0213_0231_1320_3012

Counting sequence:
1, 1, 2, 6, 20, 62, 178, 489, 1312, 3477, 9157, 24041, 63023, 165096, 432345, 1132032, 2963858, 7759664, 20315272, 53186307, 139243822, 364545351, 954392443, 2498632211, 6541504445, 17125881402, 44836140063, 117382539114, 307311477632, 804551894162, 2106344205262, 5514480722061, 14437097961388, 37796813162601, 98953341526945, 259063211418797, 678236292730043, 1775645666771964, 4648700707586517, 12170456455988292, 31862668660379102, 83417549525149796, 218389979915071108, 571752390220064391, 1496867190745122970, 3918849182015305467, 10259680355300794423, 26860191883887078839, 70320895296360443177, 184102494005194251822, 481986586719222313467, 1261857266152472689806, 3303585211738195757228, 8648898369062114583206, 22643109895448147993770, 59280431317282329399537, 155198184056398840206328, 406314120851914191220989, 1063744178499343733458237, 2784918414646117009155377, 7291011065439007294009607, 19088114781670904872875216, 49973333279573707324617873, 130831885057050217100980296, 342522321891576943978324970, 896735080617680614833996632, 2347682919961464900523667008, 6146313679266714086737006539, 16091258117838677359687354822, 42127460674249317992325060207, 110291123904909276617287828147, 288745911040478511859538426651, 755946609216526258961327454293, 1979093916609100265024443938786, 5181335140610774536112004364695, 13564911505223223343311569158002, 35513399375058895493822703112088, 92975286619953463138156540181114, 243412460484801493920646917434182, 637262094834451018623784212124437, 1668373824018551561950705718942212, 4367859377221203667228332944705361, 11435204307645059439734293115177113, 29937753545713974651974546400829301, 78378056329496864516189346087314195, 205196415442776618896593491861116772, 537211189998832992173591129496039693, 1406437154553722357624179896627005964, 3682100273662334080698948560384981942, 9639863666433279884472665784527943692, 25237490725637505572719048793198853052, 66072608510479236833684480595068619471, 172980334805800204928334392992007009458, 452868395906921377951318698380952413091, 1185624852914963928925621702150850234095, 3104006162837970408825546408071598293567, 8126393635598947297551017522063944651073, 21275174743958871483827506158120235664214, 55699130596277667153931500952296762346227, 145822217044874129977966996698770051379222, 381767520538344722779969489144013391796292

Generating function in Maple syntax:
-(x^7-x^6-x^5+3*x^4-6*x^3+9*x^2-5*x+1)/(x^2-3*x+1)/(x-1)^3

Generating function in latex syntax:
-\frac{x^{7}-x^{6}-x^{5}+3 x^{4}-6 x^{3}+9 x^{2}-5 x +1}{\left(x^{2}-3 x +1\right) \left(x -1\right)^{3}}

Generating function in sympy syntax:
(-x**7 + x**6 + x**5 - 3*x**4 + 6*x**3 - 9*x**2 + 5*x - 1)/((x - 1)**3*(x**2 - 3*x + 1))

Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x-1)^3*F(x)+x^7-x^6-x^5+3*x^4-6*x^3+9*x^2-5*x+1 = 0

Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{7}-x^{6}-x^{5}+3 x^{4}-6 x^{3}+9 x^{2}-5 x +1 = 0

Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/10*(-42*5^(1/2)+100)*(3/2-1/2*5^(1/2))^(-n)+1/10*(42*5^(1/2)+100)*(3/2+1/2*5^(1/2))^(-n)-1/2*n^2+1/2*n-3)

Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(-42 \sqrt{5}+100\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{10}+\frac{\left(42 \sqrt{5}+100\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{10}-\frac{n^{2}}{2}+\frac{n}{2}-3 & \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) = 62
a(6) = 178
a(7) = 489
a(n+2) = 1/2*n^2-a(n)+3*a(n+1)+1/2*n+2, 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) = 20
a \! \left(5\right) = 62
a \! \left(6\right) = 178
a \! \left(7\right) = 489
a \! \left(n +2\right) = \frac{n^{2}}{2}-a \! \left(n \right)+3 a \! \left(n +1\right)+\frac{n}{2}+2, \quad n \geq 8

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15708/
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[17,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[6,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[4,x]
F[12,x] = F[13,x]+F[16,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[4,x]
F[16,x] = F[11,x]
F[17,x] = F[18,x]+F[45,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[13,x]+F[21,x]
F[21,x] = F[18,x]+F[22,x]
F[22,x] = F[23,x]+F[24,x]+F[43,x]
F[23,x] = 0
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]+F[30,x]
F[26,x] = F[14,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[14,x]
F[30,x] = F[22,x]+F[31,x]
F[31,x] = 2*F[23,x]+F[32,x]+F[36,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]+F[35,x]
F[34,x] = F[27,x]
F[35,x] = F[31,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[42,x]
F[41,x] = F[14,x]
F[42,x] = F[38,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[18,x]+F[38,x]
F[45,x] = F[23,x]+F[46,x]+F[67,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]+F[57,x]
F[48,x] = F[49,x]+F[7,x]
F[49,x] = F[23,x]+F[50,x]+F[52,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[7,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]+F[55,x]
F[54,x] = F[14,x]
F[55,x] = F[56,x]
F[56,x] = F[4,x]*F[41,x]
F[57,x] = F[45,x]+F[58,x]
F[58,x] = F[23,x]+F[59,x]+F[63,x]+F[65,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[61,x]+F[62,x]
F[61,x] = F[49,x]
F[62,x] = F[58,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[45,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[38,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[17,x]+F[69,x]
F[69,x] = F[39,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_{17}\! \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_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{16}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{11}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{45}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{18}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{43}\! \left(x \right)
F_{23}\! \left(x \right) = 0
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{30}\! \left(x \right)
F_{26}\! \left(x \right) = F_{14}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{14}\! \left(x \right)
F_{30}\! \left(x \right) = F_{22}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = 2 F_{23}\! \left(x \right)+F_{32}\! \left(x \right)+F_{36}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{35}\! \left(x \right)
F_{34}\! \left(x \right) = F_{27}\! \left(x \right)
F_{35}\! \left(x \right) = F_{31}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{42}\! \left(x \right)
F_{41}\! \left(x \right) = F_{14}\! \left(x \right)
F_{42}\! \left(x \right) = F_{38}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{18}\! \left(x \right)+F_{38}\! \left(x \right)
F_{45}\! \left(x \right) = F_{23}\! \left(x \right)+F_{46}\! \left(x \right)+F_{67}\! \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_{57}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{7}\! \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_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{7}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{55}\! \left(x \right)
F_{54}\! \left(x \right) = F_{14}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{57}\! \left(x \right) = F_{45}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{23}\! \left(x \right)+F_{59}\! \left(x \right)+F_{63}\! \left(x \right)+F_{65}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{62}\! \left(x \right)
F_{61}\! \left(x \right) = F_{49}\! \left(x \right)
F_{62}\! \left(x \right) = F_{58}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{45}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{38}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{17}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{39}\! \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_17(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_6(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_4(x))
Eq(F_12(x), F_13(x) + F_16(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_4(x))
Eq(F_16(x), F_11(x))
Eq(F_17(x), F_18(x) + F_45(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_13(x) + F_21(x))
Eq(F_21(x), F_18(x) + F_22(x))
Eq(F_22(x), F_23(x) + F_24(x) + F_43(x))
Eq(F_23(x), 0)
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x) + F_30(x))
Eq(F_26(x), F_14(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_14(x))
Eq(F_30(x), F_22(x) + F_31(x))
Eq(F_31(x), 2*F_23(x) + F_32(x) + F_36(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_35(x))
Eq(F_34(x), F_27(x))
Eq(F_35(x), F_31(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_42(x))
Eq(F_41(x), F_14(x))
Eq(F_42(x), F_38(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_18(x) + F_38(x))
Eq(F_45(x), F_23(x) + F_46(x) + F_67(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_57(x))
Eq(F_48(x), F_49(x) + F_7(x))
Eq(F_49(x), F_23(x) + F_50(x) + F_52(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_7(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_55(x))
Eq(F_54(x), F_14(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_4(x)*F_41(x))
Eq(F_57(x), F_45(x) + F_58(x))
Eq(F_58(x), F_23(x) + F_59(x) + F_63(x) + F_65(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_61(x) + F_62(x))
Eq(F_61(x), F_49(x))
Eq(F_62(x), F_58(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_45(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_38(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_17(x) + F_69(x))
Eq(F_69(x), F_39(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, 2, 1, 3], "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, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 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, 2, 1, 3], "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, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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