0132_0213_1032_2130_3102
Counting sequence:
1, 1, 2, 6, 19, 54, 142, 359, 893, 2213, 5498, 13732, 34512, 87277, 221979, 567435, 1456816, 3753906, 9702550, 25140803, 65277545, 169774385, 442145126, 1152728896, 3007915164, 7854239449, 20520200247, 53635058199, 140238173788, 366777473358, 959473489378, 2510367926447, 6569013044357, 17191302497597, 44993888594834, 117767814708700, 308263384632936, 806927849909701, 2112326891567955, 5529657687803043, 14475838717989640, 37896209198724330, 99209421623823406, 259725183725072411, 679952110778139809, 1780102561307024969, 4660297299026663102, 12200670588517164664, 31941472573966720308, 83623254552173752753, 218927288327950006383, 573156569738095114287, 1500538269130428854452, 3928449793403890129542, 10284793941107662184314, 26925897127021984302215, 70492826508264159637181, 184552438282582418753621, 483164195605507317541610, 1264939554058788720965908, 3311653259606158710063360, 8670017774801490119674525, 22698395092824323031932811, 59425157415608313666213051, 155577056689643911195172704, 407305971148149254072814114, 1066340772591534014723440582, 2791716175994070108284155379, 7308807409514224928074932953, 19134705351572329874977682273, 50095307224803471021222639638, 131151213445146007690000184752, 343358327281463424756559604172, 898923756593328059405565593737, 2353412918591540433932558282919, 6161314950777036792978245814855, 16130531835750465425457040977676, 42230280358134967202867778143166, 110560308837253285139226450287890, 289450645341377853161232595965023, 757791625543496738305834803241205, 1983924227964566357816041583316845, 5193981051625552463535915162406466, 13598018913311675562127125688458316, 35600075660806411826612648178403224, 93202208013496972215437876810267509, 244006548267254403595540469004725187, 638817436560988184483633245359147219, 1672445760956318338401800180684369848, 4378519845379511801257732090519619418, 11463113773305964193329491611730503518, 30010821470747189408419265652793323467, 78569350631275850038850014895711136017, 205697230407606110217063525598101032057, 538522340560284494620384709957389073774, 1409869791210112431640536210254210854120, 3691087032942544976254829752490633693796, 9663391307360030968952593950038672388961, 25299086888617613114103092382167751299295, 66233869357443035220042680723350124167263, 173402521181592139198768378757255320531300
Generating function in Maple syntax:
-(x^8-8*x^6+10*x^5-25*x^4+37*x^3-25*x^2+8*x-1)/(x^2-3*x+1)/(2*x-1)^2/(x-1)^2
Generating function in latex syntax:
-\frac{x^{8}-8 x^{6}+10 x^{5}-25 x^{4}+37 x^{3}-25 x^{2}+8 x -1}{\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-x**8 + 8*x**6 - 10*x**5 + 25*x**4 - 37*x**3 + 25*x**2 - 8*x + 1)/((x - 1)**2*(2*x - 1)**2*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(2*x-1)^2*(x-1)^2*F(x)+x^8-8*x^6+10*x^5-25*x^4+37*x^3-25*x^2+8*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{2} F \! \left(x \right)+x^{8}-8 x^{6}+10 x^{5}-25 x^{4}+37 x^{3}-25 x^{2}+8 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,n = 2,2,1/80*(-8*5^(1/2)+40)*(3/2-1/2*5^(1/2))^(-n)+1/80*(8*5^(1/2)+40)*(3/2+1/2*5^(1/2))^(-n)+1/80*(5*n+55)*2^n-3*n+3)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ 2 & n =2 \\ \frac{\left(-8 \sqrt{5}+40\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{80}+\frac{\left(8 \sqrt{5}+40\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{80}+\frac{\left(5 n +55\right) 2^{n}}{80}\\-3 n +3 & \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) = 54
a(6) = 142
a(7) = 359
a(8) = 893
a(n+4) = -4*a(n)+16*a(n+1)-17*a(n+2)+7*a(n+3)+3*n-6, n >= 9
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) = 54
a \! \left(6\right) = 142
a \! \left(7\right) = 359
a \! \left(8\right) = 893
a \! \left(n +4\right) = -4 a \! \left(n \right)+16 a \! \left(n +1\right)-17 a \! \left(n +2\right)+7 a \! \left(n +3\right)+3 n -6, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18364/
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[16,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[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[7,x]
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[4,x]
F[16,x] = F[17,x]+F[30,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[20,x]+F[6,x]
F[20,x] = F[17,x]+F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[27,x]
F[24,x] = F[17,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[24,x]*F[4,x]
F[27,x] = F[21,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[27,x]*F[4,x]
F[30,x] = F[31,x]+F[32,x]+F[55,x]
F[31,x] = 0
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]+F[41,x]
F[34,x] = F[11,x]+F[35,x]
F[35,x] = F[15,x]+F[31,x]+F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]
F[38,x] = F[11,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[38,x]*F[4,x]
F[41,x] = F[42,x]+F[54,x]
F[42,x] = F[43,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[45,x]+F[47,x]
F[45,x] = F[11,x]+F[46,x]
F[46,x] = F[36,x]
F[47,x] = F[42,x]+F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[42,x]+F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[4,x]*F[51,x]
F[54,x] = F[49,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[57,x]+F[60,x]
F[57,x] = F[17,x]+F[58,x]
F[58,x] = F[31,x]+F[32,x]+F[59,x]
F[59,x] = F[4,x]*F[57,x]
F[60,x] = F[61,x]+F[69,x]
F[61,x] = F[31,x]+F[62,x]+F[64,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[34,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[24,x]+F[66,x]
F[66,x] = F[61,x]+F[67,x]
F[67,x] = F[68,x]
F[68,x] = F[4,x]*F[66,x]
F[69,x] = F[70,x]
F[70,x] = F[4,x]*F[60,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_{16}\! \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_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{30}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{6}\! \left(x \right)
F_{20}\! \left(x \right) = F_{17}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{27}\! \left(x \right)
F_{24}\! \left(x \right) = F_{17}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{21}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{55}\! \left(x \right)
F_{31}\! \left(x \right) = 0
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{41}\! \left(x \right)
F_{34}\! \left(x \right) = F_{11}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{15}\! \left(x \right)+F_{31}\! \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_{38}\! \left(x \right) = F_{11}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{54}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \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_{11}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{36}\! \left(x \right)
F_{47}\! \left(x \right) = F_{42}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{42}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{54}\! \left(x \right) = F_{49}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{60}\! \left(x \right)
F_{57}\! \left(x \right) = F_{17}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{69}\! \left(x \right)
F_{61}\! \left(x \right) = F_{31}\! \left(x \right)+F_{62}\! \left(x \right)+F_{64}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{34}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{24}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{61}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \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_16(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_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_7(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_30(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_20(x) + F_6(x))
Eq(F_20(x), F_17(x) + F_21(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_27(x))
Eq(F_24(x), F_17(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_24(x)*F_4(x))
Eq(F_27(x), F_21(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_27(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_32(x) + F_55(x))
Eq(F_31(x), 0)
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_41(x))
Eq(F_34(x), F_11(x) + F_35(x))
Eq(F_35(x), F_15(x) + F_31(x) + F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_11(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_38(x)*F_4(x))
Eq(F_41(x), F_42(x) + F_54(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_47(x))
Eq(F_45(x), F_11(x) + F_46(x))
Eq(F_46(x), F_36(x))
Eq(F_47(x), F_42(x) + F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_42(x) + F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_4(x)*F_51(x))
Eq(F_54(x), F_49(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_57(x) + F_60(x))
Eq(F_57(x), F_17(x) + F_58(x))
Eq(F_58(x), F_31(x) + F_32(x) + F_59(x))
Eq(F_59(x), F_4(x)*F_57(x))
Eq(F_60(x), F_61(x) + F_69(x))
Eq(F_61(x), F_31(x) + F_62(x) + F_64(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_34(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_24(x) + F_66(x))
Eq(F_66(x), F_61(x) + F_67(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_4(x)*F_66(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_4(x)*F_60(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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