0231_1203_1320
Counting sequence:
1, 1, 2, 6, 21, 75, 267, 948, 3367, 11988, 42842, 153783, 554624, 2009904, 7318260, 26768537, 98339812, 362754125, 1343255139, 4991749328, 18611597401, 69605800781, 261059418738, 981689563163, 3700548668353, 13981017142012, 52932518715339, 200795652996605, 763091546936820, 2904942078878827, 11076166385821174, 42295050292602173, 161732923969661431, 619270363007009067, 2374118081591797867, 9112434524523490771, 35014584539387498083, 134685407001592399849, 518591795825771665201, 1998676800907383472725, 7709944076137493600713, 29766855055597236737703, 115019214629490164014025, 444782206360878182712001, 1721269599357850309100567, 6665930648306667367665309, 25832705794744498177079819, 100176040302388444732881097, 388714804855669768543231253, 1509248170440410792941092429, 5863302902115338030256037733, 22791160157391961011464485789, 88638795484159169307453517085, 344910135487347218391285251169, 1342779254897620648125404062769, 5230125838762292883059579528953, 20380798774138869053827814859425, 79455668777422770411800646986901, 309896466982055449366301192077265, 1209177812435698618647245320286161, 4719973751484804954074940506546165, 18431437835009149385261097900408681, 72001854826030320964694036880020222, 281376522878159228249768624098821262, 1099985205487478565478904075985850553, 4301658885767786691056520413126977335, 16827932780100163532109952767750501140, 65851697822474221193632405348897232473, 257774830969830979632645743045964589667, 1009363191289749913956866707121978759466, 3953521371777683576209804819246949260019, 15489835522209536552066685521399104486129, 60706077069635405417963012559726158677106, 237978036758482826618607388260060214341717, 933163539973825662029582241342162865179593, 3660090108585275078866552164368426578944850, 14359390208815314157160738111212687871491145, 56349180845753761432598030128085534610745713, 221179004245464986005155287622855898106137112, 868364908997608839731724595806038399069193171, 3410045676904044971630194906864476020929514909, 13394153546471814531118994651840776579943006214, 52621735114485409910855606707886319315315145771, 206779566355339855789207029656675891755170151497, 812719110319322158328704224327415858288009818318, 3194932042598063699077526650520207957183144786355, 12562297012799884630337019606307499270526985093145, 49403852379482005200827256991536546364162790384618, 194327834599898109028434483400882640877109235070807, 764521705834704329698054738627722342153508826397641, 3008316160073460317583448226701063581097193866659434, 11839524842181148142165487611026262322187937053829979, 46603717038424689510590880248848220088969597577829421, 183476618400038353596684123844800455230622231716164202, 722459048313110874841460030305663358689861333381298753, 2845224748322425862198805744208070735503704748066859675, 11206998161836823280702785326899021883101798740717651890, 44149927327720829201546456267833389214040655617073796463, 173955148281042698280475408670690020334027724368023807707, 685503779279083941818035919747798089186699273913869870586, 2701757751471951377881848237492887375210201661232130742619
Generating function in Maple syntax:
1/2*((2*x^4-5*x^3+8*x^2-5*x+1)*(1-4*x)^(1/2)+2*x^5-10*x^4+21*x^3-18*x^2+7*x-1)/x^2/(x^2-3*x+1)/(x-1)
Generating function in latex syntax:
\frac{\left(2 x^{4}-5 x^{3}+8 x^{2}-5 x +1\right) \sqrt{1-4 x}+2 x^{5}-10 x^{4}+21 x^{3}-18 x^{2}+7 x -1}{2 x^{2} \left(x^{2}-3 x +1\right) \left(x -1\right)}
Generating function in sympy syntax:
(x**5 - 5*x**4 + 21*x**3/2 - 9*x**2 + 7*x/2 + sqrt(1 - 4*x)*(2*x**4 - 5*x**3 + 8*x**2 - 5*x + 1)/2 - 1/2)/(x**2*(x - 1)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
x^2*(x-1)^2*(x^2-3*x+1)^2*F(x)^2-(x-1)*(x^2-3*x+1)*(2*x^5-10*x^4+21*x^3-18*x^2+7*x-1)*F(x)+x^8-6*x^7+25*x^6-61*x^5+93*x^4-82*x^3+40*x^2-10*x+1 = 0
Implicit equation for the generating function in latex syntax:
x^{2} \left(x -1\right)^{2} \left(x^{2}-3 x +1\right)^{2} F \! \left(x \right)^{2}-\left(x -1\right) \left(x^{2}-3 x +1\right) \left(2 x^{5}-10 x^{4}+21 x^{3}-18 x^{2}+7 x -1\right) F \! \left(x \right)+x^{8}-6 x^{7}+25 x^{6}-61 x^{5}+93 x^{4}-82 x^{3}+40 x^{2}-10 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 75
a(6) = 267
a(7) = 948
a(8) = 3367
a(n+8) = -4*(1+2*n)/(10+n)*a(n)-(396+157*n)/(10+n)*a(2+n)+2*(35+27*n)/(10+n)*a(n+1)+34*(31+8*n)/(10+n)*a(n+3)-(1496+291*n)/(10+n)*a(n+4)+6*(197+31*n)/(10+n)*a(n+5)-2*(257+34*n)/(10+n)*a(n+6)+(114+13*n)/(10+n)*a(n+7), 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) = 21
a \! \left(5\right) = 75
a \! \left(6\right) = 267
a \! \left(7\right) = 948
a \! \left(8\right) = 3367
a \! \left(n +8\right) = -\frac{4 \left(1+2 n \right) a \! \left(n \right)}{10+n}-\frac{\left(396+157 n \right) a \! \left(2+n \right)}{10+n}+\frac{2 \left(35+27 n \right) a \! \left(n +1\right)}{10+n}+\frac{34 \left(31+8 n \right) a \! \left(n +3\right)}{10+n}-\frac{\left(1496+291 n \right) a \! \left(n +4\right)}{10+n}+\frac{6 \left(197+31 n \right) a \! \left(n +5\right)}{10+n}-\frac{2 \left(257+34 n \right) a \! \left(n +6\right)}{10+n}+\frac{\left(114+13 n \right) a \! \left(n +7\right)}{10+n}, \quad n \geq 9
Specification 1
Strategy pack name: row_placements
Tree: http://permpal.com/tree/361/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[14,x]*F[3,x]
F[3,x] = F[1,x]+F[4,x]+F[42,x]
F[4,x] = F[14,x]*F[5,x]
F[5,x] = F[1,x]+F[24,x]+F[4,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[14,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]*F[14,x]*F[3,x]
F[10,x] = F[11,x]+F[15,x]
F[11,x] = F[1,x]+F[12,x]
F[12,x] = F[13,x]
F[13,x] = F[11,x]*F[14,x]
F[14,x] = x
F[15,x] = F[16,x]+F[19,x]
F[16,x] = F[17,x]
F[17,x] = F[14,x]*F[18,x]
F[18,x] = F[1,x]+F[16,x]
F[19,x] = F[20,x]+F[21,x]+F[23,x]
F[20,x] = 0
F[21,x] = F[14,x]*F[22,x]
F[22,x] = F[12,x]+F[19,x]
F[23,x] = F[14,x]*F[15,x]
F[24,x] = F[14,x]*F[25,x]
F[25,x] = F[1,x]+F[26,x]+F[34,x]+F[36,x]+F[40,x]
F[26,x] = F[14,x]*F[27,x]
F[27,x] = F[1,x]+F[26,x]+F[28,x]+F[34,x]
F[28,x] = F[14,x]*F[29,x]
F[29,x] = F[1,x]+F[28,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[14,x]*F[29,x]*F[32,x]
F[32,x] = F[1,x]+F[33,x]
F[33,x] = F[14,x]*F[29,x]
F[34,x] = F[35,x]
F[35,x] = F[14,x]*F[27,x]*F[32,x]
F[36,x] = F[14,x]*F[37,x]
F[37,x] = F[1,x]+F[30,x]+F[36,x]+F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[14,x]*F[32,x]*F[37,x]
F[40,x] = F[41,x]
F[41,x] = F[14,x]*F[25,x]*F[32,x]
F[42,x] = F[14,x]*F[37,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_{14}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{42}\! \left(x \right)
F_{4}\! \left(x \right) = F_{14}\! \left(x \right) F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{24}\! \left(x \right)+F_{4}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{14}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{14}\! \left(x \right) F_{3}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{15}\! \left(x \right)
F_{11}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{11}\! \left(x \right) F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = x
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{19}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right) F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{23}\! \left(x \right)
F_{20}\! \left(x \right) = 0
F_{21}\! \left(x \right) = F_{14}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right)+F_{19}\! \left(x \right)
F_{23}\! \left(x \right) = F_{14}\! \left(x \right) F_{15}\! \left(x \right)
F_{24}\! \left(x \right) = F_{14}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{1}\! \left(x \right)+F_{26}\! \left(x \right)+F_{34}\! \left(x \right)+F_{36}\! \left(x \right)+F_{40}\! \left(x \right)
F_{26}\! \left(x \right) = F_{14}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{1}\! \left(x \right)+F_{26}\! \left(x \right)+F_{28}\! \left(x \right)+F_{34}\! \left(x \right)
F_{28}\! \left(x \right) = F_{14}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{1}\! \left(x \right)+F_{28}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{14}\! \left(x \right) F_{29}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{1}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{14}\! \left(x \right) F_{29}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{14}\! \left(x \right) F_{27}\! \left(x \right) F_{32}\! \left(x \right)
F_{36}\! \left(x \right) = F_{14}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{1}\! \left(x \right)+F_{30}\! \left(x \right)+F_{36}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{14}\! \left(x \right) F_{32}\! \left(x \right) F_{37}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{14}\! \left(x \right) F_{25}\! \left(x \right) F_{32}\! \left(x \right)
F_{42}\! \left(x \right) = F_{14}\! \left(x \right) F_{37}\! \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_14(x)*F_3(x))
Eq(F_3(x), F_1(x) + F_4(x) + F_42(x))
Eq(F_4(x), F_14(x)*F_5(x))
Eq(F_5(x), F_1(x) + F_24(x) + F_4(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_14(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x)*F_14(x)*F_3(x))
Eq(F_10(x), F_11(x) + F_15(x))
Eq(F_11(x), F_1(x) + F_12(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_11(x)*F_14(x))
Eq(F_14(x), x)
Eq(F_15(x), F_16(x) + F_19(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_14(x)*F_18(x))
Eq(F_18(x), F_1(x) + F_16(x))
Eq(F_19(x), F_20(x) + F_21(x) + F_23(x))
Eq(F_20(x), 0)
Eq(F_21(x), F_14(x)*F_22(x))
Eq(F_22(x), F_12(x) + F_19(x))
Eq(F_23(x), F_14(x)*F_15(x))
Eq(F_24(x), F_14(x)*F_25(x))
Eq(F_25(x), F_1(x) + F_26(x) + F_34(x) + F_36(x) + F_40(x))
Eq(F_26(x), F_14(x)*F_27(x))
Eq(F_27(x), F_1(x) + F_26(x) + F_28(x) + F_34(x))
Eq(F_28(x), F_14(x)*F_29(x))
Eq(F_29(x), F_1(x) + F_28(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_14(x)*F_29(x)*F_32(x))
Eq(F_32(x), F_1(x) + F_33(x))
Eq(F_33(x), F_14(x)*F_29(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_14(x)*F_27(x)*F_32(x))
Eq(F_36(x), F_14(x)*F_37(x))
Eq(F_37(x), F_1(x) + F_30(x) + F_36(x) + F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_14(x)*F_32(x)*F_37(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_14(x)*F_25(x)*F_32(x))
Eq(F_42(x), F_14(x)*F_37(x))
Pack JSON:
{"expansion_strats": [[{"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": "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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "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, 1, 2], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, 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