0132_0213_1032_2103
Counting sequence:
1, 1, 2, 6, 20, 67, 223, 740, 2454, 8138, 26989, 89509, 296858, 984534, 3265220, 10829143, 35914987, 119112500, 395038086, 1310148722, 4345124521, 14410659481, 47793131282, 158506513926, 525688823540, 1743453517147, 5782185259303, 19176689280260, 63599728348374, 210929289559418, 699549610500469, 2320064978048029, 7694524336185578, 25518985597534614, 84634032914113700, 280689822090706063, 930911283706221907, 3087378842869200980, 10239330305941868166, 33958866226065337442, 112625001919373393041, 373522218701249742961, 1238791080895019782562, 4108466016937152339846, 13625778610007498543060, 45190064117251275506227, 149873409319994629646383, 497057024812314560063780, 1648495800797859271782294, 5467256812785816311190698, 18132225173085644824149949, 60135750155104491741262549, 199440962827054167221434298, 661448432102181801727892694, 2193701946323966651971586180, 7275439770884726992128590983, 24129086427839723238259880827, 80024415042523178704529305460, 265401967125837499163498730246, 880208922699839134084959326162, 2919223832403109346564692165561, 9681642123703337111341709890441, 32109286438068278302017002193842, 106490847563733739329431543237766, 353178219538304324743182558118580, 1171319954812023502808635289007307, 3884697188672587659186039252201463, 12883646509806561660656230485311300, 42728773782846507878278871548064214, 141710509334137522559922305315141978, 469984665551122278295754538913615429, 1558709984821074711290054684617093069, 5169481038135358713301324305717803018, 17144648115350744456395089125522298774, 56860438568360462567020694371758722660, 188578351240202820495554414824109443903, 625422445761109228077301334840129851747, 2074221314850577965296334153165154465940, 6879180771557778869851312801693651864326, 22814888531400191912073177195688717874882, 75665861384588288069811065797573668802081, 250946769746078413348503318473350423741921, 832268080923717230944353704960486374605122, 2760227434787578509719227447281158532967686, 9154328594817675710658366828251566492510100, 30360444565447836726630507216618804181060387, 100690791756529628976195751816778361747964543, 333942262363814087838137815261536853401982820, 1107523663755784544279812982845078189186954134, 3673115996449424414686448459261905976679653258, 12181934856019171073688044740884888464351536909, 40401538361364993099196011588554954577532899589, 133992204133182105509316174070045005331415051738, 444386810420002006805962564475502880959422992854, 1473814380118541089886891617089089251900522563140, 4887923709957236936240104992878596164866194224823, 16210859736923224980512780293076825460493013094667, 53763517805088169791511662036203897526944895722420, 178307375036645917685282481476785600983703835510406, 591358625522271287852154569603006253367503778743602, 1961248231643352639440841007565474722125823510662601
Generating function in Maple syntax:
-(2*x^3-4*x^2+4*x-1)/(x^4-5*x^3+7*x^2-5*x+1)
Generating function in latex syntax:
-\frac{2 x^{3}-4 x^{2}+4 x -1}{x^{4}-5 x^{3}+7 x^{2}-5 x +1}
Generating function in sympy syntax:
(-2*x**3 + 4*x**2 - 4*x + 1)/(x**4 - 5*x**3 + 7*x**2 - 5*x + 1)
Implicit equation for the generating function in Maple syntax:
(x^4-5*x^3+7*x^2-5*x+1)*F(x)+2*x^3-4*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{4}-5 x^{3}+7 x^{2}-5 x +1\right) F \! \left(x \right)+2 x^{3}-4 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
11/1155200*((5^(1/2)+10)*(14+10*5^(1/2))^(1/2)+19*5^(1/2)+95)*((((I*5^(1/2)-15/11*I)*(-70+50*5^(1/2))^(1/2)+190/11*5^(1/2)-570/11)*(14+10*5^(1/2))^(1/2)+(-84/11*I*5^(1/2)+80/11*I)*(-70+50*5^(1/2))^(1/2)-1140/11*5^(1/2)+380)*(5/4-1/4*5^(1/2)-1/4*I*(-14+10*5^(1/2))^(1/2))^(-n)+(((-I*5^(1/2)+15/11*I)*(-70+50*5^(1/2))^(1/2)+190/11*5^(1/2)-570/11)*(14+10*5^(1/2))^(1/2)+(84/11*I*5^(1/2)-80/11*I)*(-70+50*5^(1/2))^(1/2)-1140/11*5^(1/2)+380)*(5/4-1/4*5^(1/2)+1/4*I*(-14+10*5^(1/2))^(1/2))^(-n)+(760/11*5^(1/2)-760/11*(14+10*5^(1/2))^(1/2)+3800/11)*(5/4+1/4*5^(1/2)-1/4*(14+10*5^(1/2))^(1/2))^(-n)+3040/11*(5/4+1/4*5^(1/2)+1/4*(14+10*5^(1/2))^(1/2))^(-n))
Explicit closed form in latex syntax:
\frac{11 \left(\left(\sqrt{5}+10\right) \sqrt{14+10 \sqrt{5}}+19 \sqrt{5}+95\right) \left(\left(\left(\left(\mathrm{I} \sqrt{5}-\frac{15 \,\mathrm{I}}{11}\right) \sqrt{-70+50 \sqrt{5}}+\frac{190 \sqrt{5}}{11}-\frac{570}{11}\right) \sqrt{14+10 \sqrt{5}}+\left(-\frac{84 \,\mathrm{I} \sqrt{5}}{11}+\frac{80 \,\mathrm{I}}{11}\right) \sqrt{-70+50 \sqrt{5}}-\frac{1140 \sqrt{5}}{11}+380\right) \left(\frac{5}{4}-\frac{\sqrt{5}}{4}-\frac{\mathrm{I} \sqrt{-14+10 \sqrt{5}}}{4}\right)^{-n}+\left(\left(\left(-\mathrm{I} \sqrt{5}+\frac{15 \,\mathrm{I}}{11}\right) \sqrt{-70+50 \sqrt{5}}+\frac{190 \sqrt{5}}{11}-\frac{570}{11}\right) \sqrt{14+10 \sqrt{5}}+\left(\frac{84 \,\mathrm{I} \sqrt{5}}{11}-\frac{80 \,\mathrm{I}}{11}\right) \sqrt{-70+50 \sqrt{5}}-\frac{1140 \sqrt{5}}{11}+380\right) \left(\frac{5}{4}-\frac{\sqrt{5}}{4}+\frac{\mathrm{I} \sqrt{-14+10 \sqrt{5}}}{4}\right)^{-n}+\left(\frac{760 \sqrt{5}}{11}-\frac{760 \sqrt{14+10 \sqrt{5}}}{11}+\frac{3800}{11}\right) \left(\frac{5}{4}+\frac{\sqrt{5}}{4}-\frac{\sqrt{14+10 \sqrt{5}}}{4}\right)^{-n}+\frac{3040 \left(\frac{5}{4}+\frac{\sqrt{5}}{4}+\frac{\sqrt{14+10 \sqrt{5}}}{4}\right)^{-n}}{11}\right)}{1155200}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n+4) = -a(n)+5*a(n+1)-7*a(n+2)+5*a(n+3), n >= 4
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(n +4\right) = -a \! \left(n \right)+5 a \! \left(n +1\right)-7 a \! \left(n +2\right)+5 a \! \left(n +3\right), \quad n \geq 4
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15264/
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[16,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[13,x]*F[4,x]
F[16,x] = F[17,x]+F[2,x]
F[17,x] = F[18,x]+F[19,x]+F[47,x]
F[18,x] = 0
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[31,x]
F[21,x] = F[11,x]+F[22,x]
F[22,x] = F[15,x]+F[18,x]+F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[28,x]
F[25,x] = F[11,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[25,x]*F[4,x]
F[28,x] = F[14,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[28,x]*F[4,x]
F[31,x] = F[32,x]+F[35,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[2,x]+F[32,x]
F[35,x] = 2*F[18,x]+F[36,x]+F[43,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[41,x]
F[38,x] = F[32,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[38,x]*F[4,x]
F[41,x] = F[42,x]+F[45,x]
F[42,x] = F[43,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[17,x]+F[42,x]
F[45,x] = F[46,x]
F[46,x] = F[4,x]*F[41,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[52,x]
F[49,x] = F[2,x]+F[50,x]
F[50,x] = F[18,x]+F[19,x]+F[51,x]
F[51,x] = F[4,x]*F[49,x]
F[52,x] = F[17,x]+F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[4,x]*F[52,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_{16}\! \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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{2}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{47}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{31}\! \left(x \right)
F_{21}\! \left(x \right) = F_{11}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)+F_{18}\! \left(x \right)+F_{23}\! \left(x \right)
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_{28}\! \left(x \right)
F_{25}\! \left(x \right) = F_{11}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{14}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{35}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{2}\! \left(x \right)+F_{32}\! \left(x \right)
F_{35}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{36}\! \left(x \right)+F_{43}\! \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_{41}\! \left(x \right)
F_{38}\! \left(x \right) = F_{32}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{45}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{17}\! \left(x \right)+F_{42}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{52}\! \left(x \right)
F_{49}\! \left(x \right) = F_{2}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{17}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \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_16(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_13(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_2(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_47(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_31(x))
Eq(F_21(x), F_11(x) + F_22(x))
Eq(F_22(x), F_15(x) + F_18(x) + F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_28(x))
Eq(F_25(x), F_11(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_25(x)*F_4(x))
Eq(F_28(x), F_14(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_28(x)*F_4(x))
Eq(F_31(x), F_32(x) + F_35(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_2(x) + F_32(x))
Eq(F_35(x), 2*F_18(x) + F_36(x) + F_43(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_41(x))
Eq(F_38(x), F_32(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_38(x)*F_4(x))
Eq(F_41(x), F_42(x) + F_45(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_17(x) + F_42(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_4(x)*F_41(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_52(x))
Eq(F_49(x), F_2(x) + F_50(x))
Eq(F_50(x), F_18(x) + F_19(x) + F_51(x))
Eq(F_51(x), F_4(x)*F_49(x))
Eq(F_52(x), F_17(x) + F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_4(x)*F_52(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, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "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, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "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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