0132_0213_1230_1302_1320
Counting sequence:
1, 1, 2, 6, 19, 56, 154, 403, 1022, 2546, 6291, 15520, 38390, 95451, 238838, 601622, 1525203, 3889176, 9967570, 25655699, 66270366, 171675226, 445759587, 1159554816, 3020714734, 7878059131, 20564159414, 53715419838, 140383465987, 367036681496, 959928321226, 2511149075155, 6570316133630, 17193386735426, 44997007486323, 117771944096032, 308267411686118, 806927416415451, 2112309010290230, 5529587833423334, 14475630723247731, 37895656471695576, 99208042426761922, 259721877411447251, 679944401610476574, 1780084948755969706, 4660257683656783299, 12200582574270651648, 31941278973652660702, 83622832200124824571, 218926373308424263286, 573154599047822843790, 1500534046414491464995, 3928440785247913362776, 10284774799261237104058, 26925856592120607186067, 70492740935818500060158, 184552258132039853614226, 483163817292528408562515, 1264938761408086623581344, 3311651602253717947685846, 8670014315992030728921051, 22698387886992195778891382, 59425142427511084746741686, 155577025560572894319595539, 407305906584236934568213656, 1066340638852141026317312434, 2791715899292102487107832083, 7308806837663853673533904542, 19134704170978597672488401146, 50095304789829836836938919971, 131151208427626091701301260800, 343358316951277799084886908110, 898923735342664414158137456251, 2353412874909627050659991588918, 6161314861052035729392613611102, 16130531651578111287750156032323, 42230279980345549378443066292376, 110560308062785021447960818875050, 289450643754662455221534893040979, 757791622294508177895378168729086, 1983924221315473670043943483821026, 5193981038025135892783558580860467, 13598018885506379931011238665502496, 35600075603986895424657882596397254, 93202207897440088872386865662944731, 244006548030304935410959346015968694, 638817436077417844437621160354800518, 1672445759969834849855988554255907507, 4378519843367859205478554508818443096, 11463113769205287761515960596268831906, 30010821462391094059621805307375812435, 78569350614254174363374183771141432350, 205697230372943788898149882066370385226, 538522340489721912026893186108877078403, 1409869791066511386510284254352095360512, 3691087032650391126056107315421361378238, 9663391306765819748595090881880196678651, 25299086887409383633332668005466230974710, 66233869354986962178172072493343390541614, 173402521176600764954013418495855205412451
Generating function in Maple syntax:
-(3*x^6-6*x^5-x^4+18*x^3-18*x^2+7*x-1)/(2*x-1)/(x^2-3*x+1)/(x^2+x-1)/(x-1)^2
Generating function in latex syntax:
-\frac{3 x^{6}-6 x^{5}-x^{4}+18 x^{3}-18 x^{2}+7 x -1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-3*x**6 + 6*x**5 + x**4 - 18*x**3 + 18*x**2 - 7*x + 1)/((x - 1)**2*(2*x - 1)*(x**2 - 3*x + 1)*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^2-3*x+1)*(x^2+x-1)*(x-1)^2*F(x)+3*x^6-6*x^5-x^4+18*x^3-18*x^2+7*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+3 x^{6}-6 x^{5}-x^{4}+18 x^{3}-18 x^{2}+7 x -1 = 0
Explicit closed form in Maple syntax:
1/10*(5-5^(1/2))*(3/2-1/2*5^(1/2))^(-n)+1/10*(5+5^(1/2))*(3/2+1/2*5^(1/2))^(-n)+1/10*(18*5^(1/2)-40)*(-1/2*5^(1/2)-1/2)^(-n)+1/10*(-18*5^(1/2)-40)*(1/2*5^(1/2)-1/2)^(-n)+2*n+3*2^n+5
Explicit closed form in latex syntax:
\frac{\left(5-\sqrt{5}\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{10}+\frac{\left(5+\sqrt{5}\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{10}+\frac{\left(18 \sqrt{5}-40\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{10}+\frac{\left(-18 \sqrt{5}-40\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{10}+2 n +3 \,2^{n}+5
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 56
a(6) = 154
a(n+5) = -2*a(n)+5*a(n+1)+4*a(n+2)-11*a(n+3)+6*a(n+4)-2*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) = 56
a \! \left(6\right) = 154
a \! \left(n +5\right) = -2 a \! \left(n \right)+5 a \! \left(n +1\right)+4 a \! \left(n +2\right)-11 a \! \left(n +3\right)+6 a \! \left(n +4\right)-2 n -3, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18449/
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[23,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[21,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[1,x]+F[12,x]
F[18,x] = F[14,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[12,x]*F[14,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[13,x]
F[23,x] = F[24,x]+F[35,x]
F[24,x] = F[25,x]
F[25,x] = F[12,x]*F[26,x]
F[26,x] = F[17,x]+F[27,x]
F[27,x] = F[24,x]+F[28,x]
F[28,x] = F[29,x]+F[30,x]+F[34,x]
F[29,x] = 0
F[30,x] = F[12,x]*F[31,x]
F[31,x] = F[32,x]+F[33,x]
F[32,x] = F[12,x]
F[33,x] = F[28,x]
F[34,x] = F[12,x]*F[24,x]
F[35,x] = F[29,x]+F[36,x]+F[61,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[38,x]+F[41,x]
F[38,x] = F[39,x]+F[6,x]
F[39,x] = F[40,x]
F[40,x] = F[10,x]*F[12,x]
F[41,x] = F[35,x]+F[42,x]
F[42,x] = 2*F[29,x]+F[43,x]+F[47,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[45,x]+F[46,x]
F[45,x] = F[39,x]
F[46,x] = F[42,x]
F[47,x] = F[12,x]*F[48,x]
F[48,x] = F[29,x]+F[49,x]+F[59,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[51,x]+F[53,x]
F[51,x] = F[10,x]+F[52,x]
F[52,x] = F[40,x]
F[53,x] = F[48,x]+F[54,x]
F[54,x] = 2*F[29,x]+F[47,x]+F[55,x]
F[55,x] = F[12,x]*F[56,x]
F[56,x] = F[57,x]+F[58,x]
F[57,x] = F[52,x]
F[58,x] = F[54,x]
F[59,x] = F[12,x]*F[60,x]
F[60,x] = F[24,x]+F[48,x]
F[61,x] = F[12,x]*F[62,x]
F[62,x] = F[60,x]+F[63,x]
F[63,x] = F[64,x]+F[70,x]
F[64,x] = F[29,x]+F[30,x]+F[65,x]
F[65,x] = F[12,x]*F[66,x]
F[66,x] = F[27,x]+F[67,x]
F[67,x] = F[64,x]+F[68,x]
F[68,x] = F[69,x]
F[69,x] = F[12,x]*F[64,x]
F[70,x] = F[71,x]
F[71,x] = F[12,x]*F[63,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_{23}\! \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_{21}\! \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_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{35}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{17}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{24}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)+F_{34}\! \left(x \right)
F_{29}\! \left(x \right) = 0
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right)
F_{33}\! \left(x \right) = F_{28}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{35}\! \left(x \right) = F_{29}\! \left(x \right)+F_{36}\! \left(x \right)+F_{61}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{41}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{6}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{41}\! \left(x \right) = F_{35}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = 2 F_{29}\! \left(x \right)+F_{43}\! \left(x \right)+F_{47}\! \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_{46}\! \left(x \right)
F_{45}\! \left(x \right) = F_{39}\! \left(x \right)
F_{46}\! \left(x \right) = F_{42}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{29}\! \left(x \right)+F_{49}\! \left(x \right)+F_{59}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{53}\! \left(x \right)
F_{51}\! \left(x \right) = F_{10}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{40}\! \left(x \right)
F_{53}\! \left(x \right) = F_{48}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = 2 F_{29}\! \left(x \right)+F_{47}\! \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_{57}\! \left(x \right)+F_{58}\! \left(x \right)
F_{57}\! \left(x \right) = F_{52}\! \left(x \right)
F_{58}\! \left(x \right) = F_{54}\! \left(x \right)
F_{59}\! \left(x \right) = F_{12}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{24}\! \left(x \right)+F_{48}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{60}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{70}\! \left(x \right)
F_{64}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{12}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{27}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{64}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \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_23(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_21(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_1(x) + F_12(x))
Eq(F_18(x), F_14(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_12(x)*F_14(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_12(x)*F_13(x))
Eq(F_23(x), F_24(x) + F_35(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_12(x)*F_26(x))
Eq(F_26(x), F_17(x) + F_27(x))
Eq(F_27(x), F_24(x) + F_28(x))
Eq(F_28(x), F_29(x) + F_30(x) + F_34(x))
Eq(F_29(x), 0)
Eq(F_30(x), F_12(x)*F_31(x))
Eq(F_31(x), F_32(x) + F_33(x))
Eq(F_32(x), F_12(x))
Eq(F_33(x), F_28(x))
Eq(F_34(x), F_12(x)*F_24(x))
Eq(F_35(x), F_29(x) + F_36(x) + F_61(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_38(x) + F_41(x))
Eq(F_38(x), F_39(x) + F_6(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_10(x)*F_12(x))
Eq(F_41(x), F_35(x) + F_42(x))
Eq(F_42(x), 2*F_29(x) + F_43(x) + F_47(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_46(x))
Eq(F_45(x), F_39(x))
Eq(F_46(x), F_42(x))
Eq(F_47(x), F_12(x)*F_48(x))
Eq(F_48(x), F_29(x) + F_49(x) + F_59(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_53(x))
Eq(F_51(x), F_10(x) + F_52(x))
Eq(F_52(x), F_40(x))
Eq(F_53(x), F_48(x) + F_54(x))
Eq(F_54(x), 2*F_29(x) + F_47(x) + F_55(x))
Eq(F_55(x), F_12(x)*F_56(x))
Eq(F_56(x), F_57(x) + F_58(x))
Eq(F_57(x), F_52(x))
Eq(F_58(x), F_54(x))
Eq(F_59(x), F_12(x)*F_60(x))
Eq(F_60(x), F_24(x) + F_48(x))
Eq(F_61(x), F_12(x)*F_62(x))
Eq(F_62(x), F_60(x) + F_63(x))
Eq(F_63(x), F_64(x) + F_70(x))
Eq(F_64(x), F_29(x) + F_30(x) + F_65(x))
Eq(F_65(x), F_12(x)*F_66(x))
Eq(F_66(x), F_27(x) + F_67(x))
Eq(F_67(x), F_64(x) + F_68(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_12(x)*F_64(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_12(x)*F_63(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:
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