0213_0321_1032_3120
Counting sequence:
1, 1, 2, 6, 20, 61, 166, 414, 973, 2200, 4853, 10539, 22658, 48387, 102840, 217772, 459755, 968066, 2033535, 4262385, 8916200, 18616437, 38802602, 80746906, 167780185, 348136956, 721431961, 1493186119, 3087024158, 6375361335, 13153359788, 27112007064, 55834604823, 114890409542, 236223240515, 485331349037, 996432463140, 2044404489801, 4191888144830, 8589934663590, 17592186124325, 36009005898592, 73667279159677, 150633093114531, 307863255897850, 628920651220651, 1284229581388256, 2621235720777924, 5348024557677443, 10907155347728970, 22236523160350023, 45317471250642089, 92323792361341088, 188025284442984797, 382805968326780690, 779122735535407602, 1585267068834751025, 3224577333197637636, 6557241057451832225, 13330654897017087295, 27093655358261353750, 55052001844977425247, 111833385946864672852, 227125536407549406256, 461168601842739380015, 936172261740760373646, 1900014639592084487051, 3855369511405297001829, 7821419487252850644828, 15864199903390215197105, 32171121664549458875558, 65227687044636975423614, 132226261520350066947453, 267994297902852366898152, 543072145530009200655365, 1100311390508627335933467, 2228956979914472542071410, 4514582357623380825567571, 9142501510835633135059720, 18511676612849009239105500, 37476700408053504417384475, 75860095180817980714384402, 153533579091057905189338127, 310693935640959697901226081, 628641426199607170849038680, 1271789962234589891792815941, 2572594144139930883776756346, 5203216727621363967937493834, 10522490333925732336644770313, 21277094425217473474831017740, 43018416365166964552746996393, 86965287759797964311666019639, 175787485578523999035678299918, 355288791274904138896051433607, 718005222785520559441494956540, 1450865726042465682181776626632, 2931442013027780490961129332295, 5922305147941259235117413595606, 11963452539653914976625139951315, 24164589566850622966030908450205, 48804548108786831957623077156500
Generating function in Maple syntax:
(x^8-2*x^7-10*x^6+31*x^5-49*x^4+48*x^3-27*x^2+8*x-1)/(2*x-1)^2/(x-1)^5
Generating function in latex syntax:
\frac{x^{8}-2 x^{7}-10 x^{6}+31 x^{5}-49 x^{4}+48 x^{3}-27 x^{2}+8 x -1}{\left(2 x -1\right)^{2} \left(x -1\right)^{5}}
Generating function in sympy syntax:
(x**8 - 2*x**7 - 10*x**6 + 31*x**5 - 49*x**4 + 48*x**3 - 27*x**2 + 8*x - 1)/((x - 1)**5*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
-(2*x-1)^2*(x-1)^5*F(x)+x^8-2*x^7-10*x^6+31*x^5-49*x^4+48*x^3-27*x^2+8*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right)^{2} \left(x -1\right)^{5} F \! \left(x \right)+x^{8}-2 x^{7}-10 x^{6}+31 x^{5}-49 x^{4}+48 x^{3}-27 x^{2}+8 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,3/8*2^n*n+2^n-1/24*n^2-7/12*n-1-5/12*n^3+1/24*n^4)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{3 \,2^{n} n}{8}+2^{n}-\frac{n^{2}}{24}-\frac{7 n}{12}-1-\frac{5 n^{3}}{12}+\frac{n^{4}}{24} & \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) = 166
a(7) = 414
a(8) = 973
a(n+2) = 1/24*n^4-3/4*n^3+59/24*n^2-4*a(n)+4*a(n+1)+1/4*n-1, 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) = 166
a \! \left(7\right) = 414
a \! \left(8\right) = 973
a \! \left(n +2\right) = \frac{n^{4}}{24}-\frac{3 n^{3}}{4}+\frac{59 n^{2}}{24}-4 a \! \left(n \right)+4 a \! \left(n +1\right)+\frac{n}{4}-1, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21364/
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[14,x]*F[4,x]
F[4,x] = F[40,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[14,x]*F[8,x]
F[8,x] = F[18,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[14,x]
F[12,x] = F[13,x]+F[9,x]
F[13,x] = F[14,x]+F[15,x]
F[14,x] = x
F[15,x] = F[16,x]
F[16,x] = F[14,x]*F[17,x]
F[17,x] = F[13,x]
F[18,x] = F[19,x]+F[22,x]
F[19,x] = F[20,x]
F[20,x] = F[14,x]*F[21,x]
F[21,x] = F[1,x]+F[19,x]
F[22,x] = F[23,x]+F[24,x]+F[32,x]
F[23,x] = 0
F[24,x] = F[14,x]*F[25,x]
F[25,x] = F[26,x]+F[29,x]
F[26,x] = F[27,x]
F[27,x] = F[14,x]*F[28,x]
F[28,x] = F[1,x]+F[26,x]
F[29,x] = F[23,x]+F[24,x]+F[30,x]
F[30,x] = F[14,x]*F[31,x]
F[31,x] = F[19,x]+F[29,x]
F[32,x] = F[14,x]*F[33,x]
F[33,x] = F[18,x]+F[34,x]
F[34,x] = F[35,x]+F[37,x]
F[35,x] = F[36,x]
F[36,x] = F[14,x]*F[19,x]
F[37,x] = F[38,x]
F[38,x] = F[14,x]*F[39,x]
F[39,x] = F[34,x]
F[40,x] = F[41,x]
F[41,x] = F[14,x]*F[42,x]
F[42,x] = F[43,x]+F[66,x]
F[43,x] = F[44,x]+F[45,x]
F[44,x] = F[28,x]*F[5,x]
F[45,x] = F[46,x]
F[46,x] = F[14,x]*F[28,x]*F[47,x]*F[53,x]
F[47,x] = F[28,x]+F[48,x]
F[48,x] = F[19,x]+F[49,x]
F[49,x] = F[23,x]+F[50,x]+F[52,x]
F[50,x] = F[14,x]*F[51,x]
F[51,x] = F[26,x]+F[49,x]
F[52,x] = F[14,x]*F[48,x]
F[53,x] = F[54,x]+F[61,x]
F[54,x] = F[55,x]+F[59,x]
F[55,x] = F[28,x]*F[56,x]
F[56,x] = F[1,x]+F[57,x]
F[57,x] = F[58,x]
F[58,x] = F[14,x]*F[21,x]*F[56,x]
F[59,x] = F[60,x]
F[60,x] = F[26,x]^2*F[14,x]*F[21,x]
F[61,x] = F[62,x]+F[65,x]
F[62,x] = F[63,x]+F[64,x]
F[63,x] = F[26,x]*F[28,x]
F[64,x] = F[30,x]
F[65,x] = F[26,x]^2
F[66,x] = F[67,x]
F[67,x] = F[28,x]^2*F[14,x]*F[68,x]
F[68,x] = F[54,x]+F[69,x]
F[69,x] = F[65,x]+F[70,x]
F[70,x] = F[28,x]*F[57,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_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{40}\! \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_{14}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{18}\! \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_{14}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{15}\! \left(x \right)
F_{14}\! \left(x \right) = x
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{14}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{13}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{22}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{14}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{32}\! \left(x \right)
F_{23}\! \left(x \right) = 0
F_{24}\! \left(x \right) = F_{14}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{29}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{14}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{1}\! \left(x \right)+F_{26}\! \left(x \right)
F_{29}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{14}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{19}\! \left(x \right)+F_{29}\! \left(x \right)
F_{32}\! \left(x \right) = F_{14}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{18}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{37}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{14}\! \left(x \right) F_{19}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{14}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{34}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{14}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{66}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{45}\! \left(x \right)
F_{44}\! \left(x \right) = F_{28}\! \left(x \right) F_{5}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{14}\! \left(x \right) F_{28}\! \left(x \right) F_{47}\! \left(x \right) F_{53}\! \left(x \right)
F_{47}\! \left(x \right) = F_{28}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{19}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{23}\! \left(x \right)+F_{50}\! \left(x \right)+F_{52}\! \left(x \right)
F_{50}\! \left(x \right) = F_{14}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{26}\! \left(x \right)+F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{14}\! \left(x \right) F_{48}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{61}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{59}\! \left(x \right)
F_{55}\! \left(x \right) = F_{28}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{1}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{14}\! \left(x \right) F_{21}\! \left(x \right) F_{56}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{26} \left(x \right)^{2} F_{14}\! \left(x \right) F_{21}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{65}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{64}\! \left(x \right)
F_{63}\! \left(x \right) = F_{26}\! \left(x \right) F_{28}\! \left(x \right)
F_{64}\! \left(x \right) = F_{30}\! \left(x \right)
F_{65}\! \left(x \right) = F_{26} \left(x \right)^{2}
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{28} \left(x \right)^{2} F_{14}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{54}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{65}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{28}\! \left(x \right) F_{57}\! \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_14(x)*F_4(x))
Eq(F_4(x), F_40(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_14(x)*F_8(x))
Eq(F_8(x), F_18(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_14(x))
Eq(F_12(x), F_13(x) + F_9(x))
Eq(F_13(x), F_14(x) + F_15(x))
Eq(F_14(x), x)
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_14(x)*F_17(x))
Eq(F_17(x), F_13(x))
Eq(F_18(x), F_19(x) + F_22(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_14(x)*F_21(x))
Eq(F_21(x), F_1(x) + F_19(x))
Eq(F_22(x), F_23(x) + F_24(x) + F_32(x))
Eq(F_23(x), 0)
Eq(F_24(x), F_14(x)*F_25(x))
Eq(F_25(x), F_26(x) + F_29(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_14(x)*F_28(x))
Eq(F_28(x), F_1(x) + F_26(x))
Eq(F_29(x), F_23(x) + F_24(x) + F_30(x))
Eq(F_30(x), F_14(x)*F_31(x))
Eq(F_31(x), F_19(x) + F_29(x))
Eq(F_32(x), F_14(x)*F_33(x))
Eq(F_33(x), F_18(x) + F_34(x))
Eq(F_34(x), F_35(x) + F_37(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_14(x)*F_19(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_14(x)*F_39(x))
Eq(F_39(x), F_34(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_14(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_66(x))
Eq(F_43(x), F_44(x) + F_45(x))
Eq(F_44(x), F_28(x)*F_5(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_14(x)*F_28(x)*F_47(x)*F_53(x))
Eq(F_47(x), F_28(x) + F_48(x))
Eq(F_48(x), F_19(x) + F_49(x))
Eq(F_49(x), F_23(x) + F_50(x) + F_52(x))
Eq(F_50(x), F_14(x)*F_51(x))
Eq(F_51(x), F_26(x) + F_49(x))
Eq(F_52(x), F_14(x)*F_48(x))
Eq(F_53(x), F_54(x) + F_61(x))
Eq(F_54(x), F_55(x) + F_59(x))
Eq(F_55(x), F_28(x)*F_56(x))
Eq(F_56(x), F_1(x) + F_57(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_14(x)*F_21(x)*F_56(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_14(x)*F_21(x)*F_26(x)**2)
Eq(F_61(x), F_62(x) + F_65(x))
Eq(F_62(x), F_63(x) + F_64(x))
Eq(F_63(x), F_26(x)*F_28(x))
Eq(F_64(x), F_30(x))
Eq(F_65(x), F_26(x)**2)
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_14(x)*F_28(x)**2*F_68(x))
Eq(F_68(x), F_54(x) + F_69(x))
Eq(F_69(x), F_65(x) + F_70(x))
Eq(F_70(x), F_28(x)*F_57(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, 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": [3, 1, 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, 2, 1, 3], "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": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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