0132_0321_1032_1203_1230
Counting sequence:
1, 1, 2, 6, 19, 53, 142, 371, 953, 2417, 6070, 15127, 37469, 92361, 226790, 555147, 1355505, 3303009, 8035238, 19520927, 47371989, 114854713, 278261046, 673736083, 1630451785, 3944076817, 9537479766, 23056785063, 55726547341, 134660874665, 325350286534, 785965427483, 1898489101025, 4585359548609, 11074040036422, 26743103297839, 64579573984901, 155940905973273, 376538695342742, 909172915481379, 2195193763950777, 5300178918673521, 12796788551879030, 30896229923594039, 74594196201392061, 180094517931028105, 434803023272748198, 1049740146895124395, 2534362481900196817, 6118623440369917729, 14771926021988831718, 35663108803045180095, 86099410265474389813, 207864462608784355577, 501833402032623892726, 1211541399773193724595, 2924936467777334509097, 7061454867724509077201, 17047927268019645332374, 41157471533350385079751, 99363194593893586167533, 239884509239483898766185, 579133510109554066402694, 1398154123531977396977211, 3375446945320279591168449, 8149058390466078040936833, 19673584478839518596287622, 47496268853319281080003151, 114666205195826412449276133, 276828845265668769364519897, 668324227768557277950244950, 1613477964885569978808867907, 3895281485705270542655697049, 9404043592627257678295694641, 22703373983622079127597951670, 54810802185196002390193328727, 132324999604663256821388070685, 319460843895820861859776393289, 771246772398901672194554703718, 1861954558698817589556113493707, 4495156229806923617921237077169, 10852267698333438358627498420193, 26199692986515347401634055461926, 63251656391447227294811252432863, 152703011209575990257087846505365, 368657689690931584340649517799097, 890018412352103912001712026814646, 2148694557916468914470723860850643, 5187407615227700753196460327360521, 12523509962457188445370245673260945, 30234427888312713692950153989261014, 72992366435423887929296958282540263, 176219162151843033747596879815856205, 425430693524475043816596336437282089, 1027080554771523298165000789736479302, 2479591814208981993715020390002358619, 5986264205472407992731886517925432481, 14452120269719639393452483322221695553, 34890504834043369608184232955105767622, 84233130116069744266915708817907118959, 203356765422709589456205169761867781957
Generating function in Maple syntax:
(x^7+4*x^6-4*x^5-2*x^4-2*x^3+8*x^2-5*x+1)/(2*x-1)/(x^2+2*x-1)/(x-1)^2
Generating function in latex syntax:
\frac{x^{7}+4 x^{6}-4 x^{5}-2 x^{4}-2 x^{3}+8 x^{2}-5 x +1}{\left(2 x -1\right) \left(x^{2}+2 x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(x**7 + 4*x**6 - 4*x**5 - 2*x**4 - 2*x**3 + 8*x**2 - 5*x + 1)/((x - 1)**2*(2*x - 1)*(x**2 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
-(2*x-1)*(x^2+2*x-1)*(x-1)^2*F(x)+x^7+4*x^6-4*x^5-2*x^4-2*x^3+8*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right) \left(x^{2}+2 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{7}+4 x^{6}-4 x^{5}-2 x^{4}-2 x^{3}+8 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/8*(2^(1/2)+10)*(-1-2^(1/2))^(-n)+1/8*(-2^(1/2)+10)*(2^(1/2)-1)^(-n)+1/2*n-9/8*2^n-3/2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(\sqrt{2}+10\right) \left(-1-\sqrt{2}\right)^{-n}}{8}+\frac{\left(-\sqrt{2}+10\right) \left(\sqrt{2}-1\right)^{-n}}{8}+\frac{n}{2}-\frac{9 \,2^{n}}{8}-\frac{3}{2} & \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) = 53
a(6) = 142
a(7) = 371
a(n+3) = -2*a(n)-3*a(n+1)+4*a(n+2)+n-4, 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) = 19
a \! \left(5\right) = 53
a \! \left(6\right) = 142
a \! \left(7\right) = 371
a \! \left(n +3\right) = -2 a \! \left(n \right)-3 a \! \left(n +1\right)+4 a \! \left(n +2\right)+n -4, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19346/
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[22,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[17,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[1,x]+F[12,x]
F[17,x] = F[18,x]+F[19,x]+F[21,x]
F[18,x] = 0
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[10,x]
F[21,x] = F[12,x]*F[13,x]
F[22,x] = F[23,x]+F[43,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[26,x]+F[29,x]
F[26,x] = F[1,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[12,x]*F[16,x]
F[29,x] = F[23,x]+F[30,x]
F[30,x] = F[18,x]+F[31,x]+F[41,x]
F[31,x] = F[12,x]*F[32,x]
F[32,x] = F[33,x]+F[36,x]
F[33,x] = F[12,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = x^2
F[36,x] = F[37,x]+F[39,x]
F[37,x] = F[18,x]+F[31,x]+F[38,x]
F[38,x] = F[12,x]*F[23,x]
F[39,x] = F[40,x]
F[40,x] = F[12,x]*F[37,x]
F[41,x] = F[12,x]*F[42,x]
F[42,x] = F[23,x]+F[37,x]
F[43,x] = F[18,x]+F[44,x]+F[59,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[46,x]+F[51,x]
F[46,x] = F[10,x]+F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[12,x]*F[49,x]
F[49,x] = F[12,x]+F[50,x]
F[50,x] = F[48,x]
F[51,x] = F[52,x]+F[55,x]
F[52,x] = F[18,x]+F[44,x]+F[53,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[23,x]+F[52,x]
F[55,x] = F[56,x]
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[37,x]+F[58,x]
F[58,x] = F[56,x]
F[59,x] = F[12,x]*F[60,x]
F[60,x] = F[54,x]+F[61,x]
F[61,x] = F[62,x]+F[64,x]
F[62,x] = F[18,x]+F[31,x]+F[63,x]
F[63,x] = F[12,x]*F[42,x]
F[64,x] = F[18,x]+F[65,x]+F[66,x]+F[68,x]
F[65,x] = 0
F[66,x] = F[12,x]*F[67,x]
F[67,x] = F[52,x]
F[68,x] = F[12,x]*F[61,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_{22}\! \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_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{43}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{29}\! \left(x \right)
F_{26}\! \left(x \right) = F_{1}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{29}\! \left(x \right) = F_{23}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{18}\! \left(x \right)+F_{31}\! \left(x \right)+F_{41}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{36}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = x^{2}
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{39}\! \left(x \right)
F_{37}\! \left(x \right) = F_{18}\! \left(x \right)+F_{31}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{23}\! \left(x \right)+F_{37}\! \left(x \right)
F_{43}\! \left(x \right) = F_{18}\! \left(x \right)+F_{44}\! \left(x \right)+F_{59}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{51}\! \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_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{48}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{55}\! \left(x \right)
F_{52}\! \left(x \right) = F_{18}\! \left(x \right)+F_{44}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{23}\! \left(x \right)+F_{52}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{37}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{56}\! \left(x \right)
F_{59}\! \left(x \right) = F_{12}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{54}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{64}\! \left(x \right)
F_{62}\! \left(x \right) = F_{18}\! \left(x \right)+F_{31}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{64}\! \left(x \right) = F_{18}\! \left(x \right)+F_{65}\! \left(x \right)+F_{66}\! \left(x \right)+F_{68}\! \left(x \right)
F_{65}\! \left(x \right) = 0
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{52}\! \left(x \right)
F_{68}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \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_22(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_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_12(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_21(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_10(x))
Eq(F_21(x), F_12(x)*F_13(x))
Eq(F_22(x), F_23(x) + F_43(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_26(x) + F_29(x))
Eq(F_26(x), F_1(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_12(x)*F_16(x))
Eq(F_29(x), F_23(x) + F_30(x))
Eq(F_30(x), F_18(x) + F_31(x) + F_41(x))
Eq(F_31(x), F_12(x)*F_32(x))
Eq(F_32(x), F_33(x) + F_36(x))
Eq(F_33(x), F_12(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), x**2)
Eq(F_36(x), F_37(x) + F_39(x))
Eq(F_37(x), F_18(x) + F_31(x) + F_38(x))
Eq(F_38(x), F_12(x)*F_23(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_12(x)*F_37(x))
Eq(F_41(x), F_12(x)*F_42(x))
Eq(F_42(x), F_23(x) + F_37(x))
Eq(F_43(x), F_18(x) + F_44(x) + F_59(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_51(x))
Eq(F_46(x), F_10(x) + F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_12(x)*F_49(x))
Eq(F_49(x), F_12(x) + F_50(x))
Eq(F_50(x), F_48(x))
Eq(F_51(x), F_52(x) + F_55(x))
Eq(F_52(x), F_18(x) + F_44(x) + F_53(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_23(x) + F_52(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_37(x) + F_58(x))
Eq(F_58(x), F_56(x))
Eq(F_59(x), F_12(x)*F_60(x))
Eq(F_60(x), F_54(x) + F_61(x))
Eq(F_61(x), F_62(x) + F_64(x))
Eq(F_62(x), F_18(x) + F_31(x) + F_63(x))
Eq(F_63(x), F_12(x)*F_42(x))
Eq(F_64(x), F_18(x) + F_65(x) + F_66(x) + F_68(x))
Eq(F_65(x), 0)
Eq(F_66(x), F_12(x)*F_67(x))
Eq(F_67(x), F_52(x))
Eq(F_68(x), F_12(x)*F_61(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, 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, 2, 3, 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, 1, 3, 2], "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, 2, 3, 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