0213_0231_0312_0321_1032_1302_2031

Counting sequence:
1, 1, 2, 6, 17, 50, 155, 498, 1644, 5547, 19048, 66353, 233897, 832764, 2990280, 10816480, 39376263, 144152644, 530362834, 1959989705, 7272243026, 27080036325, 101170080321, 379098308172, 1424424073978, 5365603545063, 20258393162201, 76651983343255, 290607467823779, 1103811982478031, 4199856398607672, 16005752187923033, 61090819710711464, 233503812252365100, 893703906536621878, 3424850677984007464, 13140363229723308527, 50473424781007420354, 194080547595402138308, 747035478535128486165, 2878182843214812753952, 11099234654419389777839, 42839680176427783449179, 165485634934959713882338, 639762590831695343226552, 2475178412740088128988313, 9583165078862471090943369, 37128962074918767234801577, 143947939541482931721911743, 558438173865741150994707827, 2167757272882919189618174976, 8419798041327105192461370656, 32721886209213704299997791564, 127236390440754505214155323421, 495007686728477392981498803535, 1926778781983552524821196343534, 7503488788140212666248084214783, 29234701621036658015353055694479, 113954399573066482723228295173067, 444379330077371504955119033983432, 1733646614381811174629048027045603, 6766206364922608204712798407209012, 26418131488618843813879194355619649, 103187164994548391757405661090070465, 403190950795415130533993136106008498, 1575986207445887935781885223592503319, 6162342651666594928329596947762688511, 24103856647035967544554532661666273975, 94312695461239062296032225580086327866, 369141236854268036100971925818017316373, 1445272658252864821824167664694589259152, 5660281488708580980419813139930994753777, 22174490532753328971775160178837045054806, 86894674222097237734199301157234467072044, 340606715689289591769926632071085159357878, 1335459055360822537141150312901331816022838, 5237477654824488864376082567403247503689575, 20545894623940590372393812926862351204958486, 80618802606574254190508091744533239440682130, 316412257521644597034985077888875800197287877, 1242147820572326712643190529338321168751759510, 4877460336152167520285756273890994887987966405, 19156327013746221994904424456874308350638225187, 75253448555848504543856162589009231346776401014, 295688178307474702417993083563985519776464650266, 1162071123756894650752078335641566383058285365575, 4567941457373160810037279727793355647285022149451, 17959546799899751859674451910030399151676132594269, 70624480017475346074926012714621539323322309207389, 277778282366410451217426063385197907784135023514601, 1092754491587280359176414602989728601685583510227958, 4299583751987008129550874426006149736536216561247876, 16920297630931940874246667405505257242964950824125682, 66598689414199767665865757421342476282636894160606617, 262178967063934179824082190778889373375308737537639957, 1032292865481963751680978180338099560622605772906809124, 4065176411766915702594114305921694684267364443488879555, 16011270304134060158231642388012473777047627512993264021, 63072591615120118090834256294373542987494340779254907401, 248497871862565595833711333118306675967059182680410875730, 979197824126789322903228735559028559439223020035342103955

Generating function in Maple syntax:
((-1/2*x+1/2)*(-4*x+1)^(1/2)+x^3+1/2*x-1/2)*(x-1)/x/(x^5+x^3-2*x+1)

Generating function in latex syntax:
\frac{\left(\left(-\frac{x}{2}+\frac{1}{2}\right) \sqrt{-4 x +1}+x^{3}+\frac{x}{2}-\frac{1}{2}\right) \left(x -1\right)}{x \left(x^{5}+x^{3}-2 x +1\right)}

Generating function in sympy syntax:
(x - 1)*(x**3 + x/2 + (1/2 - x/2)*sqrt(1 - 4*x) - 1/2)/(x*(x**5 + x**3 - 2*x + 1))

Implicit equation for the generating function in Maple syntax:
x*(x^5+x^3-2*x+1)*F(x)^2-(x-1)*(2*x^3+x-1)*F(x)+(x-1)^2 = 0

Implicit equation for the generating function in latex syntax:
x \left(x^{5}+x^{3}-2 x +1\right) F \! \left(x \right)^{2}-\left(x -1\right) \left(2 x^{3}+x -1\right) F \! \left(x \right)+\left(x -1\right)^{2} = 0

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 17
a(5) = 50
a(6) = 155
a(n+7) = -2*(7+2*n)/(8+n)*a(n)+(31+5*n)/(8+n)*a(1+n)-(22+5*n)/(8+n)*a(n+2)+(31+5*n)/(8+n)*a(n+3)+(20+7*n)/(8+n)*a(n+4)-2*(38+7*n)/(8+n)*a(n+5)+(47+7*n)/(8+n)*a(n+6), 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) = 17
a \! \left(5\right) = 50
a \! \left(6\right) = 155
a \! \left(n +7\right) = -\frac{2 \left(7+2 n \right) a \! \left(n \right)}{8+n}+\frac{\left(31+5 n \right) a \! \left(1+n \right)}{8+n}-\frac{\left(22+5 n \right) a \! \left(n +2\right)}{8+n}+\frac{\left(31+5 n \right) a \! \left(n +3\right)}{8+n}+\frac{\left(20+7 n \right) a \! \left(n +4\right)}{8+n}-\frac{2 \left(38+7 n \right) a \! \left(n +5\right)}{8+n}+\frac{\left(47+7 n \right) a \! \left(n +6\right)}{8+n}, \quad n \geq 7

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/23054/
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[11,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[0,x]*F[11,x]*F[7,x]
F[7,x] = F[17,x]+F[8,x]
F[8,x] = F[15,x]*F[9,x]
F[9,x] = F[10,x]+F[12,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = x
F[12,x] = F[13,x]
F[13,x] = F[14,x]^2*F[11,x]
F[14,x] = F[1,x]+F[12,x]
F[15,x] = F[1,x]+F[16,x]
F[16,x] = F[11,x]*F[15,x]
F[17,x] = F[14,x]*F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[11,x]*F[12,x]*F[14,x]*F[15,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_{11}\! \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_{0}\! \left(x \right) F_{11}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{17}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{15}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{12}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = x
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14} \left(x \right)^{2} F_{11}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{11}\! \left(x \right) F_{15}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right) F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{11}\! \left(x \right) F_{12}\! \left(x \right) F_{14}\! \left(x \right) F_{15}\! \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_11(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_0(x)*F_11(x)*F_7(x))
Eq(F_7(x), F_17(x) + F_8(x))
Eq(F_8(x), F_15(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_12(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), x)
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_11(x)*F_14(x)**2)
Eq(F_14(x), F_1(x) + F_12(x))
Eq(F_15(x), F_1(x) + F_16(x))
Eq(F_16(x), F_11(x)*F_15(x))
Eq(F_17(x), F_14(x)*F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_11(x)*F_12(x)*F_14(x)*F_15(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": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 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, 0, 3, 2], "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": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 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, 0, 3, 2], "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": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1, 3], "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, 3, 1, 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, 0, 3, 2], "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": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "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, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[1, 2]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 2], [0, 2]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 2], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 2], [0, 2]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 2]]}, {"patt": [1, 0, 2], "pos": [[0, 2], [0, 0], [0, 2]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 2], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 0], [0, 2]], [[1, 1]]], "strategy_class": "FactorStrategy", "workable": true}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", 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