0123_0213_0231_1230_3102
Counting sequence:
1, 1, 2, 6, 19, 54, 134, 302, 640, 1306, 2608, 5151, 10130, 19914, 39211, 77393, 153138, 303701, 603435, 1200799, 2392319, 4770421, 9518953, 19003996, 37955109, 75827209, 151523344, 302838642, 605345625, 1210160656, 2419470068, 4837571476, 9672938690, 19342322884, 38678908566, 77348550493, 154682126108, 309340043996, 618640942985, 1237218576119, 2474334746564, 4948503833151, 9896739663153, 19793045733261, 39585389948649, 79169644873639, 158337453302679, 316671935244482, 633339062802135, 1266670346687331, 2533328106914678, 5066635848610176, 10133238745713875, 20266424174890754, 40532762081943746, 81065384579736618, 162130543307727828, 324260721179823166, 648520851072554060, 1297040745422692139, 2594079942836209334, 5188157380941187006, 10376310709142354103, 20752614860813872301, 41505219111417335014, 83010421055153902969, 166020814332417141639, 332041583719263399947, 664083094715065837795, 1328166071761100455081, 2656331953129709393813, 5312663598197896760376, 10625326697941780732185, 21250652589368027451149, 42501303873766508952876, 85002605636047938847518, 170005207855641253646173, 340010410183342805200836, 680020811422291285336160, 1360041608372248544651984, 2720083193327768739419254, 5440166348766475104201736, 10880332636227159485220178, 21760665173259465874031241, 43521330186018287929956704, 87042660112341078946963824, 174085319804486017164519349, 348170638929080396688412787, 696341276758073015008611480, 1392682551736166614010294531, 2785365100592266033647452973, 5570730196524485456916946201, 11141460385508857109085003245, 22282920758817553803045469819, 44565841497894833386219935331, 89131682963849232137446864318, 178263365876017755420032369643, 356526731668414367350213146631, 713053463201526882355716239410, 1426106926184130768876873859404, 2852213852014036689574482206199
Generating function in Maple syntax:
-(2*x^10-4*x^9-2*x^8+12*x^7-10*x^6+x^5+15*x^4-27*x^3+20*x^2-7*x+1)/(2*x-1)/(x^2+x-1)/(-1+x)^5
Generating function in latex syntax:
-\frac{2 x^{10}-4 x^{9}-2 x^{8}+12 x^{7}-10 x^{6}+x^{5}+15 x^{4}-27 x^{3}+20 x^{2}-7 x +1}{\left(2 x -1\right) \left(x^{2}+x -1\right) \left(-1+x \right)^{5}}
Generating function in sympy syntax:
(-2*x**10 + 4*x**9 + 2*x**8 - 12*x**7 + 10*x**6 - x**5 - 15*x**4 + 27*x**3 - 20*x**2 + 7*x - 1)/((x - 1)**5*(2*x - 1)*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^2+x-1)*(-1+x)^5*F(x)+2*x^10-4*x^9-2*x^8+12*x^7-10*x^6+x^5+15*x^4-27*x^3+20*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{2}+x -1\right) \left(-1+x \right)^{5} F \! \left(x \right)+2 x^{10}-4 x^{9}-2 x^{8}+12 x^{7}-10 x^{6}+x^{5}+15 x^{4}-27 x^{3}+20 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/120*(-48*5^(1/2)+120)*(-1/2*5^(1/2)-1/2)^(-n)+1/120*(48*5^(1/2)+120)*(1/2*5^(1/2)-1/2)^(-n)+1/24*n^4-1/4*n^3-13/24*n^2-17/4*n+9/4*2^n+1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(-48 \sqrt{5}+120\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{120}+\frac{\left(48 \sqrt{5}+120\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{120}+\frac{n^{4}}{24}-\frac{n^{3}}{4}\\-\frac{13 n^{2}}{24}-\frac{17 n}{4}+\frac{9 \,2^{n}}{4}+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) = 19
a(5) = 54
a(6) = 134
a(7) = 302
a(8) = 640
a(9) = 1306
a(10) = 2608
a(n+3) = 1/24*n^4-7/12*n^3+11/24*n^2-2*a(n)-a(n+1)+3*a(n+2)+1/12*n+11, n >= 11
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) = 54
a \! \left(6\right) = 134
a \! \left(7\right) = 302
a \! \left(8\right) = 640
a \! \left(9\right) = 1306
a \! \left(10\right) = 2608
a \! \left(n +3\right) = \frac{n^{4}}{24}-\frac{7 n^{3}}{12}+\frac{11 n^{2}}{24}-2 a \! \left(n \right)-a \! \left(n +1\right)+3 a \! \left(n +2\right)+\frac{n}{12}+11, \quad n \geq 11
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16733/
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[14,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[7,x]
F[14,x] = F[15,x]+F[24,x]
F[15,x] = F[16,x]
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[6,x]
F[18,x] = F[15,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[23,x]
F[22,x] = F[15,x]
F[23,x] = F[19,x]
F[24,x] = F[25,x]+F[26,x]+F[61,x]
F[25,x] = 0
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[28,x]+F[39,x]
F[28,x] = F[29,x]+F[7,x]
F[29,x] = F[25,x]+F[30,x]+F[36,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[32,x]
F[32,x] = F[11,x]+F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[11,x]+F[33,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[13,x]+F[38,x]
F[38,x] = F[29,x]
F[39,x] = F[24,x]+F[40,x]
F[40,x] = 2*F[25,x]+F[41,x]+F[54,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]+F[50,x]
F[46,x] = F[11,x]+F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[11,x]
F[50,x] = F[43,x]+F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[43,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[56,x]+F[57,x]
F[56,x] = F[19,x]
F[57,x] = F[58,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[23,x]+F[60,x]
F[60,x] = F[58,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[63,x]+F[64,x]
F[63,x] = F[15,x]+F[43,x]
F[64,x] = F[65,x]
F[65,x] = F[25,x]+F[66,x]+F[73,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[28,x]+F[68,x]
F[68,x] = F[19,x]+F[69,x]
F[69,x] = F[70,x]
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[23,x]+F[72,x]
F[72,x] = F[69,x]
F[73,x] = F[4,x]*F[74,x]
F[74,x] = F[22,x]+F[75,x]
F[75,x] = F[65,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_{14}\! \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_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{24}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{6}\! \left(x \right)
F_{18}\! \left(x \right) = F_{15}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)
F_{23}\! \left(x \right) = F_{19}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{61}\! \left(x \right)
F_{25}\! \left(x \right) = 0
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_{39}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{7}\! \left(x \right)
F_{29}\! \left(x \right) = F_{25}\! \left(x \right)+F_{30}\! \left(x \right)+F_{36}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{11}\! \left(x \right)+F_{33}\! \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_{11}\! \left(x \right)+F_{33}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{13}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{29}\! \left(x \right)
F_{39}\! \left(x \right) = F_{24}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{41}\! \left(x \right)+F_{54}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{50}\! \left(x \right)
F_{46}\! \left(x \right) = F_{11}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{11}\! \left(x \right)
F_{50}\! \left(x \right) = F_{43}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{43}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{57}\! \left(x \right)
F_{56}\! \left(x \right) = F_{19}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{23}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{58}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{64}\! \left(x \right)
F_{63}\! \left(x \right) = F_{15}\! \left(x \right)+F_{43}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{25}\! \left(x \right)+F_{66}\! \left(x \right)+F_{73}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{28}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{19}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{23}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{69}\! \left(x \right)
F_{73}\! \left(x \right) = F_{4}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{22}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{65}\! \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_14(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_7(x))
Eq(F_14(x), F_15(x) + F_24(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_6(x))
Eq(F_18(x), F_15(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_23(x))
Eq(F_22(x), F_15(x))
Eq(F_23(x), F_19(x))
Eq(F_24(x), F_25(x) + F_26(x) + F_61(x))
Eq(F_25(x), 0)
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_28(x) + F_39(x))
Eq(F_28(x), F_29(x) + F_7(x))
Eq(F_29(x), F_25(x) + F_30(x) + F_36(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_11(x) + F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_11(x) + F_33(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_13(x) + F_38(x))
Eq(F_38(x), F_29(x))
Eq(F_39(x), F_24(x) + F_40(x))
Eq(F_40(x), 2*F_25(x) + F_41(x) + F_54(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_50(x))
Eq(F_46(x), F_11(x) + F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_11(x))
Eq(F_50(x), F_43(x) + F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_43(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_57(x))
Eq(F_56(x), F_19(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_23(x) + F_60(x))
Eq(F_60(x), F_58(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_63(x) + F_64(x))
Eq(F_63(x), F_15(x) + F_43(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_25(x) + F_66(x) + F_73(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_28(x) + F_68(x))
Eq(F_68(x), F_19(x) + F_69(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_4(x)*F_71(x))
Eq(F_71(x), F_23(x) + F_72(x))
Eq(F_72(x), F_69(x))
Eq(F_73(x), F_4(x)*F_74(x))
Eq(F_74(x), F_22(x) + F_75(x))
Eq(F_75(x), F_65(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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