0213_1230_1320_3102
Counting sequence:
1, 1, 2, 6, 20, 61, 170, 448, 1141, 2847, 7016, 17159, 41769, 101377, 245593, 594257, 1436803, 3472172, 8388057, 20259395, 48924740, 118137734, 285246799, 688706602, 1662781664, 4014466640, 9692033072, 23399047356, 56490960194, 136382314411, 329257767752, 794901374950, 1919066221027, 4633043044989, 11185167241910, 27003401687213, 65192009705127, 157387484344111, 379967080728199, 917321811381383, 2214610971406033, 5346544187688698, 12907700048193015, 31161945418978793, 75231592722463442, 181625133835121732, 438481865200234909, 1058588872014334672, 2555659621815174398, 6169908136009695710, 14895475926785847134, 35960860042897682418, 86817196098848784560, 209595252380179115353, 506007701085058450406, 1221610654915731313804, 2949229011507807809425, 7120068678888068960283, 17189366370831954487556, 41498801423056708719059, 100186969220998111465317, 241872739871610401971357, 583932448974829125267637, 1409737637838436332686141, 3403407724679479680677215, 8216553087242341264255496, 19836513899236885669438485, 47889580885833781633595663, 115615675671094841427341216, 279120932228331526009450562, 673857540128256347458124011, 1626836012485650736458750454, 3927529565100862789920556124, 9481895142689487801377843372, 22891319850483254847299152268, 55264534843661525435677035432, 133420389537815250112977017438, 322105313919306497995655749447, 777631017376451662832636987012, 1877367348672247712723302870450, 4532365714721008394069964342655, 10942098778114363695716326314489, 26416563270949896286146433342034, 63775225320014415963506104124537, 153967013910979148409299369084259, 371709253141973392673742480908875, 897385520194927033844562697007587, 2166480293531829240342283879641971, 5230346107258588394596324827111469, 12627172508049010689581543909399126, 30484691123356617313873217392260147, 73596554754762257517488393815800245, 177677800632881152089124224892087982, 428952156020524593636171478590080384, 1035582112673930391042176036930576185, 2500116381368385459341667042299660188, 6035814875410701445027362466236647162, 14571746132189788568319387809328128186, 35179307139790278935890986264154823354, 84930360411770347013249204351454863838, 205040027963330973889762087160143555148
Generating function in Maple syntax:
-(x^8+x^7-2*x^6-4*x^4-x^3+8*x^2-5*x+1)/(x^2+x-1)/(x^2+2*x-1)/(x-1)^3
Generating function in latex syntax:
-\frac{x^{8}+x^{7}-2 x^{6}-4 x^{4}-x^{3}+8 x^{2}-5 x +1}{\left(x^{2}+x -1\right) \left(x^{2}+2 x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-x**8 - x**7 + 2*x**6 + 4*x**4 + x**3 - 8*x**2 + 5*x - 1)/((x - 1)**3*(x**2 + x - 1)*(x**2 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^2+x-1)*(x^2+2*x-1)*(x-1)^3*F(x)+x^8+x^7-2*x^6-4*x^4-x^3+8*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}+x -1\right) \left(x^{2}+2 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{8}+x^{7}-2 x^{6}-4 x^{4}-x^{3}+8 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/40*(16*5^(1/2)-40)*(-1/2*5^(1/2)-1/2)^(-n)+1/40*(-16*5^(1/2)-40)*(1/2*5^(1/2)-1/2)^(-n)+1/40*(-20*2^(1/2)+15)*(-1-2^(1/2))^(-n)+1/40*(20*2^(1/2)+15)*(2^(1/2)-1)^(-n)-1/4*n^2-3/4*n+13/4)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(16 \sqrt{5}-40\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{40}+\frac{\left(-16 \sqrt{5}-40\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{40}+\\\frac{\left(-20 \sqrt{2}+15\right) \left(-1-\sqrt{2}\right)^{-n}}{40}+\frac{\left(20 \sqrt{2}+15\right) \left(\sqrt{2}-1\right)^{-n}}{40}-\frac{n^{2}}{4}-\frac{3 n}{4}\\+\frac{13}{4} & \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) = 61
a(6) = 170
a(7) = 448
a(8) = 1141
a(n+1) = -1/3*a(n)+a(n+3)-1/3*a(n+4)-1/6*(n+5)*(n-4), 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) = 20
a \! \left(5\right) = 61
a \! \left(6\right) = 170
a \! \left(7\right) = 448
a \! \left(8\right) = 1141
a \! \left(n +1\right) = -\frac{a \! \left(n \right)}{3}+a \! \left(n +3\right)-\frac{a \! \left(n +4\right)}{3}-\frac{\left(n +5\right) \left(n -4\right)}{6}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15762/
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[5,x]
F[4,x] = x
F[5,x] = F[15,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[6,x]
F[10,x] = F[11,x]+F[7,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[14,x]
F[14,x] = F[7,x]
F[15,x] = F[16,x]+F[7,x]
F[16,x] = F[17,x]+F[18,x]+F[43,x]
F[17,x] = 0
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[20,x]+F[32,x]
F[20,x] = F[21,x]+F[7,x]
F[21,x] = F[17,x]+F[22,x]+F[25,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[20,x]+F[24,x]
F[24,x] = F[11,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[14,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[31,x]
F[31,x] = F[28,x]+F[7,x]
F[32,x] = F[16,x]+F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[31,x]+F[39,x]
F[39,x] = F[36,x]+F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[36,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[45,x]+F[46,x]
F[45,x] = F[36,x]+F[7,x]
F[46,x] = F[28,x]+F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[50,x]
F[50,x] = F[11,x]+F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[50,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_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{15}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \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_{10}\! \left(x \right)+F_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{7}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{7}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{7}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)+F_{43}\! \left(x \right)
F_{17}\! \left(x \right) = 0
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{32}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{7}\! \left(x \right)
F_{21}\! \left(x \right) = F_{17}\! \left(x \right)+F_{22}\! \left(x \right)+F_{25}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{20}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{11}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{14}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{28}\! \left(x \right)+F_{7}\! \left(x \right)
F_{32}\! \left(x \right) = F_{16}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{31}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{36}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{36}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{46}\! \left(x \right)
F_{45}\! \left(x \right) = F_{36}\! \left(x \right)+F_{7}\! \left(x \right)
F_{46}\! \left(x \right) = F_{28}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{11}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{50}\! \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_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_15(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_6(x))
Eq(F_10(x), F_11(x) + F_7(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_7(x))
Eq(F_15(x), F_16(x) + F_7(x))
Eq(F_16(x), F_17(x) + F_18(x) + F_43(x))
Eq(F_17(x), 0)
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_20(x) + F_32(x))
Eq(F_20(x), F_21(x) + F_7(x))
Eq(F_21(x), F_17(x) + F_22(x) + F_25(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_20(x) + F_24(x))
Eq(F_24(x), F_11(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_14(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_28(x) + F_7(x))
Eq(F_32(x), F_16(x) + F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_31(x) + F_39(x))
Eq(F_39(x), F_36(x) + F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_36(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_46(x))
Eq(F_45(x), F_36(x) + F_7(x))
Eq(F_46(x), F_28(x) + F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_11(x) + F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_50(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": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "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, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "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, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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