+5 x^{6}-12 x^{5}+23 x^{4}-33 x^{3}+24 x^{2}-8 x +1 = 0

Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/6900*(-115*((I+1/23*23^(1/2))*3^(1/2)+3/23*I*23^(1/2)+1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)+4600-230*2^(2/3)*((I+4/23*23^(1/2))*3^(1/2)-12/23*I*23^(1/2)-1)*(11+3*23^(1/2)*3^(1/2))^(1/3))*(11/600*2^(1/3)*((I-3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)+1)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/6900*(230*((I-4/23*23^(1/2))*3^(1/2)-12/23*I*23^(1/2)+1)*2^(2/3)*(11+3*23^(1/2)*3^(1/2))^(1/3)+4600+115*((I-1/23*23^(1/2))*3^(1/2)+3/23*I*23^(1/2)-1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3))*(-11/600*2^(1/3)*((I+3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)-1)*(11+3*23^(1/2)*3^(1/2))^(2/3)+1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/6900*((80*3^(1/2)*2^(2/3)*23^(1/2)-460*2^(2/3))*(11+3*23^(1/2)*3^(1/2))^(1/3)+4600+(10*23^(1/2)*3^(1/2)*2^(1/3)+230*2^(1/3))*(11+3*23^(1/2)*3^(1/2))^(2/3))*(1/300*2^(1/3)*(3*23^(1/2)*3^(1/2)-11)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/6*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/6900*(1380*n-828*5^(1/2)-6900)*(3/2-1/2*5^(1/2))^(-n)+1/5*(n+3/5*5^(1/2)-5)*(3/2+1/2*5^(1/2))^(-n))

Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(-115 \left(\left(\mathrm{I}+\frac{\sqrt{23}}{23}\right) \sqrt{3}+\frac{3 \,\mathrm{I} \sqrt{23}}{23}+1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+4600-230 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}+\frac{4 \sqrt{23}}{23}\right) \sqrt{3}-\frac{12 \,\mathrm{I} \sqrt{23}}{23}-1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{11 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}-\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{6900}\\+\\\frac{\left(230 \left(\left(\mathrm{I}-\frac{4 \sqrt{23}}{23}\right) \sqrt{3}-\frac{12 \,\mathrm{I} \sqrt{23}}{23}+1\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+4600+115 \left(\left(\mathrm{I}-\frac{\sqrt{23}}{23}\right) \sqrt{3}+\frac{3 \,\mathrm{I} \sqrt{23}}{23}-1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{11 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}+\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}-1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}+\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{6900}\\+\\\frac{\left(\left(80 \sqrt{3}\, 2^{\frac{2}{3}} \sqrt{23}-460 \,2^{\frac{2}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+4600+\left(10 \sqrt{23}\, \sqrt{3}\, 2^{\frac{1}{3}}+230 \,2^{\frac{1}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{2^{\frac{1}{3}} \left(3 \sqrt{23}\, \sqrt{3}-11\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{300}-\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{2}{3}\right)^{-n}}{6900}\\+\frac{\left(1380 n -828 \sqrt{5}-6900\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{6900}+\frac{\left(n +\frac{3 \sqrt{5}}{5}-5\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{5} & \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) = 65
a(6) = 206
a(7) = 641
a(n+7) = a(n)-8*a(n+1)+26*a(n+2)-47*a(n+3)+52*a(n+4)-31*a(n+5)+9*a(n+6), 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) = 20
a \! \left(5\right) = 65
a \! \left(6\right) = 206
a \! \left(7\right) = 641
a \! \left(n +7\right) = a \! \left(n \right)-8 a \! \left(n +1\right)+26 a \! \left(n +2\right)-47 a \! \left(n +3\right)+52 a \! \left(n +4\right)-31 a \! \left(n +5\right)+9 a \! \left(n +6\right), \quad n \geq 8

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21386/
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[14,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[14,x]*F[7,x]
F[7,x] = F[25,x]+F[8,x]
F[8,x] = F[15,x]*F[18,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[14,x]*F[9,x]
F[12,x] = F[1,x]+F[13,x]
F[13,x] = F[12,x]*F[14,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[9,x]
F[18,x] = F[1,x]+F[19,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[23,x]
F[22,x] = F[14,x]*F[21,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[14,x]*F[21,x]
F[25,x] = F[12,x]*F[26,x]
F[26,x] = F[27,x]+F[28,x]
F[27,x] = F[12,x]*F[2,x]
F[28,x] = F[16,x]*F[19,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_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{14}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{25}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{15}\! \left(x \right) F_{18}\! \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_{14}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = F_{1}\! \left(x \right)+F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \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_{9}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \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_{23}\! \left(x \right)
F_{22}\! \left(x \right) = F_{14}\! \left(x \right) F_{21}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right) F_{21}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{2}\! \left(x \right)
F_{28}\! \left(x \right) = F_{16}\! \left(x \right) F_{19}\! \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_14(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_14(x)*F_7(x))
Eq(F_7(x), F_25(x) + F_8(x))
Eq(F_8(x), F_15(x)*F_18(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_14(x)*F_9(x))
Eq(F_12(x), F_1(x) + F_13(x))
Eq(F_13(x), F_12(x)*F_14(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)*F_9(x))
Eq(F_18(x), F_1(x) + F_19(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_23(x))
Eq(F_22(x), F_14(x)*F_21(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_14(x)*F_21(x))
Eq(F_25(x), F_12(x)*F_26(x))
Eq(F_26(x), F_27(x) + F_28(x))
Eq(F_27(x), F_12(x)*F_2(x))
Eq(F_28(x), F_16(x)*F_19(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": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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