0231_0321_1032_1203_1320
Counting sequence:
1, 1, 2, 6, 19, 55, 151, 405, 1075, 2838, 7468, 19611, 51432, 134781, 353044, 924530, 2420781, 6338115, 16593945, 43444193, 113739213, 297774146, 779584062, 2040979031, 5343354194, 13989084905, 36623902086, 95882623150, 251023969415, 657189287423, 1720543895483, 4504442401981, 11792783313767, 30873907543006, 80828939319344, 211612910419555, 554009791944316, 1450416465418885, 3797239604318360, 9941302347542778, 26026667438317153, 68138699967416491, 178389432463940797, 467029597424415081, 1222699359809314369, 3201068482003538730, 8380506086201313346, 21940449776600413695, 57440843243599941030, 150382079954199423633, 393705396618998345098, 1030734109902795627926, 2698496933089388556027, 7064756689365370058631, 18495773135006721639519, 48422562715654794880805, 126771915011957663025051, 331893182320218194217830, 868907631948696919653300, 2274829713525872564768363, 5955581508628920774679568, 15591914812360889759299661, 40820162928453748503250332, 106868573973000355750483906, 279785558990547318748235669, 732488102998641600494259155, 1917678750005377482734579681, 5020548147017490847709519665, 13143965691047095060394021045, 34411348926123794333472587218, 90090081087324287940023786438, 235858894335849069486598820071, 617486601920222920519772723962, 1616600911424819692072719404281, 4232316132354236155698385543694, 11080347485637888775022437284030, 29008726324559430169368926368111, 75945831488040401733084341882575, 198828768139561775029884099344515, 520540472930644923356567956218573, 1362792650652372995039819769381583, 3567837479026474061762891351999406, 9340719786427049190248854286692792, 24454321880254673508983671508158131, 64022245854336971336702160237863844, 167612415682756240501122809205518805, 438815001193931750166666267378781216, 1148832587899039009998875992930916810, 3007682762503185279829961711414064585, 7874215699610516829491009141311375803, 20614964336328365208643065712520165253, 53970677309374578796438187996249226041, 141297067591795371180671498276227622697, 369920525466011534745576306832433755706, 968464508806239233056057422221073761994, 2535473000952706164422595959830787651855, 6637954494051879260211730457271289319246, 17378390481202931616212595411983080435745, 45497216949556915588426055778677952122130, 119113260367467815149065571924050776069158, 311842564152846529858770659993474376228323
Generating function in Maple syntax:
(3*x^6-4*x^5+8*x^4-15*x^3+14*x^2-6*x+1)/(x^2-3*x+1)/(x-1)^4
Generating function in latex syntax:
\frac{3 x^{6}-4 x^{5}+8 x^{4}-15 x^{3}+14 x^{2}-6 x +1}{\left(x^{2}-3 x +1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(3*x**6 - 4*x**5 + 8*x**4 - 15*x**3 + 14*x**2 - 6*x + 1)/((x - 1)**4*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x-1)^4*F(x)-3*x^6+4*x^5-8*x^4+15*x^3-14*x^2+6*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x -1\right)^{4} F \! \left(x \right)-3 x^{6}+4 x^{5}-8 x^{4}+15 x^{3}-14 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/30*(114*5^(1/2)-240)*(3/2-1/2*5^(1/2))^(-n)+1/30*(-114*5^(1/2)-240)*(3/2+1/2*5^(1/2))^(-n)-1/6*n^3+2*n^2-59/6*n+14)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(114 \sqrt{5}-240\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{30}+\frac{\left(-114 \sqrt{5}-240\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{30}-\frac{n^{3}}{6}+2 n^{2}\\-\frac{59 n}{6}+14 & \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) = 55
a(6) = 151
a(n+2) = 1/6*n^3-3/2*n^2-a(n)+3*a(n+1)+16/3*n-3, n >= 7
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) = 55
a \! \left(6\right) = 151
a \! \left(n +2\right) = \frac{n^{3}}{6}-\frac{3 n^{2}}{2}-a \! \left(n \right)+3 a \! \left(n +1\right)+\frac{16 n}{3}-3, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20757/
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[10,x]*F[4,x]
F[4,x] = F[19,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[10,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[11,x]
F[10,x] = x
F[11,x] = F[12,x]+F[13,x]+F[17,x]
F[12,x] = 0
F[13,x] = F[10,x]*F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[10,x]*F[16,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[10,x]*F[18,x]
F[18,x] = F[9,x]
F[19,x] = F[20,x]+F[30,x]
F[20,x] = F[21,x]
F[21,x] = F[10,x]*F[22,x]
F[22,x] = F[16,x]+F[23,x]
F[23,x] = F[20,x]+F[24,x]
F[24,x] = F[12,x]+F[25,x]+F[29,x]
F[25,x] = F[10,x]*F[26,x]
F[26,x] = F[27,x]+F[28,x]
F[27,x] = F[14,x]
F[28,x] = F[24,x]
F[29,x] = F[10,x]*F[23,x]
F[30,x] = F[12,x]+F[31,x]+F[52,x]
F[31,x] = F[10,x]*F[32,x]
F[32,x] = F[33,x]+F[38,x]
F[33,x] = F[14,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[10,x]*F[36,x]
F[36,x] = F[10,x]+F[37,x]
F[37,x] = F[35,x]
F[38,x] = F[39,x]+F[42,x]
F[39,x] = F[12,x]+F[31,x]+F[40,x]
F[40,x] = F[10,x]*F[41,x]
F[41,x] = F[20,x]+F[39,x]
F[42,x] = F[43,x]
F[43,x] = F[10,x]*F[44,x]
F[44,x] = F[45,x]+F[51,x]
F[45,x] = F[12,x]+F[46,x]+F[50,x]
F[46,x] = F[10,x]*F[47,x]
F[47,x] = F[48,x]+F[49,x]
F[48,x] = F[10,x]
F[49,x] = F[45,x]
F[50,x] = F[10,x]*F[20,x]
F[51,x] = F[43,x]
F[52,x] = F[10,x]*F[53,x]
F[53,x] = F[19,x]+F[54,x]
F[54,x] = F[45,x]+F[55,x]
F[55,x] = F[12,x]+F[56,x]+F[57,x]+F[58,x]
F[56,x] = 0
F[57,x] = F[10,x]*F[39,x]
F[58,x] = F[10,x]*F[59,x]
F[59,x] = F[54,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_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{19}\! \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_{10}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = x
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{17}\! \left(x \right)
F_{12}\! \left(x \right) = 0
F_{13}\! \left(x \right) = F_{10}\! \left(x \right) F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{10}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{10}\! \left(x \right) F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{9}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{30}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{10}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{16}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{20}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right)+F_{25}\! \left(x \right)+F_{29}\! \left(x \right)
F_{25}\! \left(x \right) = F_{10}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)
F_{27}\! \left(x \right) = F_{14}\! \left(x \right)
F_{28}\! \left(x \right) = F_{24}\! \left(x \right)
F_{29}\! \left(x \right) = F_{10}\! \left(x \right) F_{23}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right)+F_{31}\! \left(x \right)+F_{52}\! \left(x \right)
F_{31}\! \left(x \right) = F_{10}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{38}\! \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_{10}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{10}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{35}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{42}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right)+F_{31}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{10}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{20}\! \left(x \right)+F_{39}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{10}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{51}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right)+F_{46}\! \left(x \right)+F_{50}\! \left(x \right)
F_{46}\! \left(x \right) = F_{10}\! \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_{10}\! \left(x \right)
F_{49}\! \left(x \right) = F_{45}\! \left(x \right)
F_{50}\! \left(x \right) = F_{10}\! \left(x \right) F_{20}\! \left(x \right)
F_{51}\! \left(x \right) = F_{43}\! \left(x \right)
F_{52}\! \left(x \right) = F_{10}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{19}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{45}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right)+F_{56}\! \left(x \right)+F_{57}\! \left(x \right)+F_{58}\! \left(x \right)
F_{56}\! \left(x \right) = 0
F_{57}\! \left(x \right) = F_{10}\! \left(x \right) F_{39}\! \left(x \right)
F_{58}\! \left(x \right) = F_{10}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{54}\! \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_10(x)*F_4(x))
Eq(F_4(x), F_19(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_10(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_11(x))
Eq(F_10(x), x)
Eq(F_11(x), F_12(x) + F_13(x) + F_17(x))
Eq(F_12(x), 0)
Eq(F_13(x), F_10(x)*F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_10(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_10(x)*F_18(x))
Eq(F_18(x), F_9(x))
Eq(F_19(x), F_20(x) + F_30(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_10(x)*F_22(x))
Eq(F_22(x), F_16(x) + F_23(x))
Eq(F_23(x), F_20(x) + F_24(x))
Eq(F_24(x), F_12(x) + F_25(x) + F_29(x))
Eq(F_25(x), F_10(x)*F_26(x))
Eq(F_26(x), F_27(x) + F_28(x))
Eq(F_27(x), F_14(x))
Eq(F_28(x), F_24(x))
Eq(F_29(x), F_10(x)*F_23(x))
Eq(F_30(x), F_12(x) + F_31(x) + F_52(x))
Eq(F_31(x), F_10(x)*F_32(x))
Eq(F_32(x), F_33(x) + F_38(x))
Eq(F_33(x), F_14(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_10(x)*F_36(x))
Eq(F_36(x), F_10(x) + F_37(x))
Eq(F_37(x), F_35(x))
Eq(F_38(x), F_39(x) + F_42(x))
Eq(F_39(x), F_12(x) + F_31(x) + F_40(x))
Eq(F_40(x), F_10(x)*F_41(x))
Eq(F_41(x), F_20(x) + F_39(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_10(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_51(x))
Eq(F_45(x), F_12(x) + F_46(x) + F_50(x))
Eq(F_46(x), F_10(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_49(x))
Eq(F_48(x), F_10(x))
Eq(F_49(x), F_45(x))
Eq(F_50(x), F_10(x)*F_20(x))
Eq(F_51(x), F_43(x))
Eq(F_52(x), F_10(x)*F_53(x))
Eq(F_53(x), F_19(x) + F_54(x))
Eq(F_54(x), F_45(x) + F_55(x))
Eq(F_55(x), F_12(x) + F_56(x) + F_57(x) + F_58(x))
Eq(F_56(x), 0)
Eq(F_57(x), F_10(x)*F_39(x))
Eq(F_58(x), F_10(x)*F_59(x))
Eq(F_59(x), F_54(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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 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