0213_1320_2130_3021
Counting sequence:
1, 1, 2, 6, 20, 64, 193, 555, 1544, 4203, 11284, 30033, 79501, 209724, 552016, 1450796, 3809032, 9993258, 26204205, 68685725, 179985028, 471532661, 1235138592, 3234933251, 8471760105, 22184543414, 58090261068, 152103019630, 398252355204, 1042721158148, 2730045340649, 7147683303343, 18713541444816, 48994014777919, 128270650378076, 335822231329621, 879204633551957, 2301808849202608, 6026256273802040, 15777028691688720, 41304967240226768, 108138147906908446, 283110026236323157, 741193030313700321, 1940471263728045740, 5080225158916961289, 13300213009115874568, 34820431460616722055, 91161116557106396689, 238662988579446663018, 624827989918721966164, 1635821262651695965778, 4282636360986319373356, 11212088946207169019064, 29353632729435001392561, 76848813745697462553971, 201192817514856641036632, 526729656813270970065683, 1378996188953753288153700, 3610258982105582932354265, 9451780901478183584797533, 24745084010559343973784356, 64783471706660600640015072, 169605332262343962553145716, 444032527386214296233155800, 1162492254507984944573751314, 3043444245361112574342917693, 7967840500022096852164599205, 20860077291598666129570031124, 54612391448560877831383750813, 142977097201657919954257686704, 374318900451560787210742189963, 979979604743320252036674592121, 2565619914958991589616692949534, 6716880142494837758248226925596, 17585020517247888167997633105846, 46038181418693559711483962886932, 120529523757722256897932836480108, 315550389892252142845271708336185, 826121645994592035363796612023735, 2162814548242639690697946774652448, 5662321999035558491633701005692391, 14824151449468498694010470829863212, 38810132350578863410012340658692061, 101606245604685943175255809495717285, 266008604468314669394213604527379864, 696419567809929471564302037484170568, 1823250098980816558412526574720533080, 4773330729171205829900945820268132384, 12496742088610172183745647153265169158, 32716895536814053226246668173889879269, 85653944522141472504815702437129367273, 224244938030229334307843129274947906236, 587080869569784470457999065662613601329, 1536997670681599957144724828262691276056, 4023912142479967161133316940225056857871, 10534738756768205046569509034611672429857, 27580304127844455019203776248008346562258, 72206173626804774092298951878210139379620, 189038216752649095420207343724215615679130, 494908476631300968493351607969623795715388
Generating function in Maple syntax:
(x^7-5*x^6+20*x^5-39*x^4+43*x^3-26*x^2+8*x-1)/(2*x-1)/(x^2-3*x+1)/(x-1)^4
Generating function in latex syntax:
\frac{x^{7}-5 x^{6}+20 x^{5}-39 x^{4}+43 x^{3}-26 x^{2}+8 x -1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(x**7 - 5*x**6 + 20*x**5 - 39*x**4 + 43*x**3 - 26*x**2 + 8*x - 1)/((x - 1)**4*(2*x - 1)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^2-3*x+1)*(x-1)^4*F(x)-x^7+5*x^6-20*x^5+39*x^4-43*x^3+26*x^2-8*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{4} F \! \left(x \right)-x^{7}+5 x^{6}-20 x^{5}+39 x^{4}-43 x^{3}+26 x^{2}-8 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/30*(24*5^(1/2)-30)*(3/2-1/2*5^(1/2))^(-n)+1/30*(-24*5^(1/2)-30)*(3/2+1/2*5^(1/2))^(-n)-1/6*n^3+1/2*n^2-7/3*n-2^(n-1)+3)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(24 \sqrt{5}-30\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{30}+\frac{\left(-24 \sqrt{5}-30\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{30}-\frac{n^{3}}{6}+\frac{n^{2}}{2}-\frac{7 n}{3}\\-2^{n -1}+3 & \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) = 64
a(6) = 193
a(7) = 555
a(n+3) = -1/6*n^3+1/2*n^2+2*a(n)-7*a(n+1)+5*a(n+2)-1/3*n+2, 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) = 64
a \! \left(6\right) = 193
a \! \left(7\right) = 555
a \! \left(n +3\right) = -\frac{n^{3}}{6}+\frac{n^{2}}{2}+2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)-\frac{n}{3}+2, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15767/
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[16,x]*F[4,x]
F[4,x] = F[23,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[16,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[20,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[16,x]
F[12,x] = F[13,x]+F[17,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[16,x]
F[16,x] = x
F[17,x] = F[10,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[16,x]*F[17,x]
F[20,x] = F[21,x]
F[21,x] = F[16,x]*F[22,x]
F[22,x] = F[9,x]
F[23,x] = F[10,x]+F[24,x]
F[24,x] = F[25,x]+F[26,x]+F[48,x]
F[25,x] = 0
F[26,x] = F[16,x]*F[27,x]
F[27,x] = F[28,x]+F[45,x]
F[28,x] = F[29,x]+F[6,x]
F[29,x] = F[25,x]+F[30,x]+F[31,x]
F[30,x] = F[16,x]*F[28,x]
F[31,x] = F[16,x]*F[32,x]
F[32,x] = F[33,x]+F[37,x]
F[33,x] = F[14,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[16,x]*F[36,x]
F[36,x] = F[33,x]
F[37,x] = F[38,x]+F[42,x]
F[38,x] = F[39,x]
F[39,x] = F[16,x]*F[40,x]
F[40,x] = F[14,x]+F[41,x]
F[41,x] = F[39,x]
F[42,x] = F[43,x]
F[43,x] = F[16,x]*F[44,x]
F[44,x] = F[37,x]
F[45,x] = F[24,x]+F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[16,x]*F[45,x]
F[48,x] = F[16,x]*F[49,x]
F[49,x] = F[23,x]+F[50,x]
F[50,x] = F[51,x]+F[54,x]
F[51,x] = F[52,x]
F[52,x] = F[16,x]*F[53,x]
F[53,x] = F[40,x]
F[54,x] = 2*F[25,x]+F[55,x]+F[66,x]
F[55,x] = F[16,x]*F[56,x]
F[56,x] = F[57,x]
F[57,x] = F[34,x]+F[58,x]
F[58,x] = 2*F[25,x]+F[59,x]+F[60,x]
F[59,x] = F[16,x]*F[57,x]
F[60,x] = F[16,x]*F[61,x]
F[61,x] = F[62,x]
F[62,x] = F[41,x]+F[63,x]
F[63,x] = F[64,x]
F[64,x] = F[16,x]*F[65,x]
F[65,x] = F[62,x]
F[66,x] = F[16,x]*F[67,x]
F[67,x] = F[50,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_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{23}\! \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_{16}\! \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_{20}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{17}\! \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_{16}\! \left(x \right)
F_{16}\! \left(x \right) = x
F_{17}\! \left(x \right) = F_{10}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{16}\! \left(x \right) F_{17}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{16}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{9}\! \left(x \right)
F_{23}\! \left(x \right) = F_{10}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{48}\! \left(x \right)
F_{25}\! \left(x \right) = 0
F_{26}\! \left(x \right) = F_{16}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{45}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{6}\! \left(x \right)
F_{29}\! \left(x \right) = F_{25}\! \left(x \right)+F_{30}\! \left(x \right)+F_{31}\! \left(x \right)
F_{30}\! \left(x \right) = F_{16}\! \left(x \right) F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{16}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{37}\! \left(x \right)
F_{33}\! \left(x \right) = F_{14}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{16}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{33}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{42}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{16}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{14}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{39}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{16}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{37}\! \left(x \right)
F_{45}\! \left(x \right) = F_{24}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{16}\! \left(x \right) F_{45}\! \left(x \right)
F_{48}\! \left(x \right) = F_{16}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{23}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{54}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{16}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{40}\! \left(x \right)
F_{54}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{55}\! \left(x \right)+F_{66}\! \left(x \right)
F_{55}\! \left(x \right) = F_{16}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{34}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{59}\! \left(x \right)+F_{60}\! \left(x \right)
F_{59}\! \left(x \right) = F_{16}\! \left(x \right) F_{57}\! \left(x \right)
F_{60}\! \left(x \right) = F_{16}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{41}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{16}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{62}\! \left(x \right)
F_{66}\! \left(x \right) = F_{16}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{50}\! \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_16(x)*F_4(x))
Eq(F_4(x), F_23(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_16(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_20(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_16(x))
Eq(F_12(x), F_13(x) + F_17(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_16(x))
Eq(F_16(x), x)
Eq(F_17(x), F_10(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_16(x)*F_17(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_16(x)*F_22(x))
Eq(F_22(x), F_9(x))
Eq(F_23(x), F_10(x) + F_24(x))
Eq(F_24(x), F_25(x) + F_26(x) + F_48(x))
Eq(F_25(x), 0)
Eq(F_26(x), F_16(x)*F_27(x))
Eq(F_27(x), F_28(x) + F_45(x))
Eq(F_28(x), F_29(x) + F_6(x))
Eq(F_29(x), F_25(x) + F_30(x) + F_31(x))
Eq(F_30(x), F_16(x)*F_28(x))
Eq(F_31(x), F_16(x)*F_32(x))
Eq(F_32(x), F_33(x) + F_37(x))
Eq(F_33(x), F_14(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_16(x)*F_36(x))
Eq(F_36(x), F_33(x))
Eq(F_37(x), F_38(x) + F_42(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_16(x)*F_40(x))
Eq(F_40(x), F_14(x) + F_41(x))
Eq(F_41(x), F_39(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_16(x)*F_44(x))
Eq(F_44(x), F_37(x))
Eq(F_45(x), F_24(x) + F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_16(x)*F_45(x))
Eq(F_48(x), F_16(x)*F_49(x))
Eq(F_49(x), F_23(x) + F_50(x))
Eq(F_50(x), F_51(x) + F_54(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_16(x)*F_53(x))
Eq(F_53(x), F_40(x))
Eq(F_54(x), 2*F_25(x) + F_55(x) + F_66(x))
Eq(F_55(x), F_16(x)*F_56(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_34(x) + F_58(x))
Eq(F_58(x), 2*F_25(x) + F_59(x) + F_60(x))
Eq(F_59(x), F_16(x)*F_57(x))
Eq(F_60(x), F_16(x)*F_61(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_41(x) + F_63(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_16(x)*F_65(x))
Eq(F_65(x), F_62(x))
Eq(F_66(x), F_16(x)*F_67(x))
Eq(F_67(x), F_50(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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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