0213_0231_1032_3012
Counting sequence:
1, 1, 2, 6, 20, 62, 183, 526, 1490, 4184, 11680, 32462, 89897, 248186, 683330, 1876842, 5143628, 14068286, 38407515, 104678854, 284856530, 774048572, 2100535192, 5693137166, 15412370765, 41678878322, 112595479298, 303886301646, 819428436740, 2207715299774, 5943301325055, 15987623971582, 42976292615762, 115445984657696, 309919131677200, 831479090551502, 2229463710190193, 5974581070480106, 16002341022420290, 42838957529830194, 114626016645048380, 306567032106199742, 819547413698227107, 2189964271361625526, 5849540253481405010, 15618351985993712132, 41685407342138343688, 111217849456426033166, 296628187637515286165, 790866873871241702114, 2107912868611199920130, 5616492875452206481302, 14960488753579974692852, 39838121229301534681406, 106054395470531525434695, 282253478946899912788462, 750990762127781930344658, 1997644555944680159321576, 5312435430473347673434240, 14124213561268395957347534, 37543368205943850361893305, 99770828088606152521820762, 265080764203391589220749122, 704141471920076563798478586, 1870042025608844966131662380, 4965409719663970977809383742, 13181764103603599032978138603, 34987188387050906674957106086, 92846141475040831587707751890, 246342951494642639321777792204, 653491518961108529482645980280, 1733266307788776227207609572622, 4596402705653278543279510653917, 12187093010515141297967456714066, 32308234628676332645493134546690, 85636534582522336627786237776222, 226954781937131929910818176274148, 601389125976588731134251792672446, 1593343127417538844907735109129615, 4220870535803026269806763755249374, 11179828787147630781750188910830930, 29608066467581140060938891019347632, 78402162234263520540314115519312112, 207582344449271340992251261612784846, 549538852137068529593935456169530625, 1454632230818426409829794661373075402, 3849957915864169158858671404531012418, 10188443724555464281595648627486643906, 26959379805600414871513338827147648540, 71328513127858970674850131826345807294, 188698605337017876993169284219839560755, 499145822724705609483148639564632731542, 1320201976602590419151617163100488416082, 3491470928538029101879233516884662006676, 9232780159610895780513334860370556291560, 24412566780637891467834994815530353440782, 64543442522733079413487069367313485061413, 170627630578509015915938248878385698262466, 451030536245206674973768364262677418252290, 1192127369463401459780376869302172385554214, 3150650659031456350052951632826583417063380
Generating function in Maple syntax:
-(x^7-4*x^6+6*x^5-10*x^4+20*x^3-18*x^2+7*x-1)/(x-1)^2/(x^2-3*x+1)^2
Generating function in latex syntax:
-\frac{x^{7}-4 x^{6}+6 x^{5}-10 x^{4}+20 x^{3}-18 x^{2}+7 x -1}{\left(x -1\right)^{2} \left(x^{2}-3 x +1\right)^{2}}
Generating function in sympy syntax:
(-x**7 + 4*x**6 - 6*x**5 + 10*x**4 - 20*x**3 + 18*x**2 - 7*x + 1)/((x - 1)**2*(x**2 - 3*x + 1)**2)
Implicit equation for the generating function in Maple syntax:
(x-1)^2*(x^2-3*x+1)^2*F(x)+x^7-4*x^6+6*x^5-10*x^4+20*x^3-18*x^2+7*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right)^{2} \left(x^{2}-3 x +1\right)^{2} F \! \left(x \right)+x^{7}-4 x^{6}+6 x^{5}-10 x^{4}+20 x^{3}-18 x^{2}+7 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/25*((5*n-19)*5^(1/2)-10*n+50)*(3/2-1/2*5^(1/2))^(-n)+1/25*((-5*n+19)*5^(1/2)-10*n+50)*(3/2+1/2*5^(1/2))^(-n)-n+1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(\left(5 n -19\right) \sqrt{5}-10 n +50\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{25}+\\\frac{\left(\left(-5 n +19\right) \sqrt{5}-10 n +50\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{25}-n +1 & \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) = 183
a(7) = 526
a(n+4) = -a(n)+6*a(n+1)-11*a(n+2)+6*a(n+3)-n-1, 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) = 183
a \! \left(7\right) = 526
a \! \left(n +4\right) = -a \! \left(n \right)+6 a \! \left(n +1\right)-11 a \! \left(n +2\right)+6 a \! \left(n +3\right)-n -1, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15698/
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[35,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[34,x]
F[23,x] = 0
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]+F[29,x]
F[26,x] = F[27,x]+F[4,x]
F[27,x] = F[28,x]
F[28,x] = F[14,x]*F[4,x]
F[29,x] = F[30,x]+F[32,x]
F[30,x] = F[23,x]+F[24,x]+F[31,x]
F[31,x] = F[18,x]*F[4,x]
F[32,x] = F[33,x]
F[33,x] = F[22,x]*F[4,x]
F[34,x] = F[21,x]*F[4,x]
F[35,x] = F[23,x]+F[36,x]+F[63,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[47,x]
F[38,x] = F[39,x]+F[42,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[1,x]+F[39,x]
F[42,x] = F[23,x]+F[43,x]+F[45,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[39,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[14,x]
F[47,x] = F[48,x]+F[51,x]
F[48,x] = F[23,x]+F[36,x]+F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[18,x]+F[48,x]
F[51,x] = F[23,x]+F[52,x]+F[59,x]+F[61,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]+F[56,x]
F[54,x] = F[55,x]
F[55,x] = F[39,x]*F[4,x]
F[56,x] = F[57,x]
F[57,x] = 2*F[23,x]+F[52,x]+F[58,x]
F[58,x] = F[4,x]*F[48,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[48,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[22,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[17,x]+F[65,x]
F[65,x] = F[66,x]
F[66,x] = F[23,x]+F[24,x]+F[67,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[21,x]+F[69,x]
F[69,x] = F[66,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_{35}\! \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_{34}\! \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_{29}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{32}\! \left(x \right)
F_{30}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{23}\! \left(x \right)+F_{36}\! \left(x \right)+F_{63}\! \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_{47}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{42}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{1}\! \left(x \right)+F_{39}\! \left(x \right)
F_{42}\! \left(x \right) = F_{23}\! \left(x \right)+F_{43}\! \left(x \right)+F_{45}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{39}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{14}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{51}\! \left(x \right)
F_{48}\! \left(x \right) = F_{23}\! \left(x \right)+F_{36}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{18}\! \left(x \right)+F_{48}\! \left(x \right)
F_{51}\! \left(x \right) = F_{23}\! \left(x \right)+F_{52}\! \left(x \right)+F_{59}\! \left(x \right)+F_{61}\! \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_{56}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = 2 F_{23}\! \left(x \right)+F_{52}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{48}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{22}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{17}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{21}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{66}\! \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_35(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_34(x))
Eq(F_23(x), 0)
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x) + F_29(x))
Eq(F_26(x), F_27(x) + F_4(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_14(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_32(x))
Eq(F_30(x), F_23(x) + F_24(x) + F_31(x))
Eq(F_31(x), F_18(x)*F_4(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_22(x)*F_4(x))
Eq(F_34(x), F_21(x)*F_4(x))
Eq(F_35(x), F_23(x) + F_36(x) + F_63(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_47(x))
Eq(F_38(x), F_39(x) + F_42(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_1(x) + F_39(x))
Eq(F_42(x), F_23(x) + F_43(x) + F_45(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_39(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_14(x))
Eq(F_47(x), F_48(x) + F_51(x))
Eq(F_48(x), F_23(x) + F_36(x) + F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_18(x) + F_48(x))
Eq(F_51(x), F_23(x) + F_52(x) + F_59(x) + F_61(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_56(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_39(x)*F_4(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), 2*F_23(x) + F_52(x) + F_58(x))
Eq(F_58(x), F_4(x)*F_48(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_48(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_22(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_17(x) + F_65(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_23(x) + F_24(x) + F_67(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_21(x) + F_69(x))
Eq(F_69(x), F_66(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:
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