0132_0321_1320_2130_3021
Counting sequence:
1, 1, 2, 6, 19, 56, 152, 389, 965, 2373, 5868, 14688, 37249, 95508, 246856, 641433, 1672179, 4367671, 11420582, 29880220, 78201685, 204700800, 535869746, 1402869035, 3672687793, 9615132635, 25172634006, 65902676328, 172535282099, 451703034042, 1182573657452, 3096017745087, 8105479349493, 21220420035087, 55555780442080, 145446920926162, 380784981913633, 996908024327110, 2609939090507516, 6832909246554339, 17888788648424419, 46833456697888049, 122611581444298490, 321001287633944414, 840392281456337717, 2200175556733724514, 5760134388743330310, 15080227609494584517, 39480548439738548833, 103361417709718977861, 270603704689416072586, 708449696358526680186, 1854745384386161139987, 4855786456799953621516, 12712613986013696292704, 33282055501241131486441, 87133552517709694032037, 228118602051887946083049, 597222253637954139269300, 1563548158861974466325108, 4093422222947969253821953, 10716718509981933288738248, 28056733306997830605435984, 73453481411011558520020893, 192303710926036844946446291, 503457651367098976310464443, 1318069243175260083975376814, 3450750078158681275605333456, 9034180991300783742829480917, 23651792895743669952871106580, 61921197695930226115770923770, 162111800192047008394427782735, 424414202880210799067497518481, 1111130808448585388808048783295, 2908978222465545367356631696478, 7615803858948050713261827961020, 19938433354378606772428832563891, 52199496204187769604024648760238, 136660055258184702039645091325684, 357780669570366336514910601329027, 936681953452914307505086687198037, 2452265190788376586000349433144147, 6420113618912215450495961583370728, 16808075665948269765487535286273222, 44004113378932593845966644242831265, 115204264470849511772412397407584922, 301608680033615941471270547943171268, 789621775629998312641399246382957895, 2067256646856378996452927191164406851, 5412148164939138676717382327066532949, 14169187847961037033699219789988914754, 37095415378943972424380277042851269234, 97117058288870880239441611338513164725, 254255759487668668293944556972633585174, 665650220174135124642392059579329909902, 1742694901034736705633231621765295288649, 4562434482930074992257302805716491786945, 11944608547755488271138676795384112447177, 31271391160336389821158727580435774326418, 81869564933253681192337505945923135548846, 214337303639424653755853790257333553425171
Generating function in Maple syntax:
-(x^2-x+1)*(x^5-10*x^4+23*x^3-19*x^2+7*x-1)/(x^2-3*x+1)/(x-1)^6
Generating function in latex syntax:
-\frac{\left(x^{2}-x +1\right) \left(x^{5}-10 x^{4}+23 x^{3}-19 x^{2}+7 x -1\right)}{\left(x^{2}-3 x +1\right) \left(x -1\right)^{6}}
Generating function in sympy syntax:
(-x**2 + x - 1)*(x**5 - 10*x**4 + 23*x**3 - 19*x**2 + 7*x - 1)/((x - 1)**6*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x-1)^6*F(x)+(x^2-x+1)*(x^5-10*x^4+23*x^3-19*x^2+7*x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x -1\right)^{6} F \! \left(x \right)+\left(x^{2}-x +1\right) \left(x^{5}-10 x^{4}+23 x^{3}-19 x^{2}+7 x -1\right) = 0
Explicit closed form in Maple syntax:
1/120*(72*5^(1/2)-120)*(3/2-1/2*5^(1/2))^(-n)+1/120*(-72*5^(1/2)-120)*(3/2+1/2*5^(1/2))^(-n)+1/120*n^5-5/24*n^3+n^2-14/5*n+3
Explicit closed form in latex syntax:
\frac{\left(72 \sqrt{5}-120\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{120}+\frac{\left(-72 \sqrt{5}-120\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{120}+\frac{n^{5}}{120}-\frac{5 n^{3}}{24}+n^{2}-\frac{14 n}{5}+3
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 56
a(6) = 152
a(7) = 389
a(n+2) = -a(n)+3*a(n+1)-1/120*n*(n^4+5*n^3-35*n^2-5*n-86), n >= 8
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) = 19
a \! \left(5\right) = 56
a \! \left(6\right) = 152
a \! \left(7\right) = 389
a \! \left(n +2\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)-\frac{n \left(n^{4}+5 n^{3}-35 n^{2}-5 n -86\right)}{120}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19555/
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[12,x]*F[4,x]
F[4,x] = F[20,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,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[9,x]
F[12,x] = x
F[13,x] = F[10,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[19,x]
F[15,x] = 0
F[16,x] = F[12,x]*F[17,x]
F[17,x] = F[10,x]+F[18,x]
F[18,x] = F[16,x]
F[19,x] = F[12,x]*F[13,x]
F[20,x] = F[21,x]+F[27,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[24,x]+F[9,x]
F[24,x] = F[21,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[12,x]*F[24,x]
F[27,x] = F[15,x]+F[28,x]+F[53,x]
F[28,x] = F[12,x]*F[29,x]
F[29,x] = F[30,x]+F[40,x]
F[30,x] = F[31,x]+F[6,x]
F[31,x] = F[15,x]+F[16,x]+F[32,x]
F[32,x] = F[12,x]*F[33,x]
F[33,x] = F[13,x]+F[34,x]
F[34,x] = F[18,x]+F[35,x]
F[35,x] = 2*F[15,x]+F[36,x]+F[39,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[18,x]+F[38,x]
F[38,x] = F[36,x]
F[39,x] = F[12,x]*F[34,x]
F[40,x] = F[27,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[44,x]+F[52,x]
F[44,x] = F[15,x]+F[45,x]+F[50,x]
F[45,x] = F[12,x]*F[46,x]
F[46,x] = F[47,x]+F[48,x]
F[47,x] = F[10,x]+F[14,x]
F[48,x] = F[44,x]+F[49,x]
F[49,x] = F[42,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[21,x]+F[44,x]
F[52,x] = F[42,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[51,x]+F[55,x]
F[55,x] = F[56,x]+F[59,x]
F[56,x] = F[57,x]
F[57,x] = F[12,x]*F[58,x]
F[58,x] = F[17,x]
F[59,x] = 2*F[15,x]+F[60,x]+F[63,x]
F[60,x] = F[12,x]*F[61,x]
F[61,x] = F[62,x]
F[62,x] = F[14,x]+F[35,x]
F[63,x] = F[12,x]*F[55,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{20}\! \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_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \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_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{19}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{10}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{27}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{9}\! \left(x \right)
F_{24}\! \left(x \right) = F_{21}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{27}\! \left(x \right) = F_{15}\! \left(x \right)+F_{28}\! \left(x \right)+F_{53}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{40}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{6}\! \left(x \right)
F_{31}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{13}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{18}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{36}\! \left(x \right)+F_{39}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{18}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{36}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{40}\! \left(x \right) = F_{27}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{52}\! \left(x \right)
F_{44}\! \left(x \right) = F_{15}\! \left(x \right)+F_{45}\! \left(x \right)+F_{50}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{48}\! \left(x \right)
F_{47}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{48}\! \left(x \right) = F_{44}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{42}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{21}\! \left(x \right)+F_{44}\! \left(x \right)
F_{52}\! \left(x \right) = F_{42}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{51}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{59}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{17}\! \left(x \right)
F_{59}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{60}\! \left(x \right)+F_{63}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{14}\! \left(x \right)+F_{35}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_20(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_12(x)*F_8(x))
Eq(F_8(x), F_13(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_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_10(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_19(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_12(x)*F_17(x))
Eq(F_17(x), F_10(x) + F_18(x))
Eq(F_18(x), F_16(x))
Eq(F_19(x), F_12(x)*F_13(x))
Eq(F_20(x), F_21(x) + F_27(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_12(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_9(x))
Eq(F_24(x), F_21(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_12(x)*F_24(x))
Eq(F_27(x), F_15(x) + F_28(x) + F_53(x))
Eq(F_28(x), F_12(x)*F_29(x))
Eq(F_29(x), F_30(x) + F_40(x))
Eq(F_30(x), F_31(x) + F_6(x))
Eq(F_31(x), F_15(x) + F_16(x) + F_32(x))
Eq(F_32(x), F_12(x)*F_33(x))
Eq(F_33(x), F_13(x) + F_34(x))
Eq(F_34(x), F_18(x) + F_35(x))
Eq(F_35(x), 2*F_15(x) + F_36(x) + F_39(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_18(x) + F_38(x))
Eq(F_38(x), F_36(x))
Eq(F_39(x), F_12(x)*F_34(x))
Eq(F_40(x), F_27(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_52(x))
Eq(F_44(x), F_15(x) + F_45(x) + F_50(x))
Eq(F_45(x), F_12(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_48(x))
Eq(F_47(x), F_10(x) + F_14(x))
Eq(F_48(x), F_44(x) + F_49(x))
Eq(F_49(x), F_42(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_21(x) + F_44(x))
Eq(F_52(x), F_42(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_51(x) + F_55(x))
Eq(F_55(x), F_56(x) + F_59(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_12(x)*F_58(x))
Eq(F_58(x), F_17(x))
Eq(F_59(x), 2*F_15(x) + F_60(x) + F_63(x))
Eq(F_60(x), F_12(x)*F_61(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_14(x) + F_35(x))
Eq(F_63(x), F_12(x)*F_55(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": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "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": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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