0123_1230_1302
Counting sequence:
1, 1, 2, 6, 21, 74, 251, 817, 2570, 7872, 23621, 69746, 203317, 586561, 1677746, 4764474, 13447701, 37756922, 105525935, 293751217, 814815866, 2253030516, 6212189957, 17084948066, 46878966121, 128359477249, 350785734626, 956949006702, 2606328856725, 7087888997162, 19248624864611, 52205994352561, 141422097730922, 382670508697896, 1034367318398021, 2793154906744274, 7535489892543901, 20311768782766657, 54704786000684690, 147219043842194082, 395896739849688981, 1063887904055340506, 2857072763037944087, 7667831028874858801, 20566616464314121946, 55132106142433608924, 147709769157357520901, 395537315134385300162, 1058642450465666602705, 2832070745117472830977, 7572871637231458636226, 20240769014756164992726, 54076808099230115341077, 144417548511334156112906, 385532144427779199245771, 1028819912522624005286449, 2744494369398947661351626, 7318722496700162949320592, 19510284247520518253528645, 51993904325292379041468722, 138518139839830603494569221, 368918874449190398083110721, 982266850161239506053686642, 2614608416740033899333848586, 6957710255521880614416633621, 18510251175509157432066476474, 49232077892593221723765293951, 130911357541349848377026475697, 348019085227106715278722846010, 924971794587842132435571130692, 2457856897384384795085869038341, 6529654797661335757019242392866, 17343314597039727875505598873657, 46055854397682140563901570019841, 122278737707239466499877417827746, 324588260653510284292361587597374, 861455260930480689763421750724501, 2285877270240025669799915977722602, 6064506577418688670656148716122675, 16086532796801223510425985458537137, 42663432789711439014374350989484586, 113129898167727281825697242212012536, 299936318522926514187965356422844997, 795083095234026395620942594683302546, 2107315023868618969597845749731111021, 5584464108607255269058800853195395649, 14796881641969954808314192083807123026, 39200991705117608312204475021396749682, 103839721796811059626525696575943407765, 275024247767785139905731305869943542682, 728317615430524878607517830928857145831, 1928476306213261482129012325438027602481, 5105669832353695007899239951054454859162, 13515661070591657875715679806211624860076, 35774138489508346181568995497370054178821, 94678101848450749021807922496781153164674, 250541384297308938379983423395831617322401, 662918355317353810253711686088405719335937, 1753849377234920687291730998659603390278530, 4639564558854035092217650062863365979745702, 12272012951146896916239146085059161284103701
Generating function in Maple syntax:
-(2*x^5-11*x^4+20*x^3-18*x^2+7*x-1)/(-1+x)^2/(x^2-3*x+1)^2
Generating function in latex syntax:
-\frac{2 x^{5}-11 x^{4}+20 x^{3}-18 x^{2}+7 x -1}{\left(-1+x \right)^{2} \left(x^{2}-3 x +1\right)^{2}}
Generating function in sympy syntax:
(-2*x**5 + 11*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:
(-1+x)^2*(x^2-3*x+1)^2*F(x)+2*x^5-11*x^4+20*x^3-18*x^2+7*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(-1+x \right)^{2} \left(x^{2}-3 x +1\right)^{2} F \! \left(x \right)+2 x^{5}-11 x^{4}+20 x^{3}-18 x^{2}+7 x -1 = 0
Explicit closed form in Maple syntax:
1/50*(10*n-21*5^(1/2)+25)*(3/2-1/2*5^(1/2))^(-n)+1/50*(10*n+21*5^(1/2)+25)*(3/2+1/2*5^(1/2))^(-n)+n
Explicit closed form in latex syntax:
\frac{\left(10 n -21 \sqrt{5}+25\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{50}+\frac{\left(10 n +21 \sqrt{5}+25\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{50}+n
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 74
a(n+4) = -a(n)+6*a(n+1)-11*a(n+2)+6*a(n+3)+n+2, 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) = 21
a \! \left(5\right) = 74
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 +2, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14666/
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[21,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[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[6,x]
F[14,x] = F[15,x]+F[16,x]+F[20,x]
F[15,x] = 0
F[16,x] = F[12,x]*F[17,x]
F[17,x] = F[18,x]+F[19,x]
F[18,x] = F[10,x]
F[19,x] = F[14,x]
F[20,x] = F[12,x]*F[13,x]
F[21,x] = F[22,x]+F[6,x]
F[22,x] = F[15,x]+F[23,x]+F[46,x]
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[25,x]+F[30,x]
F[25,x] = F[26,x]+F[6,x]
F[26,x] = F[27,x]
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[10,x]+F[29,x]
F[29,x] = F[27,x]
F[30,x] = F[22,x]+F[31,x]
F[31,x] = 2*F[15,x]+F[32,x]+F[36,x]
F[32,x] = F[12,x]*F[33,x]
F[33,x] = F[34,x]+F[35,x]
F[34,x] = F[26,x]
F[35,x] = F[31,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[38,x]+F[41,x]
F[38,x] = F[15,x]+F[20,x]+F[39,x]
F[39,x] = F[12,x]*F[40,x]
F[40,x] = F[28,x]+F[37,x]
F[41,x] = 2*F[15,x]+F[36,x]+F[42,x]
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[44,x]+F[45,x]
F[44,x] = F[29,x]
F[45,x] = F[41,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[48,x]+F[49,x]
F[48,x] = F[38,x]+F[6,x]
F[49,x] = F[50,x]+F[60,x]
F[50,x] = F[15,x]+F[16,x]+F[51,x]
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[13,x]+F[53,x]
F[53,x] = F[50,x]+F[54,x]
F[54,x] = F[15,x]+F[55,x]+F[56,x]+F[59,x]
F[55,x] = 0
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[19,x]+F[58,x]
F[58,x] = F[54,x]
F[59,x] = F[12,x]*F[53,x]
F[60,x] = F[15,x]+F[42,x]+F[59,x]+F[61,x]
F[61,x] = F[12,x]*F[62,x]
F[62,x] = F[63,x]+F[64,x]
F[63,x] = F[38,x]
F[64,x] = F[60,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_{21}\! \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_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{6}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{20}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)
F_{18}\! \left(x \right) = F_{10}\! \left(x \right)
F_{19}\! \left(x \right) = F_{14}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{6}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)+F_{23}\! \left(x \right)+F_{46}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{30}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{6}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{10}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{27}\! \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_{15}\! \left(x \right)+F_{32}\! \left(x \right)+F_{36}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{35}\! \left(x \right)
F_{34}\! \left(x \right) = F_{26}\! \left(x \right)
F_{35}\! \left(x \right) = F_{31}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{41}\! \left(x \right)
F_{38}\! \left(x \right) = F_{15}\! \left(x \right)+F_{20}\! \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_{28}\! \left(x \right)+F_{37}\! \left(x \right)
F_{41}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{36}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{45}\! \left(x \right)
F_{44}\! \left(x \right) = F_{29}\! \left(x \right)
F_{45}\! \left(x \right) = F_{41}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{49}\! \left(x \right)
F_{48}\! \left(x \right) = F_{38}\! \left(x \right)+F_{6}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{60}\! \left(x \right)
F_{50}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{13}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{50}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{15}\! \left(x \right)+F_{55}\! \left(x \right)+F_{56}\! \left(x \right)+F_{59}\! \left(x \right)
F_{55}\! \left(x \right) = 0
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{19}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{54}\! \left(x \right)
F_{59}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{60}\! \left(x \right) = F_{15}\! \left(x \right)+F_{42}\! \left(x \right)+F_{59}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \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_{38}\! \left(x \right)
F_{64}\! \left(x \right) = F_{60}\! \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_21(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_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_6(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_20(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_12(x)*F_17(x))
Eq(F_17(x), F_18(x) + F_19(x))
Eq(F_18(x), F_10(x))
Eq(F_19(x), F_14(x))
Eq(F_20(x), F_12(x)*F_13(x))
Eq(F_21(x), F_22(x) + F_6(x))
Eq(F_22(x), F_15(x) + F_23(x) + F_46(x))
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_25(x) + F_30(x))
Eq(F_25(x), F_26(x) + F_6(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_10(x) + F_29(x))
Eq(F_29(x), F_27(x))
Eq(F_30(x), F_22(x) + F_31(x))
Eq(F_31(x), 2*F_15(x) + F_32(x) + F_36(x))
Eq(F_32(x), F_12(x)*F_33(x))
Eq(F_33(x), F_34(x) + F_35(x))
Eq(F_34(x), F_26(x))
Eq(F_35(x), F_31(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_38(x) + F_41(x))
Eq(F_38(x), F_15(x) + F_20(x) + F_39(x))
Eq(F_39(x), F_12(x)*F_40(x))
Eq(F_40(x), F_28(x) + F_37(x))
Eq(F_41(x), 2*F_15(x) + F_36(x) + F_42(x))
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_45(x))
Eq(F_44(x), F_29(x))
Eq(F_45(x), F_41(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_49(x))
Eq(F_48(x), F_38(x) + F_6(x))
Eq(F_49(x), F_50(x) + F_60(x))
Eq(F_50(x), F_15(x) + F_16(x) + F_51(x))
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_13(x) + F_53(x))
Eq(F_53(x), F_50(x) + F_54(x))
Eq(F_54(x), F_15(x) + F_55(x) + F_56(x) + F_59(x))
Eq(F_55(x), 0)
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_19(x) + F_58(x))
Eq(F_58(x), F_54(x))
Eq(F_59(x), F_12(x)*F_53(x))
Eq(F_60(x), F_15(x) + F_42(x) + F_59(x) + F_61(x))
Eq(F_61(x), F_12(x)*F_62(x))
Eq(F_62(x), F_63(x) + F_64(x))
Eq(F_63(x), F_38(x))
Eq(F_64(x), F_60(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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 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, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": 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