0132_1320_3021_3120
Counting sequence:
1, 1, 2, 6, 20, 65, 199, 577, 1607, 4355, 11596, 30542, 79927, 208422, 542539, 1411352, 3671476, 9554467, 24877899, 64817883, 168986875, 440833141, 1150630272, 3004767736, 7850043195, 20516003720, 53635055679, 140254948162, 366844579032, 959674812405, 2510904793391, 6570355217237, 17194523718207, 45001404782247, 117784994572004, 308302039332162, 807013749248591, 2112515870121322, 5530070004655139, 14476732071178188, 37898133344063180, 99213544792417031, 259733979818083267, 679970802475799791, 1780142143725611795, 4660380861910360025, 12200846510377593784, 31941842009873636972, 83624028608359689907, 218928906809066074380, 573159947437815622711, 1500545306004846600182, 3928464430102679061712, 10284824340405146911897, 26925960177416767464519, 70492957112653353355497, 184552708498560060955831, 483164754051861111454027, 1264940706980293327782204, 3311655637506761961652918, 8670022674717884698740135, 22698405180887488341808046, 59425178168195396589420987, 155577099347739581648471296, 407306058770183604193142820, 1066340952447288733391525355, 2791716544928951582475142139, 7308808165830731950166501507, 19134706901098832066579968107, 50095310397643451699265465405, 131151219938399921635762299056, 343358340563119157827436710752, 898923783746935335906025513067, 2353412974079346607650889481872, 6161315064113832381849730878959, 16130532067146423086069656381482, 42230280830371615489832299442696, 110560309800616047644634073808957, 289450647305882310035005004786719, 757791629548063515779293944376573, 1983924236124815640214788512501615, 5193981068248282483237066314531311, 13598018947161598511336742580142932, 35600075729715183544646511136563626, 93202208153732367290734861076088895, 244006548552560897024592954235284210, 638817437141272577898655249218009619, 1672445762136229938345678255197491892, 4378519847778020627373156373136563132, 11463113778180353098015676444145691855, 30010821480650709722702307851986200051, 78569350651392375677237444362821783447, 205697230448458131513481074669772006147, 538522340643226477252505188375630267297, 1409869791378472276983347927640491614824, 3691087033284216427097594708362791841492, 9663391308053277390952406903982181811811, 25299086890023912998731284374453156270612, 66233869360295249070556196876717706227559, 173402521187375795062309675401584028749278, 453973694194780829652603303282208968597120
Generating function in Maple syntax:
-(x^8+4*x^7-40*x^6+97*x^5-125*x^4+95*x^3-42*x^2+10*x-1)/(x^2-3*x+1)/(2*x-1)^2/(x-1)^4
Generating function in latex syntax:
-\frac{x^{8}+4 x^{7}-40 x^{6}+97 x^{5}-125 x^{4}+95 x^{3}-42 x^{2}+10 x -1}{\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{4}}
Generating function in sympy syntax:
(-x**8 - 4*x**7 + 40*x**6 - 97*x**5 + 125*x**4 - 95*x**3 + 42*x**2 - 10*x + 1)/((x - 1)**4*(2*x - 1)**2*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(2*x-1)^2*(x-1)^4*F(x)+x^8+4*x^7-40*x^6+97*x^5-125*x^4+95*x^3-42*x^2+10*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{4} F \! \left(x \right)+x^{8}+4 x^{7}-40 x^{6}+97 x^{5}-125 x^{4}+95 x^{3}-42 x^{2}+10 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/60*(6*5^(1/2)+30)*(3/2-1/2*5^(1/2))^(-n)+1/60*(-6*5^(1/2)+30)*(3/2+1/2*5^(1/2))^(-n)+1/60*(15*n-105)*2^n-1/6*n^3+1/2*n^2-1/3*n+2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(6 \sqrt{5}+30\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{60}+\frac{\left(-6 \sqrt{5}+30\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{60}+\frac{\left(15 n -105\right) 2^{n}}{60}\\-\frac{n^{3}}{6}+\frac{n^{2}}{2}-\frac{n}{3}+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) = 20
a(5) = 65
a(6) = 199
a(7) = 577
a(8) = 1607
a(n+4) = 1/6*n^3-n^2-4*a(n)+16*a(n+1)-17*a(n+2)+7*a(n+3)-7/6*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) = 65
a \! \left(6\right) = 199
a \! \left(7\right) = 577
a \! \left(8\right) = 1607
a \! \left(n +4\right) = \frac{n^{3}}{6}-n^{2}-4 a \! \left(n \right)+16 a \! \left(n +1\right)-17 a \! \left(n +2\right)+7 a \! \left(n +3\right)-\frac{7 n}{6}+1, \quad n \geq 9
Specification 1
Strategy pack name: insertion_point_placements
Tree: http://permpal.com/tree/21300/
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[22,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[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[8,x]
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[8,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[20,x]*F[8,x]
F[20,x] = F[11,x]+F[17,x]
F[21,x] = F[13,x]*F[8,x]
F[22,x] = F[2,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[8,x]
F[25,x] = F[26,x]+F[58,x]
F[26,x] = F[27,x]+F[52,x]
F[27,x] = F[28,x]+F[6,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[8,x]
F[30,x] = F[31,x]+F[41,x]
F[31,x] = F[32,x]+F[39,x]
F[32,x] = F[11,x]*F[33,x]
F[33,x] = F[10,x]+F[34,x]
F[34,x] = F[14,x]+F[35,x]
F[35,x] = F[18,x]+F[36,x]+F[38,x]
F[36,x] = F[37,x]*F[8,x]
F[37,x] = F[11,x]+F[35,x]
F[38,x] = F[34,x]*F[8,x]
F[39,x] = F[16,x]*F[40,x]
F[40,x] = F[21,x]
F[41,x] = F[10,x]*F[42,x]
F[42,x] = F[11,x]+F[43,x]+F[51,x]
F[43,x] = F[44,x]
F[44,x] = F[45,x]*F[8,x]
F[45,x] = F[42,x]+F[46,x]
F[46,x] = F[11,x]*F[47,x]
F[47,x] = F[10,x]+F[48,x]+F[50,x]
F[48,x] = F[49,x]
F[49,x] = F[10,x]*F[47,x]*F[8,x]
F[50,x] = F[47,x]*F[8,x]
F[51,x] = F[42,x]*F[8,x]
F[52,x] = F[53,x]*F[6,x]
F[53,x] = F[10,x]+F[54,x]
F[54,x] = F[55,x]+F[57,x]
F[55,x] = F[56,x]
F[56,x] = F[33,x]*F[8,x]
F[57,x] = F[44,x]
F[58,x] = F[23,x]+F[59,x]
F[59,x] = F[60,x]
F[60,x] = F[14,x]*F[55,x]*F[61,x]*F[8,x]
F[61,x] = F[5,x]+F[6,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_{22}\! \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_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{8}\! \left(x \right)
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_{16}\! \left(x \right) F_{8}\! \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_{20}\! \left(x \right) F_{8}\! \left(x \right)
F_{20}\! \left(x \right) = F_{11}\! \left(x \right)+F_{17}\! \left(x \right)
F_{21}\! \left(x \right) = F_{13}\! \left(x \right) F_{8}\! \left(x \right)
F_{22}\! \left(x \right) = F_{2}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{8}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{58}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{52}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{6}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{8}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{41}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{39}\! \left(x \right)
F_{32}\! \left(x \right) = F_{11}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{10}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{14}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{18}\! \left(x \right)+F_{36}\! \left(x \right)+F_{38}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{8}\! \left(x \right)
F_{37}\! \left(x \right) = F_{11}\! \left(x \right)+F_{35}\! \left(x \right)
F_{38}\! \left(x \right) = F_{34}\! \left(x \right) F_{8}\! \left(x \right)
F_{39}\! \left(x \right) = F_{16}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{21}\! \left(x \right)
F_{41}\! \left(x \right) = F_{10}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{11}\! \left(x \right)+F_{43}\! \left(x \right)+F_{51}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right) F_{8}\! \left(x \right)
F_{45}\! \left(x \right) = F_{42}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{11}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{10}\! \left(x \right)+F_{48}\! \left(x \right)+F_{50}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{10}\! \left(x \right) F_{47}\! \left(x \right) F_{8}\! \left(x \right)
F_{50}\! \left(x \right) = F_{47}\! \left(x \right) F_{8}\! \left(x \right)
F_{51}\! \left(x \right) = F_{42}\! \left(x \right) F_{8}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right) F_{6}\! \left(x \right)
F_{53}\! \left(x \right) = F_{10}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{57}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{33}\! \left(x \right) F_{8}\! \left(x \right)
F_{57}\! \left(x \right) = F_{44}\! \left(x \right)
F_{58}\! \left(x \right) = F_{23}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{14}\! \left(x \right) F_{55}\! \left(x \right) F_{61}\! \left(x \right) F_{8}\! \left(x \right)
F_{61}\! \left(x \right) = F_{5}\! \left(x \right)+F_{6}\! \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_22(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_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_8(x))
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_8(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_20(x)*F_8(x))
Eq(F_20(x), F_11(x) + F_17(x))
Eq(F_21(x), F_13(x)*F_8(x))
Eq(F_22(x), F_2(x) + F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_8(x))
Eq(F_25(x), F_26(x) + F_58(x))
Eq(F_26(x), F_27(x) + F_52(x))
Eq(F_27(x), F_28(x) + F_6(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_8(x))
Eq(F_30(x), F_31(x) + F_41(x))
Eq(F_31(x), F_32(x) + F_39(x))
Eq(F_32(x), F_11(x)*F_33(x))
Eq(F_33(x), F_10(x) + F_34(x))
Eq(F_34(x), F_14(x) + F_35(x))
Eq(F_35(x), F_18(x) + F_36(x) + F_38(x))
Eq(F_36(x), F_37(x)*F_8(x))
Eq(F_37(x), F_11(x) + F_35(x))
Eq(F_38(x), F_34(x)*F_8(x))
Eq(F_39(x), F_16(x)*F_40(x))
Eq(F_40(x), F_21(x))
Eq(F_41(x), F_10(x)*F_42(x))
Eq(F_42(x), F_11(x) + F_43(x) + F_51(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_45(x)*F_8(x))
Eq(F_45(x), F_42(x) + F_46(x))
Eq(F_46(x), F_11(x)*F_47(x))
Eq(F_47(x), F_10(x) + F_48(x) + F_50(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_10(x)*F_47(x)*F_8(x))
Eq(F_50(x), F_47(x)*F_8(x))
Eq(F_51(x), F_42(x)*F_8(x))
Eq(F_52(x), F_53(x)*F_6(x))
Eq(F_53(x), F_10(x) + F_54(x))
Eq(F_54(x), F_55(x) + F_57(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_33(x)*F_8(x))
Eq(F_57(x), F_44(x))
Eq(F_58(x), F_23(x) + F_59(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_14(x)*F_55(x)*F_61(x)*F_8(x))
Eq(F_61(x), F_5(x) + F_6(x))
Pack JSON:
{"expansion_strats": [[{"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}, {"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": true, "maxreqlen": 1, "one_cell_only": true, "strategy_class": "CellInsertionFactory"}], "iterative": false, "name": "insertion_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": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "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, 1, 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": [3, 0, 2, 1], "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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