01234_01243_01324_01342_01423_01432_02134_02143_02341_02431_03142_03241_10234_10243_10324_10342_10423_10432_12034_12043_12340_12430_13042_13240_20134_20143_20314_20341_20413_20431_21034_21043_21340_21430_23041_23140_30124_30142_30214_30241_30412_30421_31024_31042_31240_31420_32041_32140_40123_40132_40213_40231_40312_40321_41023_41032_41230_41320_42031_42130
Counting sequence:
1, 1, 2, 6, 24, 60, 150, 336, 728, 1512, 3060, 6050, 11748, 22464, 42406, 79170, 146400, 268464, 488682, 883728, 1588780, 2841300, 5057052, 8961766, 15818664, 27820800, 48766250, 85217886, 148492008, 258062532, 447379230, 773797200, 1335498848, 2300294304, 3954576516, 6786435530, 11626605900, 19887228864, 33965890702, 57928666458, 98663738160, 167828014200, 285129002802, 483853218576, 820171474804, 1388789291340, 2349249561900, 3970106334286, 6703061204688, 11307303447552, 19057918020050, 32094986013750, 54008009368248, 90813727952844, 152591225337126, 256214686677840, 429918296330600, 720917776424664, 1208129089111572, 2023383189645938, 3386795972185140, 5665712046667200, 9472891817376502, 15830000634963186, 26439720329602944, 44138473008127680, 73649347961753850, 122833829736043536, 204772017889758268, 341218475484277572, 568341417126601020, 946250678824400950, 1574810496221699448, 2619874286437411584, 4356796909532820986, 7242580974354046350, 12035464944474885000, 19993092453250049364, 33200805843513850542, 55115357823203001168, 91465151035953716720, 151740399698070598800, 251659152282421823652, 417247211651288488346, 691586515776635325084, 1145974596848861008320, 1898374082420306096350, 3143907524923077733386, 5205250377761215447248, 8615887480163310323592, 14257635122197760653890, 23587750321896984022800, 39013910222063939746948, 64513213713409740046524, 106653802348995446443596, 176280648783423232881310, 291296763871109711133600, 481250760856440704446464, 794902590014499108819362, 1312698571448344850167398, 2167337658762096173357400
Generating function in Maple syntax:
(x^7+4*x^6+5*x^5-11*x^4-5*x^3+x^2+2*x-1)/(x^2+x-1)^3
Generating function in latex syntax:
\frac{x^{7}+4 x^{6}+5 x^{5}-11 x^{4}-5 x^{3}+x^{2}+2 x -1}{\left(x^{2}+x -1\right)^{3}}
Generating function in sympy syntax:
(x**7 + 4*x**6 + 5*x**5 - 11*x**4 - 5*x**3 + x**2 + 2*x - 1)/(x**2 + x - 1)**3
Implicit equation for the generating function in Maple syntax:
-(x^2+x-1)^3*F(x)+x^7+4*x^6+5*x^5-11*x^4-5*x^3+x^2+2*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{2}+x -1\right)^{3} F \! \left(x \right)+x^{7}+4 x^{6}+5 x^{5}-11 x^{4}-5 x^{3}+x^{2}+2 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/10*n*(-1+n)*((5+5^(1/2))*(-1/2*5^(1/2)-1/2)^(-n)-(1/2*5^(1/2)-1/2)^(-n)*(5^(1/2)-5)))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{n \left(-1+n \right) \left(\left(5+\sqrt{5}\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}-\left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n} \left(\sqrt{5}-5\right)\right)}{10} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 24
a(5) = 60
a(6) = 150
a(7) = 336
a(n+1) = -1/3*a(n)+5/3*a(n+3)-a(n+5)+1/3*a(n+6), n >= 8
Recurrence in latex format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 24
a(5) = 60
a(6) = 150
a(7) = 336
a{\left(n + 1 \right)} = - \frac{a{\left(n \right)}}{3} + \frac{5 a{\left(n + 3 \right)}}{3} - a{\left(n + 5 \right)} + \frac{a{\left(n + 6 \right)}}{3}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/24614/
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[34,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[16,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[14,x]+F[15,x]
F[14,x] = F[1,x]+F[4,x]
F[15,x] = F[4,x]
F[16,x] = F[17,x]+F[7,x]
F[17,x] = F[18,x]+F[19,x]+F[30,x]
F[18,x] = 0
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[25,x]
F[21,x] = F[22,x]+F[4,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[4,x]
F[25,x] = F[26,x]+F[27,x]
F[26,x] = F[19,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[26,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[32,x]+F[33,x]
F[32,x] = F[26,x]+F[7,x]
F[33,x] = F[26,x]
F[34,x] = F[35,x]+F[7,x]
F[35,x] = F[18,x]+F[36,x]+F[47,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[42,x]
F[38,x] = F[11,x]+F[39,x]
F[39,x] = F[18,x]+F[23,x]+F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[15,x]
F[42,x] = F[17,x]+F[43,x]
F[43,x] = F[18,x]+F[28,x]+F[44,x]+F[45,x]
F[44,x] = 0
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[33,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[16,x]+F[49,x]
F[49,x] = F[35,x]+F[50,x]
F[50,x] = F[18,x]+F[51,x]+F[55,x]+F[62,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[53,x]+F[54,x]
F[53,x] = F[22,x]
F[54,x] = F[27,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[25,x]+F[57,x]
F[57,x] = F[58,x]+F[59,x]
F[58,x] = 2*F[18,x]+F[51,x]+F[55,x]
F[59,x] = F[60,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[58,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[64,x]+F[65,x]
F[64,x] = F[35,x]+F[58,x]
F[65,x] = F[58,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_{34}\! \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_{16}\! \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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{15}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{7}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{30}\! \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_{25}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{4}\! \left(x \right)
F_{22}\! \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_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)
F_{26}\! \left(x \right) = F_{19}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{26}\! \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_{33}\! \left(x \right)
F_{32}\! \left(x \right) = F_{26}\! \left(x \right)+F_{7}\! \left(x \right)
F_{33}\! \left(x \right) = F_{26}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{7}\! \left(x \right)
F_{35}\! \left(x \right) = F_{18}\! \left(x \right)+F_{36}\! \left(x \right)+F_{47}\! \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_{42}\! \left(x \right)
F_{38}\! \left(x \right) = F_{11}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{18}\! \left(x \right)+F_{23}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{15}\! \left(x \right)
F_{42}\! \left(x \right) = F_{17}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{18}\! \left(x \right)+F_{28}\! \left(x \right)+F_{44}\! \left(x \right)+F_{45}\! \left(x \right)
F_{44}\! \left(x \right) = 0
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{33}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{16}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{35}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{18}\! \left(x \right)+F_{51}\! \left(x \right)+F_{55}\! \left(x \right)+F_{62}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{54}\! \left(x \right)
F_{53}\! \left(x \right) = F_{22}\! \left(x \right)
F_{54}\! \left(x \right) = F_{27}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{25}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{59}\! \left(x \right)
F_{58}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{51}\! \left(x \right)+F_{55}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{58}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{65}\! \left(x \right)
F_{64}\! \left(x \right) = F_{35}\! \left(x \right)+F_{58}\! \left(x \right)
F_{65}\! \left(x \right) = F_{58}\! \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_34(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_16(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_15(x))
Eq(F_14(x), F_1(x) + F_4(x))
Eq(F_15(x), F_4(x))
Eq(F_16(x), F_17(x) + F_7(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_30(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_25(x))
Eq(F_21(x), F_22(x) + F_4(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_4(x))
Eq(F_25(x), F_26(x) + F_27(x))
Eq(F_26(x), F_19(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_26(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_32(x) + F_33(x))
Eq(F_32(x), F_26(x) + F_7(x))
Eq(F_33(x), F_26(x))
Eq(F_34(x), F_35(x) + F_7(x))
Eq(F_35(x), F_18(x) + F_36(x) + F_47(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_42(x))
Eq(F_38(x), F_11(x) + F_39(x))
Eq(F_39(x), F_18(x) + F_23(x) + F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_15(x))
Eq(F_42(x), F_17(x) + F_43(x))
Eq(F_43(x), F_18(x) + F_28(x) + F_44(x) + F_45(x))
Eq(F_44(x), 0)
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_33(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_16(x) + F_49(x))
Eq(F_49(x), F_35(x) + F_50(x))
Eq(F_50(x), F_18(x) + F_51(x) + F_55(x) + F_62(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_54(x))
Eq(F_53(x), F_22(x))
Eq(F_54(x), F_27(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_25(x) + F_57(x))
Eq(F_57(x), F_58(x) + F_59(x))
Eq(F_58(x), 2*F_18(x) + F_51(x) + F_55(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_58(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_64(x) + F_65(x))
Eq(F_64(x), F_35(x) + F_58(x))
Eq(F_65(x), F_58(x))
Pack JSON:
{"name": "point_placements", "initial_strats": [{"class_module": "tilings.strategies.factor", "strategy_class": "FactorFactory", "interleaving": null, "unions": false, "ignore_parent": true, "workable": true, "tracked": false}, {"class_module": "tilings.strategies.requirement_insertion", "strategy_class": "PointCorroborationFactory", "ignore_parent": true}], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "strategy_class": "RowColumnSeparationStrategy", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "strategy_class": "InsertionEncodingVerificationStrategy", "ignore_parent": false}, {"class_module": "tilings.strategies.verification", "strategy_class": "OneByOneVerificationStrategy", "ignore_parent": false, "basis": [], "symmetry": false}, {"class_module": "tilings.strategies.verification", "strategy_class": "LocallyFactorableVerificationStrategy", "ignore_parent": false, "basis": [], "symmetry": false}], "expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "strategy_class": "CellInsertionFactory", "ignore_parent": false, "maxreqlen": 1, "extra_basis": [], "one_cell_only": false}, {"class_module": "tilings.strategies.requirement_placement", "strategy_class": "PatternPlacementFactory", "partial": false, "ignore_parent": false, "dirs": [0, 1, 2, 3], "point_only": false}]], "symmetries": [], "iterative": false}
Specification JSON:
{"root": {"class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 4, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 4, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 4, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 4, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 4, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 4, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 4, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, 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