0132_0213_1032_1320_2301
Counting sequence:
1, 1, 2, 6, 19, 52, 131, 320, 780, 1921, 4795, 12115, 30903, 79383, 204948, 531040, 1379567, 3590762, 9359319, 24420894, 63771376, 166630007, 435593087, 1139099201, 2979605659, 7795521517, 20398568056, 53383402910, 139718083395, 365703735216, 957258900955, 2505804528220, 6559617808388, 17171975150253, 44954160153363, 117686210336655, 308095880915567, 806584252533827, 2111622516939884, 5528214578800764, 14472883780499911, 37890161884782262, 99197052118022447, 259699894957645946, 679900433731647624, 1779997008190772707, 4660081794747634231, 12200230783866070525, 31940575372478472435, 83621424964825151961, 218923558784508609840, 573148969913723947450, 1500522788006709790523, 3928418268206498559548, 10284729764812972179603, 26925766522632790584120, 70492560795886144805692, 184551897850627134323417, 483163096727198239171499, 1264937320273373545232459, 3311648719977734320637671, 8670008551429453264934767, 22698376357849873170707332, 59425119369198661640302616, 155576979443903102536467039, 407305814350824627541669122, 1066340454385198743233721031, 2791715530358027528449896790, 7308806099795395694696818400, 19134702695241183260802302031, 50095301838354201498033622575, 131151202524673516053945685305, 343358305145370536305097724683, 898923711730846472143936125893, 2353412827685985638691887984168, 6161314766604743961062082548310, 16130531462683513278755069166067, 42230279602556329943724544025032, 110560307307206544689461401126539, 289450642243505440398745335859604, 757791619272194049054945959535156, 1983924215270845251862435248987397, 5193981025935878796725045199988867, 13598018861327865318698071176184711, 35600075555629865520139909979061279, 93202207800726027963263142062079387, 244006547836876811812732482809428956, 638817435690561594361100239570806580, 1672445759196122345042900102312287415, 4378519841820434188312263800184750926, 11463113766110437714983218763879461535, 30010821456201393946816047422728735410, 78569350601874774105822233366857061368, 205697230348184988331365272402943199163, 538522340440204310809702823292174158183, 1409869790967476183940601676373982644725, 3691087032452320720697819163630580644067, 9663391306369678937524289730119373157297, 25299086886617102010617917857930766705504, 66233869353402398931815199506079382857074, 173402521173431638459799151985120293808491
Generating function in Maple syntax:
-(2*x^3-6*x^2+4*x-1)*(x^5-2*x^4+x^3-4*x^2+4*x-1)/(2*x-1)/(x^2-3*x+1)/(x-1)^4
Generating function in latex syntax:
-\frac{\left(2 x^{3}-6 x^{2}+4 x -1\right) \left(x^{5}-2 x^{4}+x^{3}-4 x^{2}+4 x -1\right)}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(-2*x**3 + 6*x**2 - 4*x + 1)*(x**5 - 2*x**4 + x**3 - 4*x**2 + 4*x - 1)/((x - 1)**4*(2*x - 1)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^2-3*x+1)*(x-1)^4*F(x)+(2*x^3-6*x^2+4*x-1)*(x^5-2*x^4+x^3-4*x^2+4*x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{4} F \! \left(x \right)+\left(2 x^{3}-6 x^{2}+4 x -1\right) \left(x^{5}-2 x^{4}+x^{3}-4 x^{2}+4 x -1\right) = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/30*(-3*5^(1/2)+15)*(3/2-1/2*5^(1/2))^(-n)+1/30*(3*5^(1/2)+15)*(3/2+1/2*5^(1/2))^(-n)+1/6*n^3-1/2*n^2-5/3*n+2^(-1+n)+2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(-3 \sqrt{5}+15\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{30}+\frac{\left(3 \sqrt{5}+15\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{30}+\frac{n^{3}}{6}-\frac{n^{2}}{2}-\frac{5 n}{3}\\+2^{-1+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) = 19
a(5) = 52
a(6) = 131
a(7) = 320
a(8) = 780
a(n+3) = 1/6*n^3-1/2*n^2-7*a(n+1)+5*a(n+2)+2*a(n)-11/3*n+3, n >= 9
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) = 19
a \! \left(5\right) = 52
a \! \left(6\right) = 131
a \! \left(7\right) = 320
a \! \left(8\right) = 780
a \! \left(n +3\right) = \frac{n^{3}}{6}-\frac{n^{2}}{2}-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)+2 a \! \left(n \right)-\frac{11 n}{3}+3, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21616/
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[4,x]*F[8,x]
F[4,x] = F[15,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[8,x]*F[9,x]
F[8,x] = x
F[9,x] = F[10,x]+F[5,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[8,x]*F[9,x]
F[12,x] = F[1,x]+F[13,x]
F[13,x] = F[14,x]
F[14,x] = F[12,x]*F[8,x]
F[15,x] = F[16,x]+F[6,x]
F[16,x] = F[17,x]
F[17,x] = F[18,x]*F[8,x]
F[18,x] = F[15,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[8,x]
F[21,x] = F[22,x]+F[37,x]
F[22,x] = F[23,x]+F[35,x]
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[25,x]+F[27,x]
F[25,x] = F[26,x]+F[29,x]
F[26,x] = F[1,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[26,x]*F[8,x]
F[29,x] = F[13,x]+F[30,x]
F[30,x] = F[31,x]+F[32,x]+F[34,x]
F[31,x] = 0
F[32,x] = F[33,x]*F[8,x]
F[33,x] = F[27,x]+F[30,x]
F[34,x] = F[29,x]*F[8,x]
F[35,x] = F[36,x]
F[36,x] = F[12,x]^2*F[27,x]*F[8,x]
F[37,x] = F[13,x]*F[26,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_{4}\! \left(x \right) F_{8}\! \left(x \right)
F_{4}\! \left(x \right) = F_{15}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right) F_{9}\! \left(x \right)
F_{8}\! \left(x \right) = x
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{5}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \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_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{6}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{8}\! \left(x \right)
F_{18}\! \left(x \right) = F_{15}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{8}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{37}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{35}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{27}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{29}\! \left(x \right)
F_{26}\! \left(x \right) = F_{1}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{26}\! \left(x \right) F_{8}\! \left(x \right)
F_{29}\! \left(x \right) = F_{13}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{34}\! \left(x \right)
F_{31}\! \left(x \right) = 0
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{8}\! \left(x \right)
F_{33}\! \left(x \right) = F_{27}\! \left(x \right)+F_{30}\! \left(x \right)
F_{34}\! \left(x \right) = F_{29}\! \left(x \right) F_{8}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12} \left(x \right)^{2} F_{27}\! \left(x \right) F_{8}\! \left(x \right)
F_{37}\! \left(x \right) = F_{13}\! \left(x \right) F_{26}\! \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_4(x)*F_8(x))
Eq(F_4(x), F_15(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_8(x)*F_9(x))
Eq(F_8(x), x)
Eq(F_9(x), F_10(x) + F_5(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_8(x)*F_9(x))
Eq(F_12(x), F_1(x) + F_13(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_12(x)*F_8(x))
Eq(F_15(x), F_16(x) + F_6(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_18(x)*F_8(x))
Eq(F_18(x), F_15(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x)*F_8(x))
Eq(F_21(x), F_22(x) + F_37(x))
Eq(F_22(x), F_23(x) + F_35(x))
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_25(x) + F_27(x))
Eq(F_25(x), F_26(x) + F_29(x))
Eq(F_26(x), F_1(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_26(x)*F_8(x))
Eq(F_29(x), F_13(x) + F_30(x))
Eq(F_30(x), F_31(x) + F_32(x) + F_34(x))
Eq(F_31(x), 0)
Eq(F_32(x), F_33(x)*F_8(x))
Eq(F_33(x), F_27(x) + F_30(x))
Eq(F_34(x), F_29(x)*F_8(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_12(x)**2*F_27(x)*F_8(x))
Eq(F_37(x), F_13(x)*F_26(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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "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, 2, 0], "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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "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, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 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, 1, 3], "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, 2, 0], "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_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], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": []}, {"assumptions": [], 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