0213_0321_1230_1320
Counting sequence:
1, 1, 2, 6, 20, 65, 202, 604, 1753, 4974, 13872, 38177, 103986, 280948, 754217, 2014470, 5358824, 14209457, 37580842, 99188428, 261360697, 687777246, 1808000352, 4748806721, 12464634210, 32699621860, 85747477577, 224777691894, 589072137368, 1543445353169, 4043344296922, 10590882504892, 27738161587801, 72641855869518, 190224986984592, 498110414495585, 1304265170292114, 3415011513895252, 8941439386291817, 23410681034514918, 61293421215799112, 160475355048928241, 420144463680984202, 1099982225249835436, 2879851690091772025, 7539674000095236030, 19739377018379957952, 51678879267509703809, 135298122801265329858, 354217248354890727364, 927357211069359913097, 2427861703202583488982, 6356242816712156218616, 16640897146231369917713, 43566510546476829878842, 114058760593988686092700, 298610027940667988517529, 781771845645572054437422, 2046706571845560234231792, 5358350031618929786096033, 14028347918524465437660402, 36726702659096127229949236, 96151778217277613810027177, 251728668886224861619235526, 659034303381294770492732840, 1725374393443298057962763825, 4517089185931562638030548202, 11825893791540688362253635724, 30960593461515843534689420473, 81055889175551012561151851934, 212207079304012511082278794272, 555565359361811107142386261697, 1454489020327217888437636278306, 3807901745301732524714740799908, 9969216304122351239512433878217, 26099747346515247542869078955190, 68330026099045610570057484441944, 178890331687310755494968027707217, 468340970455154464501654486207642, 1226132582700467187046568367681148, 3210056783766434058437110813785817, 8404037780990324639314713114414606, 22002056584289750616510583904611152, 57602132022653811634031463937078241, 150804339586430378952827287756648530, 394810886944572566198166617382329620, 1033628321667993504867563517188089577, 2706074078910491725413222873175087014, 7084593916784992038503337047118766088, 18547707674926190750586920291334998129, 48558529115034364186693024140374997322, 127127879684413212261274025291129922412, 332825109966987378510514143124419408121, 871347450274732105116681277002387140478, 2281217240974813240571640813998158814592, 5972304272887392104141034177772721154305, 15635695578167683807094189265980864847234, 40934782462586204307941271755690786779012, 107168651811551826138957668356531708260617, 280571172976030682234644950193581535519638, 734544867124542264978917880321160837282232
Generating function in Maple syntax:
(x^5-7*x^4+19*x^3-18*x^2+7*x-1)/(x-1)/(x^2-3*x+1)/(2*x-1)^2
Generating function in latex syntax:
\frac{x^{5}-7 x^{4}+19 x^{3}-18 x^{2}+7 x -1}{\left(x -1\right) \left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2}}
Generating function in sympy syntax:
(x**5 - 7*x**4 + 19*x**3 - 18*x**2 + 7*x - 1)/((x - 1)*(2*x - 1)**2*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(x^2-3*x+1)*(2*x-1)^2*F(x)-x^5+7*x^4-19*x^3+18*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} F \! \left(x \right)-x^{5}+7 x^{4}-19 x^{3}+18 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/20*(6*5^(1/2)+10)*(3/2-1/2*5^(1/2))^(-n)+1/20*(-6*5^(1/2)+10)*(3/2+1/2*5^(1/2))^(-n)+1+1/20*(-5*n-25)*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(6 \sqrt{5}+10\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\frac{\left(-6 \sqrt{5}+10\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{20}+1+\\\frac{\left(-5 n -25\right) 2^{n}}{20} & \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) = 65
a(n+4) = -4*a(n)+16*a(n+1)-17*a(n+2)+7*a(n+3)-1, n >= 6
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) = 65
a \! \left(n +4\right) = -4 a \! \left(n \right)+16 a \! \left(n +1\right)-17 a \! \left(n +2\right)+7 a \! \left(n +3\right)-1, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15735/
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[17,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[5,x]+F[9,x]
F[9,x] = F[10,x]+F[14,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[13,x]
F[12,x] = x
F[13,x] = F[1,x]+F[12,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[9,x]
F[17,x] = F[18,x]+F[29,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[20,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[28,x]
F[23,x] = 0
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[26,x]+F[27,x]
F[26,x] = F[12,x]
F[27,x] = F[22,x]
F[28,x] = F[12,x]*F[18,x]
F[29,x] = F[23,x]+F[30,x]+F[70,x]
F[30,x] = F[12,x]*F[31,x]
F[31,x] = F[32,x]+F[47,x]
F[32,x] = F[33,x]+F[6,x]
F[33,x] = F[23,x]+F[34,x]+F[35,x]
F[34,x] = F[12,x]*F[6,x]
F[35,x] = F[12,x]*F[36,x]
F[36,x] = F[37,x]+F[41,x]
F[37,x] = F[12,x]+F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[12,x]*F[40,x]
F[40,x] = F[37,x]
F[41,x] = F[42,x]+F[44,x]
F[42,x] = F[43,x]
F[43,x] = x^2
F[44,x] = F[45,x]
F[45,x] = F[12,x]*F[46,x]
F[46,x] = F[41,x]
F[47,x] = F[29,x]+F[48,x]
F[48,x] = F[23,x]+F[49,x]+F[53,x]+F[54,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[51,x]+F[52,x]
F[51,x] = F[33,x]
F[52,x] = F[48,x]
F[53,x] = F[12,x]*F[29,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[56,x]+F[64,x]
F[56,x] = F[22,x]+F[57,x]
F[57,x] = 2*F[23,x]+F[58,x]+F[62,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[60,x]+F[61,x]
F[60,x] = F[38,x]
F[61,x] = F[57,x]
F[62,x] = F[12,x]*F[63,x]
F[63,x] = F[56,x]
F[64,x] = F[65,x]+F[67,x]
F[65,x] = F[66,x]
F[66,x] = F[12,x]*F[22,x]
F[67,x] = F[68,x]
F[68,x] = F[12,x]*F[69,x]
F[69,x] = F[64,x]
F[70,x] = F[12,x]*F[71,x]
F[71,x] = F[17,x]+F[72,x]
F[72,x] = F[73,x]+F[75,x]
F[73,x] = F[23,x]+F[24,x]+F[74,x]
F[74,x] = F[12,x]*F[21,x]
F[75,x] = 2*F[23,x]+F[58,x]+F[76,x]
F[76,x] = F[12,x]*F[77,x]
F[77,x] = F[72,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_{17}\! \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_{5}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{9}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{29}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \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_{28}\! \left(x \right)
F_{23}\! \left(x \right) = 0
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right)
F_{27}\! \left(x \right) = F_{22}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{18}\! \left(x \right)
F_{29}\! \left(x \right) = F_{23}\! \left(x \right)+F_{30}\! \left(x \right)+F_{70}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{47}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{6}\! \left(x \right)
F_{33}\! \left(x \right) = F_{23}\! \left(x \right)+F_{34}\! \left(x \right)+F_{35}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{6}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{41}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{37}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{44}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = x^{2}
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{41}\! \left(x \right)
F_{47}\! \left(x \right) = F_{29}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{23}\! \left(x \right)+F_{49}\! \left(x \right)+F_{53}\! \left(x \right)+F_{54}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{52}\! \left(x \right)
F_{51}\! \left(x \right) = F_{33}\! \left(x \right)
F_{52}\! \left(x \right) = F_{48}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{64}\! \left(x \right)
F_{56}\! \left(x \right) = F_{22}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = 2 F_{23}\! \left(x \right)+F_{58}\! \left(x \right)+F_{62}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{61}\! \left(x \right)
F_{60}\! \left(x \right) = F_{38}\! \left(x \right)
F_{61}\! \left(x \right) = F_{57}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{56}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{67}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{22}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{12}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{64}\! \left(x \right)
F_{70}\! \left(x \right) = F_{12}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{17}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)+F_{75}\! \left(x \right)
F_{73}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{12}\! \left(x \right) F_{21}\! \left(x \right)
F_{75}\! \left(x \right) = 2 F_{23}\! \left(x \right)+F_{58}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{12}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{72}\! \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_17(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_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_14(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_13(x))
Eq(F_12(x), x)
Eq(F_13(x), F_1(x) + F_12(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_9(x))
Eq(F_17(x), F_18(x) + F_29(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_20(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_28(x))
Eq(F_23(x), 0)
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_26(x) + F_27(x))
Eq(F_26(x), F_12(x))
Eq(F_27(x), F_22(x))
Eq(F_28(x), F_12(x)*F_18(x))
Eq(F_29(x), F_23(x) + F_30(x) + F_70(x))
Eq(F_30(x), F_12(x)*F_31(x))
Eq(F_31(x), F_32(x) + F_47(x))
Eq(F_32(x), F_33(x) + F_6(x))
Eq(F_33(x), F_23(x) + F_34(x) + F_35(x))
Eq(F_34(x), F_12(x)*F_6(x))
Eq(F_35(x), F_12(x)*F_36(x))
Eq(F_36(x), F_37(x) + F_41(x))
Eq(F_37(x), F_12(x) + F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_12(x)*F_40(x))
Eq(F_40(x), F_37(x))
Eq(F_41(x), F_42(x) + F_44(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), x**2)
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_12(x)*F_46(x))
Eq(F_46(x), F_41(x))
Eq(F_47(x), F_29(x) + F_48(x))
Eq(F_48(x), F_23(x) + F_49(x) + F_53(x) + F_54(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_52(x))
Eq(F_51(x), F_33(x))
Eq(F_52(x), F_48(x))
Eq(F_53(x), F_12(x)*F_29(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_64(x))
Eq(F_56(x), F_22(x) + F_57(x))
Eq(F_57(x), 2*F_23(x) + F_58(x) + F_62(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_61(x))
Eq(F_60(x), F_38(x))
Eq(F_61(x), F_57(x))
Eq(F_62(x), F_12(x)*F_63(x))
Eq(F_63(x), F_56(x))
Eq(F_64(x), F_65(x) + F_67(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_12(x)*F_22(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_12(x)*F_69(x))
Eq(F_69(x), F_64(x))
Eq(F_70(x), F_12(x)*F_71(x))
Eq(F_71(x), F_17(x) + F_72(x))
Eq(F_72(x), F_73(x) + F_75(x))
Eq(F_73(x), F_23(x) + F_24(x) + F_74(x))
Eq(F_74(x), F_12(x)*F_21(x))
Eq(F_75(x), 2*F_23(x) + F_58(x) + F_76(x))
Eq(F_76(x), F_12(x)*F_77(x))
Eq(F_77(x), F_72(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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