0123_0132_1023_1032_1320
Counting sequence:
1, 1, 2, 6, 19, 55, 151, 400, 1034, 2628, 6599, 16425, 40615, 99934, 244954, 598642, 1459607, 3552219, 8632163, 20951796, 50804662, 123096352, 298063651, 721349461, 1745003243, 4219819578, 10201541006, 24656652438, 59582272555, 143955929503, 347763399055, 840020944792, 2028881208770, 4899934370556, 11833050620063, 28574634772737, 68999514965071, 166608048599062, 402284370728098, 971314292254714, 2345187935478623, 5662240084605987, 13670767884232667, 33005975309821308, 79687117251795118, 192389007041336920, 464482725356825275, 1121389645098291373, 2707332389105112659, 6536195168575614066, 15780004213819320470, 38096766566532687390, 91974663279742817971, 222048344979148299751, 536075856923934359239, 1294209066165855621664, 3124512003880006518266, 7543269103088322918324, 18211122268241977012727, 43965657755717074825689, 106142726011600287131767, 256251686242174683294670, 618647251421507000105962, 1493548494934755367520386, 3605748852987646372392359, 8705055424299252647258475, 21015878148358003266496211, 50736848614548352169566980, 122489649164503725904084774, 295716294517626062684610448, 713922533347851423116840659, 1723561951509447329145113605, 4161047616958865252830250555, 10045659546611225785161270954, 24252371432549104662342934814, 58550411856444512334215546022, 141353214034907477263977788347, 341256877705196857759033717039, 823867045003173863090692433023, 1988991118827286508103090097576, 4801849584889225199682514772402, 11592690893068684603845080139372, 27987232579952475327792571368911, 67567158470825373683541667045041, 163121554357306661654036420650783, 393810276856845513604144817118598, 950742127413811222892533577676466, 2295294570370094866948983201681418, 5541331345525254512214545528336495, 13377957416163110582030024625459123, 32297246487336488377686857874851915, 77972451009806111640140488200255372, 188242149744888758483461913920833310, 454456752975463719377985280962047176, 1097155660647576374121227795309647723, 2648768084174136813843917705764436317, 6394691848802890682055856460082831011, 15438151821393999518708942912550496482, 37270995570819052369039932223434520358, 89980143121488429504240478381062606894, 217231282130708561788803087538997450563
Generating function in Maple syntax:
(2*x^7+x^6-4*x^5-6*x^4+2*x^3+7*x^2-5*x+1)/(x-1)/(2*x-1)/(x^2+x-1)/(x^2+2*x-1)
Generating function in latex syntax:
\frac{2 x^{7}+x^{6}-4 x^{5}-6 x^{4}+2 x^{3}+7 x^{2}-5 x +1}{\left(x -1\right) \left(2 x -1\right) \left(x^{2}+x -1\right) \left(x^{2}+2 x -1\right)}
Generating function in sympy syntax:
(2*x**7 + x**6 - 4*x**5 - 6*x**4 + 2*x**3 + 7*x**2 - 5*x + 1)/((x - 1)*(2*x - 1)*(x**2 + x - 1)*(x**2 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
-(x-1)*(2*x-1)*(x^2+x-1)*(x^2+2*x-1)*F(x)+2*x^7+x^6-4*x^5-6*x^4+2*x^3+7*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right) \left(2 x -1\right) \left(x^{2}+x -1\right) \left(x^{2}+2 x -1\right) F \! \left(x \right)+2 x^{7}+x^{6}-4 x^{5}-6 x^{4}+2 x^{3}+7 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/20*(2*5^(1/2)-10)*(-1/2*5^(1/2)-1/2)^(-n)+1/20*(-2*5^(1/2)-10)*(1/2*5^(1/2)-1/2)^(-n)+1/20*(5*2^(1/2)+30)*(-1-2^(1/2))^(-n)+1/20*(-5*2^(1/2)+30)*(2^(1/2)-1)^(-n)-2^n+1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(2 \sqrt{5}-10\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{20}+\frac{\left(-2 \sqrt{5}-10\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{20}+\\\frac{\left(5 \sqrt{2}+30\right) \left(-1-\sqrt{2}\right)^{-n}}{20}+\frac{\left(-5 \sqrt{2}+30\right) \left(\sqrt{2}-1\right)^{-n}}{20}-2^{n}+1 & \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(7) = 400
a(n+5) = 2*a(n)+5*a(n+1)-3*a(n+2)-6*a(n+3)+5*a(n+4)-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) = 19
a \! \left(5\right) = 55
a \! \left(6\right) = 151
a \! \left(7\right) = 400
a \! \left(n +5\right) = 2 a \! \left(n \right)+5 a \! \left(n +1\right)-3 a \! \left(n +2\right)-6 a \! \left(n +3\right)+5 a \! \left(n +4\right)-2, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16299/
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[19,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]+F[14,x]+F[18,x]
F[13,x] = 0
F[14,x] = F[15,x]*F[4,x]
F[15,x] = F[16,x]+F[17,x]
F[16,x] = F[4,x]
F[17,x] = F[12,x]
F[18,x] = F[4,x]*F[7,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[13,x]+F[21,x]+F[48,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[24,x]
F[23,x] = F[12,x]+F[4,x]
F[24,x] = F[25,x]+F[27,x]
F[25,x] = F[13,x]+F[21,x]+F[26,x]
F[26,x] = F[2,x]*F[4,x]
F[27,x] = F[13,x]+F[28,x]+F[29,x]+F[34,x]
F[28,x] = 0
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[31,x]+F[32,x]
F[31,x] = F[25,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[43,x]
F[38,x] = F[39,x]+F[4,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[16,x]+F[42,x]
F[42,x] = F[39,x]
F[43,x] = F[44,x]+F[45,x]
F[44,x] = F[36,x]
F[45,x] = F[46,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[44,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[50,x]+F[51,x]
F[50,x] = F[2,x]+F[25,x]
F[51,x] = F[33,x]+F[35,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_{19}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{13}\! \left(x \right) = 0
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right)
F_{18}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)+F_{48}\! \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_{24}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right)+F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{27}\! \left(x \right)
F_{25}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{13}\! \left(x \right)+F_{28}\! \left(x \right)+F_{29}\! \left(x \right)+F_{34}\! \left(x \right)
F_{28}\! \left(x \right) = 0
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)
F_{31}\! \left(x \right) = F_{25}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \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_{43}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{4}\! \left(x \right)
F_{39}\! \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_{16}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{39}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{45}\! \left(x \right)
F_{44}\! \left(x \right) = F_{36}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{44}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{51}\! \left(x \right)
F_{50}\! \left(x \right) = F_{2}\! \left(x \right)+F_{25}\! \left(x \right)
F_{51}\! \left(x \right) = F_{33}\! \left(x \right)+F_{35}\! \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_19(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x) + F_14(x) + F_18(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_15(x)*F_4(x))
Eq(F_15(x), F_16(x) + F_17(x))
Eq(F_16(x), F_4(x))
Eq(F_17(x), F_12(x))
Eq(F_18(x), F_4(x)*F_7(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_13(x) + F_21(x) + F_48(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_24(x))
Eq(F_23(x), F_12(x) + F_4(x))
Eq(F_24(x), F_25(x) + F_27(x))
Eq(F_25(x), F_13(x) + F_21(x) + F_26(x))
Eq(F_26(x), F_2(x)*F_4(x))
Eq(F_27(x), F_13(x) + F_28(x) + F_29(x) + F_34(x))
Eq(F_28(x), 0)
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_32(x))
Eq(F_31(x), F_25(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_43(x))
Eq(F_38(x), F_39(x) + F_4(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_16(x) + F_42(x))
Eq(F_42(x), F_39(x))
Eq(F_43(x), F_44(x) + F_45(x))
Eq(F_44(x), F_36(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_44(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_51(x))
Eq(F_50(x), F_2(x) + F_25(x))
Eq(F_51(x), F_33(x) + F_35(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": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "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, 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, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "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, 3, 2, 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