0132_1302_2301

Counting sequence:
1, 1, 2, 6, 21, 75, 263, 901, 3024, 9980, 32489, 104585, 333549, 1055497, 3318014, 10371474, 32261565, 99930087, 308397107, 948698509, 2910188556, 8905040456, 27189383057, 82855547621, 252056910681, 765622051345, 2322431252858, 7036374047646, 21295605438309, 64389772180995, 194524652125919, 587223274831765, 1771488295708104, 5340862269038612, 16093508490401849, 48470945878687169, 145923787385078085, 439141568884837657, 1321097100554554742, 3973128669124895274, 11945724169870216461, 35907654599180938911, 107911488358381061387, 324238501062411420061, 974062503316549190916, 2925785780141045740640, 8786965651511833161953, 26386543813750916627741, 79228063795588671470769, 237866721657752732849185, 714086863545687178617074, 2143557896357581625846070, 6434130588370731025776309, 19311600459137976138794427, 57959324785041470714456759, 173943263256673449055124389, 522003560960268315577644480, 1566473063856992821904241452, 4700649204337044743378039177, 14105218837109404042803492089, 42324354230157981232723837341, 126996183233991800175509139241, 381049995387778631841457141742, 1143313249388339206976446661826, 3430373450513479314937276843101, 10292272223727473214298864789335, 30879875312135769263129016189923, 92647746138935784399019748550061, 277964792112265116955213074193404, 833951576255261753160509990727800, 2502006500200765781948626245153329, 7506422132079268161344168194935125, 22520334345130184279895253119329289, 67563835211189569376888773380576817, 202699015267973473969916810881381994, 608116954738437142995698424581245134, 1824403636949861050698383162831173125, 5473350774607826528915748842125375347, 16420422950265589569523113329088390287, 49262250835581532469379727256175867829, 147789353934404880801679278784860576696, 443374952431438357917198759957203632196, 1330143106641660722077707211118431521497, 3990477645679772122156447256036464533041, 11971560891088935980101500705080784975861, 35915021419717686281385378551456135381305, 107745960948493907227901488769805982908518, 323240256169110077960156572202839002614106, 969727049391006689750836567379541278108589, 2909197768304761097395128884193549543283215, 8727637279236717735742033668674024657232059, 26183028174174038937942921098710871905167805, 78549392263665422031991914916796796998605044, 235648990762556220000266896369304031881574032, 706949125008912905967497204534127358418604929, 2120853067777440646333730127982919891920902029, 6362574255991063222162543558636436349271970145, 19087762566023821674608014006787831041313073217, 57263392910912070282392191654927193313536932322, 171790456854424636079798136666245561374118779494, 515372105687802852556435081037903325925303106901

Generating function in Maple syntax:
-(7*x^5-27*x^4+44*x^3-30*x^2+9*x-1)/(3*x-1)/(x-1)/(x^2-3*x+1)^2

Generating function in latex syntax:
-\frac{7 x^{5}-27 x^{4}+44 x^{3}-30 x^{2}+9 x -1}{\left(3 x -1\right) \left(x -1\right) \left(x^{2}-3 x +1\right)^{2}}

Generating function in sympy syntax:
(-7*x**5 + 27*x**4 - 44*x**3 + 30*x**2 - 9*x + 1)/((x - 1)*(3*x - 1)*(x**2 - 3*x + 1)**2)

Implicit equation for the generating function in Maple syntax:
(3*x-1)*(x-1)*(x^2-3*x+1)^2*F(x)+7*x^5-27*x^4+44*x^3-30*x^2+9*x-1 = 0

Implicit equation for the generating function in latex syntax:
\left(3 x -1\right) \left(x -1\right) \left(x^{2}-3 x +1\right)^{2} F \! \left(x \right)+7 x^{5}-27 x^{4}+44 x^{3}-30 x^{2}+9 x -1 = 0

Explicit closed form in Maple syntax:
3/20*(3+5^(1/2))*(((n-4/15)*5^(1/2)-7/3*n-2/3)*(3/2-1/2*5^(1/2))^(-n)+(-2/3*n+29/15*5^(1/2)-13/3)*(3/2+1/2*5^(1/2))^(-n)-5/3*(5^(1/2)-3)*(3^n+1))

Explicit closed form in latex syntax:
\frac{3 \left(3+\sqrt{5}\right) \left(\left(\left(n -\frac{4}{15}\right) \sqrt{5}-\frac{7 n}{3}-\frac{2}{3}\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}+\left(-\frac{2 n}{3}+\frac{29 \sqrt{5}}{15}-\frac{13}{3}\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}-\frac{5 \left(\sqrt{5}-3\right) \left(3^{n}+1\right)}{3}\right)}{20}

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 75
a(n+5) = 3*a(n)-19*a(n+1)+39*a(n+2)-29*a(n+3)+9*a(n+4)-2, n >= 6

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) = 21
a \! \left(5\right) = 75
a \! \left(n +5\right) = 3 a \! \left(n \right)-19 a \! \left(n +1\right)+39 a \! \left(n +2\right)-29 a \! \left(n +3\right)+9 a \! \left(n +4\right)-2, \quad n \geq 6

Specification 1
Strategy pack name: point_and_row_placements
Tree: http://permpal.com/tree/297/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[12,x]*F[3,x]
F[3,x] = F[0,x]+F[4,x]
F[4,x] = F[5,x]
F[5,x] = F[12,x]*F[6,x]
F[6,x] = F[22,x]+F[7,x]
F[7,x] = F[0,x]*F[8,x]
F[8,x] = F[13,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[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[18,x]+F[19,x]+F[21,x]
F[18,x] = 0
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[10,x]+F[17,x]
F[21,x] = F[12,x]*F[13,x]
F[22,x] = F[16,x]*F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[25,x]*F[8,x]
F[25,x] = F[26,x]+F[31,x]
F[26,x] = F[1,x]+F[27,x]+F[28,x]+F[30,x]
F[27,x] = F[12,x]*F[26,x]
F[28,x] = F[12,x]*F[29,x]
F[29,x] = F[23,x]+F[8,x]
F[30,x] = F[12,x]*F[25,x]
F[31,x] = F[32,x]
F[32,x] = F[12,x]*F[33,x]*F[8,x]
F[33,x] = F[1,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[12,x]*F[36,x]
F[36,x] = F[33,x]+F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[12,x]*F[16,x]*F[36,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_{12}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{0}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{12}\! \left(x \right) F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{22}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{0}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \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_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{22}\! \left(x \right) = F_{16}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right) F_{8}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{31}\! \left(x \right)
F_{26}\! \left(x \right) = F_{1}\! \left(x \right)+F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{30}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{26}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{23}\! \left(x \right)+F_{8}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right) F_{8}\! \left(x \right)
F_{33}\! \left(x \right) = F_{1}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{33}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right) F_{36}\! \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_12(x)*F_3(x))
Eq(F_3(x), F_0(x) + F_4(x))
Eq(F_4(x), F_5(x))
Eq(F_5(x), F_12(x)*F_6(x))
Eq(F_6(x), F_22(x) + F_7(x))
Eq(F_7(x), F_0(x)*F_8(x))
Eq(F_8(x), F_13(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))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_21(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_10(x) + F_17(x))
Eq(F_21(x), F_12(x)*F_13(x))
Eq(F_22(x), F_16(x)*F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_25(x)*F_8(x))
Eq(F_25(x), F_26(x) + F_31(x))
Eq(F_26(x), F_1(x) + F_27(x) + F_28(x) + F_30(x))
Eq(F_27(x), F_12(x)*F_26(x))
Eq(F_28(x), F_12(x)*F_29(x))
Eq(F_29(x), F_23(x) + F_8(x))
Eq(F_30(x), F_12(x)*F_25(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_12(x)*F_33(x)*F_8(x))
Eq(F_33(x), F_1(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_12(x)*F_36(x))
Eq(F_36(x), F_33(x) + F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_12(x)*F_16(x)*F_36(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"}, {"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": "point_and_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, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 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": [[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], [0, 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]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [2, 3, 0, 1], "pos": [[2, 1], [2, 1], [2, 1], [2, 1]]}], "requirements": [[{"patt": [0], "pos": [[1, 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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_placement", "direction": 3, "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "include_empty": true, "indices": [0], "own_col": true, "own_row": true, "strategy_class": "RequirementPlacementStrategy"}}, {"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, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [1, 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