0132_0231_1302_3021

Counting sequence:
1, 1, 2, 6, 20, 67, 222, 734, 2445, 8246, 28192, 97648, 342220, 1211852, 4330431, 15597360, 56569287, 206422059, 757303633, 2791663009, 10335108948, 38409527795, 143243285787, 535898624083, 2010684962283, 7564036684514, 28524587785580, 107810233136443, 408322476472856, 1549476235002566, 5890433107677200, 22430469034418879, 85548318960021276, 326756462052456339, 1249792500206665414, 4786493957870486275, 18354005069671692784, 70460942226009127040, 270796615778576757210, 1041814167647795849388, 4012051445723061246311, 15465024921535645986555, 59665446516716635956042, 230390705673777875221443, 890347878132796856794425, 3443430308480077183732229, 13327380709213262654609825, 51618615415043401415516660, 200061226296665989167079096, 775893880651854785214229403, 3011016195434984623604623540, 11691882574510070949687386298, 45426182535546946733438600984, 176591143399892832740109969956, 686852448559756375299217304973, 2672891392111617716410061966687, 10406733659891005045785222789854, 40537335481227435671394087004541, 157977907148556584445264956369343, 615929561555397206863669314776249, 2402441168359244988072731057868337, 9374655198334351294701094058086502, 36595929805372699455197362798238160, 142915589243467351145285290574954500, 558329299156922045780779681032459077, 2182027092366788835553480142959784374, 8530685021400155407270460801619848384, 33362384705869477497407373704455713396, 130519453025565719306131646593519662680, 510780310384099967572798253299818371276, 1999538360575477464485016363618724578823, 7829934743889860802514377160984449317985, 30670131415262483538065043516839005232762, 120170719030648546823297436421009519895685, 470981440327425051840457458204446749523679, 1846408361354382247227715650586052799133463, 7240480767590429149100062120672055742492561, 28400089331160756536391692284144476687943114, 111424772253800103715623927042912517718371018, 437271078833686082538309584526657126215082843, 1716421758449225061669059417062305409485648742, 6739053990797115189937195986130010776855441980, 26465069073720622726736276637369625693746079228, 103954643783944622261026963356339037287871051838, 408422073325729339277608546213781808392587124273, 1604969298150167028584705703965702968648873401197, 6308328597619560967737330883519270779030195568964, 24799895913182657667650328998536506355161438812553, 97514987606790996944890398828415299400931639159583, 383510183136776579989703984108980570334156580803089, 1508566886887258249898076095723612636956045611952351, 5935161849458413573298514937349080325733056583229834, 23354960198045068805821301516859370317182099470708782, 91918423432688547153869385359358249253616460504348765, 361827213879988349655065062894147207877816200563314778, 1424536308960784268639899309512872648276263340450120822, 5609419370777913405268287812887738413660475648134045736, 22091891125173358218085073849207461175843204446974333440, 87019589049020131879389936750120950652521550397219901043, 342822175987103783308443424128190176434868879337458945867, 1350787926143627511247804183534283571993173408353224179476

Generating function in Maple syntax:
((-5*x^7+23*x^6-50*x^5+66*x^4-55*x^3+28*x^2-8*x+1)*(1-4*x)^(1/2)-4*x^8+x^7+x^6+12*x^5-36*x^4+43*x^3-26*x^2+8*x-1)/(16*x^8-76*x^7+162*x^6-200*x^5+152*x^4-70*x^3+18*x^2-2*x)

Generating function in latex syntax:
\frac{\left(-5 x^{7}+23 x^{6}-50 x^{5}+66 x^{4}-55 x^{3}+28 x^{2}-8 x +1\right) \sqrt{1-4 x}-4 x^{8}+x^{7}+x^{6}+12 x^{5}-36 x^{4}+43 x^{3}-26 x^{2}+8 x -1}{16 x^{8}-76 x^{7}+162 x^{6}-200 x^{5}+152 x^{4}-70 x^{3}+18 x^{2}-2 x}

Generating function in sympy syntax:
(-4*x**8 + x**7 + x**6 + 12*x**5 - 36*x**4 + 43*x**3 - 26*x**2 + 8*x + sqrt(1 - 4*x)*(-5*x**7 + 23*x**6 - 50*x**5 + 66*x**4 - 55*x**3 + 28*x**2 - 8*x + 1) - 1)/(16*x**8 - 76*x**7 + 162*x**6 - 200*x**5 + 152*x**4 - 70*x**3 + 18*x**2 - 2*x)

Implicit equation for the generating function in Maple syntax:
x*(4*x^4-9*x^3+10*x^2-5*x+1)*(2*x-1)^2*(x-1)^4*F(x)^2+(2*x-1)*(4*x^8-x^7-x^6-12*x^5+36*x^4-43*x^3+26*x^2-8*x+1)*(x-1)^2*F(x)+x^11+8*x^10-44*x^9+148*x^8-332*x^7+508*x^6-541*x^5+399*x^4-199*x^3+64*x^2-12*x+1 = 0

Implicit equation for the generating function in latex syntax:
x \left(4 x^{4}-9 x^{3}+10 x^{2}-5 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{4} F \! \left(x \right)^{2}+\left(2 x -1\right) \left(4 x^{8}-x^{7}-x^{6}-12 x^{5}+36 x^{4}-43 x^{3}+26 x^{2}-8 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{11}+8 x^{10}-44 x^{9}+148 x^{8}-332 x^{7}+508 x^{6}-541 x^{5}+399 x^{4}-199 x^{3}+64 x^{2}-12 x +1 = 0

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 67
a(6) = 222
a(7) = 734
a(8) = 2445
a(9) = 8246
a(10) = 28192
a(11) = 97648
a(12) = 342220
a(13) = 1211852
a(14) = 4330431
a(n+11) = 80*(2*n+1)/(12+n)*a(n)-4*(224*n+335)/(12+n)*a(1+n)+2*(1273*n+3261)/(12+n)*a(n+2)-(16960+4643*n)/(12+n)*a(n+3)+(5911*n+27976)/(12+n)*a(n+4)-2*(2714*n+15715)/(12+n)*a(n+5)+(3619*n+24714)/(12+n)*a(n+6)-2*(867*n+6812)/(12+n)*a(n+7)+(581*n+5160)/(12+n)*a(n+8)-3*(43*n+426)/(12+n)*a(n+9)+(17*n+186)/(12+n)*a(n+10)+(3*n^2-16*n+6)/(12+n), n >= 15

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) = 67
a \! \left(6\right) = 222
a \! \left(7\right) = 734
a \! \left(8\right) = 2445
a \! \left(9\right) = 8246
a \! \left(10\right) = 28192
a \! \left(11\right) = 97648
a \! \left(12\right) = 342220
a \! \left(13\right) = 1211852
a \! \left(14\right) = 4330431
a \! \left(n +11\right) = \frac{80 \left(2 n +1\right) a \! \left(n \right)}{12+n}-\frac{4 \left(224 n +335\right) a \! \left(1+n \right)}{12+n}+\frac{2 \left(1273 n +3261\right) a \! \left(n +2\right)}{12+n}-\frac{\left(16960+4643 n \right) a \! \left(n +3\right)}{12+n}+\frac{\left(5911 n +27976\right) a \! \left(n +4\right)}{12+n}-\frac{2 \left(2714 n +15715\right) a \! \left(n +5\right)}{12+n}+\frac{\left(3619 n +24714\right) a \! \left(n +6\right)}{12+n}-\frac{2 \left(867 n +6812\right) a \! \left(n +7\right)}{12+n}+\frac{\left(581 n +5160\right) a \! \left(n +8\right)}{12+n}-\frac{3 \left(43 n +426\right) a \! \left(n +9\right)}{12+n}+\frac{\left(17 n +186\right) a \! \left(n +10\right)}{12+n}+\frac{3 n^{2}-16 n +6}{12+n}, \quad n \geq 15

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21242/
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[12,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[12,x]*F[7,x]
F[7,x] = F[13,x]+F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[9,x]^2*F[12,x]
F[12,x] = x
F[13,x] = F[14,x]
F[14,x] = F[15,x]*F[18,x]
F[15,x] = F[16,x]+F[25,x]
F[16,x] = F[17,x]*F[20,x]
F[17,x] = F[1,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[17,x]
F[20,x] = F[1,x]+F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[20,x]*F[23,x]
F[23,x] = F[1,x]+F[24,x]
F[24,x] = F[12,x]*F[23,x]
F[25,x] = F[23,x]*F[26,x]
F[26,x] = F[2,x]+F[27,x]
F[27,x] = F[18,x]^2
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_{12}\! \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_{12}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{13}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \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_{9} \left(x \right)^{2} F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{18}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{25}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{20}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{1}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{25}\! \left(x \right) = F_{23}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{2}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{18} \left(x \right)^{2}
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_12(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_12(x)*F_7(x))
Eq(F_7(x), F_13(x) + F_8(x))
Eq(F_8(x), F_4(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_9(x)**2)
Eq(F_12(x), x)
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_15(x)*F_18(x))
Eq(F_15(x), F_16(x) + F_25(x))
Eq(F_16(x), F_17(x)*F_20(x))
Eq(F_17(x), F_1(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_17(x))
Eq(F_20(x), F_1(x) + F_21(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_12(x)*F_20(x)*F_23(x))
Eq(F_23(x), F_1(x) + F_24(x))
Eq(F_24(x), F_12(x)*F_23(x))
Eq(F_25(x), F_23(x)*F_26(x))
Eq(F_26(x), F_2(x) + F_27(x))
Eq(F_27(x), F_18(x)**2)
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, 3, 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, 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, 3, 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, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"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, 3, 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, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "strategy_class": "RequirementInsertionStrategy"}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 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