0132_0321_1302_2130_3120
Counting sequence:
1, 1, 2, 6, 19, 52, 127, 287, 616, 1279, 2602, 5232, 10457, 20846, 41529, 82757, 165022, 329297, 657516, 1313534, 2625047, 5247432, 10491427, 20978491, 41951524, 83896307, 167784382, 335558812, 671105701, 1342197234, 2684377757, 5368735937, 10737449082, 21474871781, 42949713184, 85899391562, 171798743427, 343597441772, 687194832551, 1374389607639, 2748779150752, 5497558229287, 10995116378002, 21990232666376, 43980465233329, 87960930356662, 175921860591937, 351843721050237, 703687441953686, 1407374883746489, 2814749767317012, 5629499534441942, 11258999068674607, 22517998137121616, 45035996273996139, 90071992547724467, 180143985095159132, 360287970190005147, 720575940379672486, 1441151880758981044, 2882303761517570557, 5764607523034720442, 11529215046068989477, 23058430092137495161, 46116860184274472434, 92233720368548391117, 184467440737096190792, 368934881474191750562, 737869762948382828571, 1475739525896764941044, 2951479051793529120367, 5902958103587057431247, 11805916207174114003032, 23611832414348227094351, 47223664828696453222394, 94447329657392905421472, 188894659314785809760137, 377789318629571618375422, 755578637259143235541321, 1511157274518286469805749, 3022314549036572938264462, 6044629098073145875108897, 12089258196146291748721852, 24178516392292583495868846, 48357032784585166990080839, 96714065569170333978419672, 193428131138340667955008947, 386856262276681335908095787, 773712524553362671814174356, 1547425049106725343626232899, 3094850098213450687250247822, 6189700196426901374498171852, 12379400392853802748993910357, 24758800785707605497985273986, 49517601571415210995967883949, 99035203142830421991932982577, 198070406285660843983863054442, 396140812571321687967723068597, 792281625142643375935442963056, 1584563250285286751870882613754, 3169126500570573503741761772467
Generating function in Maple syntax:
-(2*x^7-4*x^6+7*x^5-12*x^4+18*x^3-15*x^2+6*x-1)/(2*x-1)/(x-1)^5
Generating function in latex syntax:
-\frac{2 x^{7}-4 x^{6}+7 x^{5}-12 x^{4}+18 x^{3}-15 x^{2}+6 x -1}{\left(2 x -1\right) \left(x -1\right)^{5}}
Generating function in sympy syntax:
(-2*x**7 + 4*x**6 - 7*x**5 + 12*x**4 - 18*x**3 + 15*x**2 - 6*x + 1)/((x - 1)**5*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^5*F(x)+2*x^7-4*x^6+7*x^5-12*x^4+18*x^3-15*x^2+6*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{5} F \! \left(x \right)+2 x^{7}-4 x^{6}+7 x^{5}-12 x^{4}+18 x^{3}-15 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,2-67/12*n+23/24*n^2-5/12*n^3+1/24*n^4+5*2^(-1+n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2-\frac{67 n}{12}+\frac{23 n^{2}}{24}-\frac{5 n^{3}}{12}+\frac{n^{4}}{24}+5 \,2^{-1+n} & \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) = 52
a(6) = 127
a(7) = 287
a(n+1) = -1/24*n^4+7/12*n^3-47/24*n^2+2*a(n)+77/12*n-7, 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) = 52
a \! \left(6\right) = 127
a \! \left(7\right) = 287
a \! \left(n +1\right) = -\frac{n^{4}}{24}+\frac{7 n^{3}}{12}-\frac{47 n^{2}}{24}+2 a \! \left(n \right)+\frac{77 n}{12}-7, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19538/
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[20,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[10,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[19,x]
F[15,x] = 0
F[16,x] = F[12,x]*F[17,x]
F[17,x] = F[10,x]+F[18,x]
F[18,x] = F[16,x]
F[19,x] = F[12,x]*F[13,x]
F[20,x] = F[21,x]+F[30,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[24,x]+F[29,x]
F[24,x] = F[1,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[12,x]*F[27,x]
F[27,x] = F[28,x]+F[9,x]
F[28,x] = F[10,x]
F[29,x] = F[21,x]
F[30,x] = F[15,x]+F[31,x]+F[39,x]
F[31,x] = F[12,x]*F[32,x]
F[32,x] = F[33,x]+F[38,x]
F[33,x] = F[34,x]+F[6,x]
F[34,x] = F[35,x]
F[35,x] = F[12,x]*F[36,x]
F[36,x] = F[17,x]+F[37,x]
F[37,x] = F[18,x]
F[38,x] = F[30,x]
F[39,x] = F[12,x]*F[40,x]
F[40,x] = F[41,x]+F[49,x]
F[41,x] = F[21,x]+F[42,x]
F[42,x] = F[15,x]+F[43,x]+F[48,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[45,x]+F[47,x]
F[45,x] = F[10,x]+F[46,x]
F[46,x] = F[35,x]
F[47,x] = F[42,x]
F[48,x] = F[12,x]*F[41,x]
F[49,x] = F[50,x]+F[59,x]
F[50,x] = F[15,x]+F[51,x]+F[57,x]
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[10,x]+F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[12,x]*F[56,x]
F[56,x] = F[17,x]
F[57,x] = F[12,x]*F[58,x]
F[58,x] = F[21,x]
F[59,x] = F[15,x]+F[60,x]+F[69,x]+F[71,x]
F[60,x] = F[12,x]*F[61,x]
F[61,x] = F[62,x]
F[62,x] = F[18,x]+F[63,x]
F[63,x] = F[64,x]
F[64,x] = F[12,x]*F[65,x]
F[65,x] = F[66,x]
F[66,x] = F[18,x]+F[67,x]
F[67,x] = F[68,x]
F[68,x] = F[12,x]*F[66,x]
F[69,x] = F[12,x]*F[70,x]
F[70,x] = F[42,x]
F[71,x] = F[12,x]*F[49,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_{20}\! \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_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{19}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{10}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{30}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{29}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{9}\! \left(x \right)
F_{28}\! \left(x \right) = F_{10}\! \left(x \right)
F_{29}\! \left(x \right) = F_{21}\! \left(x \right)
F_{30}\! \left(x \right) = F_{15}\! \left(x \right)+F_{31}\! \left(x \right)+F_{39}\! \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_{38}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{6}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{17}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{18}\! \left(x \right)
F_{38}\! \left(x \right) = F_{30}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{49}\! \left(x \right)
F_{41}\! \left(x \right) = F_{21}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{15}\! \left(x \right)+F_{43}\! \left(x \right)+F_{48}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{47}\! \left(x \right)
F_{45}\! \left(x \right) = F_{10}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{35}\! \left(x \right)
F_{47}\! \left(x \right) = F_{42}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{59}\! \left(x \right)
F_{50}\! \left(x \right) = F_{15}\! \left(x \right)+F_{51}\! \left(x \right)+F_{57}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{10}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{17}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{21}\! \left(x \right)
F_{59}\! \left(x \right) = F_{15}\! \left(x \right)+F_{60}\! \left(x \right)+F_{69}\! \left(x \right)+F_{71}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{18}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{12}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{18}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{12}\! \left(x \right) F_{66}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{42}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \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_20(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_10(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_19(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_12(x)*F_17(x))
Eq(F_17(x), F_10(x) + F_18(x))
Eq(F_18(x), F_16(x))
Eq(F_19(x), F_12(x)*F_13(x))
Eq(F_20(x), F_21(x) + F_30(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_12(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_29(x))
Eq(F_24(x), F_1(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_12(x)*F_27(x))
Eq(F_27(x), F_28(x) + F_9(x))
Eq(F_28(x), F_10(x))
Eq(F_29(x), F_21(x))
Eq(F_30(x), F_15(x) + F_31(x) + F_39(x))
Eq(F_31(x), F_12(x)*F_32(x))
Eq(F_32(x), F_33(x) + F_38(x))
Eq(F_33(x), F_34(x) + F_6(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_12(x)*F_36(x))
Eq(F_36(x), F_17(x) + F_37(x))
Eq(F_37(x), F_18(x))
Eq(F_38(x), F_30(x))
Eq(F_39(x), F_12(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_49(x))
Eq(F_41(x), F_21(x) + F_42(x))
Eq(F_42(x), F_15(x) + F_43(x) + F_48(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_47(x))
Eq(F_45(x), F_10(x) + F_46(x))
Eq(F_46(x), F_35(x))
Eq(F_47(x), F_42(x))
Eq(F_48(x), F_12(x)*F_41(x))
Eq(F_49(x), F_50(x) + F_59(x))
Eq(F_50(x), F_15(x) + F_51(x) + F_57(x))
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_10(x) + F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_12(x)*F_56(x))
Eq(F_56(x), F_17(x))
Eq(F_57(x), F_12(x)*F_58(x))
Eq(F_58(x), F_21(x))
Eq(F_59(x), F_15(x) + F_60(x) + F_69(x) + F_71(x))
Eq(F_60(x), F_12(x)*F_61(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_18(x) + F_63(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_12(x)*F_65(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_18(x) + F_67(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_12(x)*F_66(x))
Eq(F_69(x), F_12(x)*F_70(x))
Eq(F_70(x), F_42(x))
Eq(F_71(x), F_12(x)*F_49(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, 3, 0, 2], "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, 1, 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, 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, 3, 0, 2], "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, 1, 2, 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