0132_0231_1023_2103
Counting sequence:
1, 1, 2, 6, 20, 62, 186, 556, 1666, 4998, 14996, 44990, 134970, 404908, 1214722, 3644166, 10932500, 32797502, 98392506, 295177516, 885532546, 2656597638, 7969792916, 23909378750, 71728136250, 215184408748, 645553226242, 1936659678726, 5809979036180, 17429937108542, 52289811325626, 156869433976876, 470608301930626, 1411824905791878, 4235474717375636, 12706424152126910, 38119272456380730, 114357817369142188, 343073452107426562, 1029220356322279686, 3087661068966839060, 9262983206900517182, 27788949620701551546, 83366848862104654636, 250100546586313963906, 750301639758941891718, 2250904919276825675156, 6752714757830477025470, 20258144273491431076410, 60774432820474293229228, 182323298461422879687682, 546969895384268639063046, 1640909686152805917189140, 4922729058458417751567422, 14768187175375253254702266, 44304561526125759764106796, 132913684578377279292320386, 398741053735131837876961158, 1196223161205395513630883476, 3588669483616186540892650430, 10766008450848559622677951290, 32298025352545678868033853868, 96894076057637036604101561602, 290682228172911109812304684806, 872046684518733329436914054420, 2616140053556199988310742163262, 7848420160668599964932226489786, 23545260482005799894796679469356, 70635781446017399684390038408066, 211907344338052199053170115224198, 635722033014156597159510345672596, 1907166099042469791478531037017790, 5721498297127409374435593111053370, 17164494891382228123306779333160108, 51493484674146684369920337999480322, 154480454022440053109761013998440966, 463441362067320159329283041995322900, 1390324086201960477987849125985968702, 4170972258605881433963547377957906106, 12512916775817644301890642133873718316, 37538750327452932905671926401621154946, 112616250982358798717015779204863464838, 337848752947076396151047337614590394516, 1013546258841229188453142012843771183550, 3040638776523687565359426038531313550650, 9121916329571062696078278115593940651948, 27365748988713188088234834346781821955842, 82097246966139564264704503040345465867526, 246291740898418692794113509121036397602580, 738875222695256078382340527363109192807742, 2216625668085768235147021582089327578423226, 6649877004257304705441064746267982735269676, 19949631012771914116323194238803948205809026, 59848893038315742348969582716411844617427078, 179546679114947227046908748149235533852281236, 538640037344841681140726244447706601556843710, 1615920112034525043422178733343119804670531130, 4847760336103575130266536200029359414011593388, 14543281008310725390799608600088078242034780162, 43629843024932176172398825800264234726104340486, 130889529074796528517196477400792704178313021460
Generating function in Maple syntax:
-(x-1)*(x^3+2*x-1)/(3*x-1)/(x^2-x+1)
Generating function in latex syntax:
-\frac{\left(x -1\right) \left(x^{3}+2 x -1\right)}{\left(3 x -1\right) \left(x^{2}-x +1\right)}
Generating function in sympy syntax:
(1 - x)*(x**3 + 2*x - 1)/((3*x - 1)*(x**2 - x + 1))
Implicit equation for the generating function in Maple syntax:
(3*x-1)*(x^2-x+1)*F(x)+(x-1)*(x^3+2*x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(3 x -1\right) \left(x^{2}-x +1\right) F \! \left(x \right)+\left(x -1\right) \left(x^{3}+2 x -1\right) = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/63*(3*I*3^(1/2)+27)*(1/2+1/2*I*3^(1/2))^(-n)-1/21*I*(1/2-1/2*I*3^(1/2))^(-n)*3^(1/2)+16/63*3^n+3/7*(1/2-1/2*I*3^(1/2))^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(3 \,\mathrm{I} \sqrt{3}+27\right) \left(\frac{1}{2}+\frac{\mathrm{I} \sqrt{3}}{2}\right)^{-n}}{63}-\frac{\mathrm{I} \left(\frac{1}{2}-\frac{\mathrm{I} \sqrt{3}}{2}\right)^{-n} \sqrt{3}}{21}+\frac{16 \,3^{n}}{63}+\\\frac{3 \left(\frac{1}{2}-\frac{\mathrm{I} \sqrt{3}}{2}\right)^{-n}}{7} & \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(n+3) = 3*a(n)-4*a(n+1)+4*a(n+2), n >= 5
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(n +3\right) = 3 a \! \left(n \right)-4 a \! \left(n +1\right)+4 a \! \left(n +2\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15324/
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[20,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[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]
F[17,x] = F[18,x]+F[4,x]
F[18,x] = F[19,x]
F[19,x] = F[17,x]*F[4,x]
F[20,x] = F[2,x]+F[21,x]
F[21,x] = F[22,x]+F[23,x]+F[46,x]
F[22,x] = 0
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[32,x]
F[25,x] = F[26,x]+F[29,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[1,x]+F[26,x]
F[29,x] = F[15,x]+F[22,x]+F[30,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[29,x]+F[7,x]
F[32,x] = F[33,x]+F[36,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[2,x]+F[33,x]
F[36,x] = 2*F[22,x]+F[37,x]+F[44,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]
F[39,x] = F[40,x]+F[42,x]
F[40,x] = F[41,x]
F[41,x] = F[2,x]*F[4,x]
F[42,x] = F[43,x]
F[43,x] = F[39,x]*F[4,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[21,x]+F[36,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]+F[59,x]
F[48,x] = F[2,x]+F[49,x]
F[49,x] = F[22,x]+F[50,x]+F[58,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]+F[55,x]
F[52,x] = F[4,x]+F[53,x]
F[53,x] = F[15,x]+F[22,x]+F[54,x]
F[54,x] = F[4,x]*F[7,x]
F[55,x] = F[40,x]+F[56,x]
F[56,x] = 2*F[22,x]+F[37,x]+F[57,x]
F[57,x] = F[21,x]*F[4,x]
F[58,x] = F[4,x]*F[48,x]
F[59,x] = F[21,x]+F[60,x]
F[60,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_{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_{20}\! \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_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{2}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{46}\! \left(x \right)
F_{22}\! \left(x \right) = 0
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{32}\! \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_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{1}\! \left(x \right)+F_{26}\! \left(x \right)
F_{29}\! \left(x \right) = F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{29}\! \left(x \right)+F_{7}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{36}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{2}\! \left(x \right)+F_{33}\! \left(x \right)
F_{36}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{37}\! \left(x \right)+F_{44}\! \left(x \right)
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_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{42}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{21}\! \left(x \right)+F_{36}\! \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_{59}\! \left(x \right)
F_{48}\! \left(x \right) = F_{2}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{22}\! \left(x \right)+F_{50}\! \left(x \right)+F_{58}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{55}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{55}\! \left(x \right) = F_{40}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{37}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{59}\! \left(x \right) = F_{21}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \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_3(x))
Eq(F_3(x), F_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_20(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))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_18(x) + F_4(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_17(x)*F_4(x))
Eq(F_20(x), F_2(x) + F_21(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_46(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_32(x))
Eq(F_25(x), F_26(x) + F_29(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_1(x) + F_26(x))
Eq(F_29(x), F_15(x) + F_22(x) + F_30(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_29(x) + F_7(x))
Eq(F_32(x), F_33(x) + F_36(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_2(x) + F_33(x))
Eq(F_36(x), 2*F_22(x) + F_37(x) + F_44(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_40(x) + F_42(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_2(x)*F_4(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_39(x)*F_4(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_21(x) + F_36(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_59(x))
Eq(F_48(x), F_2(x) + F_49(x))
Eq(F_49(x), F_22(x) + F_50(x) + F_58(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_55(x))
Eq(F_52(x), F_4(x) + F_53(x))
Eq(F_53(x), F_15(x) + F_22(x) + F_54(x))
Eq(F_54(x), F_4(x)*F_7(x))
Eq(F_55(x), F_40(x) + F_56(x))
Eq(F_56(x), 2*F_22(x) + F_37(x) + F_57(x))
Eq(F_57(x), F_21(x)*F_4(x))
Eq(F_58(x), F_4(x)*F_48(x))
Eq(F_59(x), F_21(x) + F_60(x))
Eq(F_60(x), F_37(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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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