0132_1023_1032
Counting sequence:
1, 1, 2, 6, 21, 77, 286, 1066, 3977, 14841, 55386, 206702, 771421, 2878981, 10744502, 40099026, 149651601, 558507377, 2084377906, 7779004246, 29031639077, 108347552061, 404358569166, 1509086724602, 5631988329241, 21018866592361, 78443478040202, 292755045568446, 1092576704233581, 4077551771365877, 15217630381229926, 56792969753553826, 211954248632985377, 791024024778387681, 2952141850480565346, 11017543377143873702, 41118031658094929461, 153454583255235844141, 572700301362848447102, 2137346622196157944266, 7976686187421783329961, 29769398127490975375577, 111100906322542118172346, 414634227162677497313806, 1547436002328167871082877, 5775109782149993987017701, 21553003126271808076987926, 80436902722937238320934002, 300194607765477145206748081, 1120341528338971342506058321, 4181171505590408224817485202, 15604344494022661556763882486, 58236206470500238002238044741, 217340481387978290452188296477, 811125719081412923806515141166, 3027162394937673404773872268186, 11297523860669280695288973931577, 42162933047739449376382023458121, 157354208330288516810239119900906, 587253900273414617864574456145502, 2191661392763369954648058704681101, 8179391670780065200727660362578901, 30525905290356890848262582745634502, 113924229490647498192322670619959106, 425171012672233101921028099734201921, 1586759821198284909491789728316848577, 5921868272120906536046130813533192386, 22100713267285341234692733525815920966, 82480984797020458402724803289730491477, 307823225920796492376206479633106044941, 1148811918886165511102101115242693688286, 4287424449623865552032197981337668708202, 16000885879609296697026690810107981144521, 59716119068813321236074565259094255869881, 222863590395643988247271570226269042335002, 831738242513762631753011715645981913470126, 3104089379659406538764775292357658611545501, 11584619276123863523306089453784652532711877, 43234387724836047554459582522780951519302006, 161352931623220326694532240637339153544496146, 602177338768045259223669380026575662658682577, 2247356423448960710200145279468963497090234161, 8387248355027797581576911737849278325702254066, 31301636996662229616107501671928149805718782102, 116819299631621120882853094949863320897172874341, 435975561529822253915304878127525133782972715261, 1627082946487667894778366417560237214234717986702, 6072356224420849325198160792113423723155899231546, 22662341951195729406014276750893457678388878939481, 84577011580362068298858946211460406990399616526377, 315645704370252543789421508094948170283209587166026, 1178005805900648106858827086168332274142438732137726, 4396377519232339883645886836578380926286545341384877, 16407504271028711427724720260145191431003742633401781, 61233639564882505827252994204002384797728425192222246, 228527053988501311881287256555864347759909958135487202, 852874576389122741697896032019455006241911407349726561, 3182971251567989654910296871521955677207735671263419041, 11879010429882835877943291454068367702589031277703949602, 44333070467963353856862868944751515133148389439552379366, 165453271441970579549508184324937692830004526480505567861
Generating function in Maple syntax:
-(2*x^2-4*x+1)/(x-1)/(x^2-4*x+1)
Generating function in latex syntax:
-\frac{2 x^{2}-4 x +1}{\left(x -1\right) \left(x^{2}-4 x +1\right)}
Generating function in sympy syntax:
(-2*x**2 + 4*x - 1)/((x - 1)*(x**2 - 4*x + 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(x^2-4*x+1)*F(x)+2*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(x^{2}-4 x +1\right) F \! \left(x \right)+2 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
1/2-1/12*(2-3^(1/2))^(-n+1)-1/12*(2+3^(1/2))^(-n+1)+3/4*(2-3^(1/2))^(-n)+3/4*(2+3^(1/2))^(-n)-1/6*(2-3^(1/2))^(-n-1)-1/6*(2+3^(1/2))^(-n-1)
Explicit closed form in latex syntax:
\frac{1}{2}-\frac{\left(2-\sqrt{3}\right)^{-n +1}}{12}-\frac{\left(2+\sqrt{3}\right)^{-n +1}}{12}+\frac{3 \left(2-\sqrt{3}\right)^{-n}}{4}+\frac{3 \left(2+\sqrt{3}\right)^{-n}}{4}-\frac{\left(2-\sqrt{3}\right)^{-n -1}}{6}-\frac{\left(2+\sqrt{3}\right)^{-n -1}}{6}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(n+2) = -a(n)+4*a(n+1)-1, n >= 3
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(n +2\right) = -a \! \left(n \right)+4 a \! \left(n +1\right)-1, \quad n \geq 3
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14704/
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[27,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[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[7,x]
F[14,x] = F[15,x]+F[16,x]+F[26,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[19,x]+F[4,x]
F[19,x] = F[20,x]
F[20,x] = F[18,x]*F[4,x]
F[21,x] = F[22,x]+F[24,x]
F[22,x] = F[15,x]+F[16,x]+F[23,x]
F[23,x] = F[4,x]*F[7,x]
F[24,x] = F[25,x]
F[25,x] = F[21,x]*F[4,x]
F[26,x] = F[13,x]*F[4,x]
F[27,x] = F[2,x]+F[28,x]
F[28,x] = F[15,x]+F[29,x]+F[46,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[31,x]+F[32,x]
F[31,x] = F[22,x]+F[4,x]
F[32,x] = F[33,x]+F[35,x]
F[33,x] = F[15,x]+F[29,x]+F[34,x]
F[34,x] = F[2,x]*F[4,x]
F[35,x] = F[15,x]+F[36,x]+F[37,x]+F[45,x]
F[36,x] = 0
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]+F[42,x]
F[39,x] = F[33,x]+F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[39,x]*F[4,x]
F[42,x] = F[35,x]+F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[4,x]*F[42,x]
F[45,x] = F[28,x]*F[4,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]+F[51,x]
F[48,x] = F[2,x]+F[49,x]
F[49,x] = F[15,x]+F[29,x]+F[50,x]
F[50,x] = F[4,x]*F[48,x]
F[51,x] = F[28,x]+F[52,x]
F[52,x] = F[15,x]+F[36,x]+F[37,x]+F[53,x]
F[53,x] = F[4,x]*F[51,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_{27}\! \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_{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_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{26}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{21}\! \left(x \right)
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_{20}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{24}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{2}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{15}\! \left(x \right)+F_{29}\! \left(x \right)+F_{46}\! \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_{32}\! \left(x \right)
F_{31}\! \left(x \right) = F_{22}\! \left(x \right)+F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{35}\! \left(x \right)
F_{33}\! \left(x \right) = F_{15}\! \left(x \right)+F_{29}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{15}\! \left(x \right)+F_{36}\! \left(x \right)+F_{37}\! \left(x \right)+F_{45}\! \left(x \right)
F_{36}\! \left(x \right) = 0
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{42}\! \left(x \right)
F_{39}\! \left(x \right) = F_{33}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{42}\! \left(x \right) = F_{35}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{45}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{51}\! \left(x \right)
F_{48}\! \left(x \right) = F_{2}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{15}\! \left(x \right)+F_{29}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{51}\! \left(x \right) = F_{28}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{15}\! \left(x \right)+F_{36}\! \left(x \right)+F_{37}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \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_27(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_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_4(x))
Eq(F_13(x), F_14(x) + F_7(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_26(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_19(x) + F_4(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_18(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_24(x))
Eq(F_22(x), F_15(x) + F_16(x) + F_23(x))
Eq(F_23(x), F_4(x)*F_7(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_21(x)*F_4(x))
Eq(F_26(x), F_13(x)*F_4(x))
Eq(F_27(x), F_2(x) + F_28(x))
Eq(F_28(x), F_15(x) + F_29(x) + F_46(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_32(x))
Eq(F_31(x), F_22(x) + F_4(x))
Eq(F_32(x), F_33(x) + F_35(x))
Eq(F_33(x), F_15(x) + F_29(x) + F_34(x))
Eq(F_34(x), F_2(x)*F_4(x))
Eq(F_35(x), F_15(x) + F_36(x) + F_37(x) + F_45(x))
Eq(F_36(x), 0)
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x) + F_42(x))
Eq(F_39(x), F_33(x) + F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_39(x)*F_4(x))
Eq(F_42(x), F_35(x) + F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_4(x)*F_42(x))
Eq(F_45(x), F_28(x)*F_4(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_51(x))
Eq(F_48(x), F_2(x) + F_49(x))
Eq(F_49(x), F_15(x) + F_29(x) + F_50(x))
Eq(F_50(x), F_4(x)*F_48(x))
Eq(F_51(x), F_28(x) + F_52(x))
Eq(F_52(x), F_15(x) + F_36(x) + F_37(x) + F_53(x))
Eq(F_53(x), F_4(x)*F_51(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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "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": 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