0132_0213_0231_1032_3201
Counting sequence:
1, 1, 2, 6, 19, 52, 126, 290, 652, 1448, 3190, 6982, 15192, 32876, 70786, 151706, 323764, 688336, 1458414, 3080462, 6488368, 13631828, 28574074, 59769250, 124781004, 260047352, 541065766, 1124074070, 2332033672, 4831838908, 9999221490, 20669530922, 42681238372, 88046830496, 181462369246, 373662155806, 768799147104, 1580547966116, 3246995277034, 6665789244722, 13675175871868, 28037546509768, 57449482552854, 117647744173670, 240793046484664, 492581209245452, 1007152651044706, 2058285767198650, 4204532464617492, 8584986789677168, 17521817300240590, 35747322042255662, 72902018968062352, 148618787703228916, 302867074940668506, 616993148949760706, 1256504296036371244, 2558044588346444696, 5206161169240296454, 10592466323575409782, 21545220617340456168, 43811017175060188508, 89063186230878932434, 181008676223274978890, 367781959969584189124, 747093134985236844352, 1517244700062610624446, 3080606260309495124030, 6253446240987538002112, 12691359922712171516228, 25751654726898534060490, 52241179216745450181202, 105958097959387664487132, 214867674970568857228136, 435638308044724770968566, 883082532296623654966406, 1789776897007595535996184, 3626777458843887524124076, 7348002247345167952516674, 14884899154005121713575642, 30147587626639815044241268, 61050753890538773322668048, 123612665055595833113712814, 250247644660228239164184910, 506539918418529624201894384, 1025169095033205540150844052, 2074516706458703663795804986, 4197390445701992494579850210, 8491494956973155323136187532, 17176418045084651314225356088, 34739692352445983964356681190, 70253097229445330600525307542, 142053619507997386544674512712, 287202089114208223776596828156, 580593878424843348927689269426, 1173567157242540500604369772906, 2371893115270788606706722021924, 4793303832112992424409409004256, 9685642867368815270810747937694, 19569356141023291385605355742302, 39534853094617904459178431227168
Generating function in Maple syntax:
(4*x^8-9*x^7-x^6+5*x^5-6*x^4+14*x^3-14*x^2+6*x-1)/(2*x-1)^2/(x-1)^3
Generating function in latex syntax:
\frac{4 x^{8}-9 x^{7}-x^{6}+5 x^{5}-6 x^{4}+14 x^{3}-14 x^{2}+6 x -1}{\left(2 x -1\right)^{2} \left(x -1\right)^{3}}
Generating function in sympy syntax:
(4*x**8 - 9*x**7 - x**6 + 5*x**5 - 6*x**4 + 14*x**3 - 14*x**2 + 6*x - 1)/((x - 1)**3*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
-(2*x-1)^2*(x-1)^3*F(x)+4*x^8-9*x^7-x^6+5*x^5-6*x^4+14*x^3-14*x^2+6*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)+4 x^{8}-9 x^{7}-x^{6}+5 x^{5}-6 x^{4}+14 x^{3}-14 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,n = 3,6,1/16*(5*n-1)*2^n+n^2-5*n+4)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 6 & n =3 \\ \frac{\left(5 n -1\right) 2^{n}}{16}+n^{2}-5 n +4 & \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) = 126
a(7) = 290
a(8) = 652
a(n+2) = -4*a(n)+4*a(n+1)+(-2+n)*(n-7), 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) = 52
a \! \left(6\right) = 126
a \! \left(7\right) = 290
a \! \left(8\right) = 652
a \! \left(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right)+\left(-2+n \right) \left(n -7\right), \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18154/
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[21,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[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[10,x]+F[17,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[1,x]+F[18,x]
F[21,x] = F[22,x]+F[40,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[26,x]
F[25,x] = F[1,x]+F[14,x]
F[26,x] = F[18,x]+F[27,x]
F[27,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[11,x]+F[31,x]
F[31,x] = F[28,x]+F[29,x]+F[32,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[18,x]+F[31,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[33,x]+F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[18,x]+F[37,x]
F[40,x] = F[28,x]+F[41,x]+F[62,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[43,x]+F[51,x]
F[43,x] = F[11,x]+F[44,x]
F[44,x] = F[28,x]+F[45,x]+F[50,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[11,x]+F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[4,x]*F[47,x]
F[50,x] = F[13,x]*F[4,x]
F[51,x] = F[31,x]+F[52,x]
F[52,x] = F[28,x]+F[53,x]+F[58,x]+F[60,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[48,x]+F[55,x]
F[55,x] = 2*F[28,x]+F[53,x]+F[56,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[31,x]+F[55,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[57,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[27,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[64,x]+F[67,x]
F[64,x] = F[22,x]+F[65,x]
F[65,x] = F[28,x]+F[41,x]+F[66,x]
F[66,x] = F[4,x]*F[64,x]
F[67,x] = F[68,x]
F[68,x] = F[28,x]+F[34,x]+F[69,x]
F[69,x] = F[4,x]*F[70,x]
F[70,x] = F[71,x]+F[74,x]
F[71,x] = F[11,x]+F[72,x]
F[72,x] = F[28,x]+F[45,x]+F[73,x]
F[73,x] = F[17,x]*F[4,x]
F[74,x] = F[31,x]+F[75,x]
F[75,x] = F[28,x]+F[53,x]+F[58,x]+F[76,x]
F[76,x] = F[36,x]*F[4,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_{21}\! \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_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{40}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)
F_{25}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{26}\! \left(x \right) = F_{18}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \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_{11}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{18}\! \left(x \right)+F_{31}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{33}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{18}\! \left(x \right)+F_{37}\! \left(x \right)
F_{40}\! \left(x \right) = F_{28}\! \left(x \right)+F_{41}\! \left(x \right)+F_{62}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{51}\! \left(x \right)
F_{43}\! \left(x \right) = F_{11}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{28}\! \left(x \right)+F_{45}\! \left(x \right)+F_{50}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{11}\! \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_{47}\! \left(x \right)
F_{50}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{51}\! \left(x \right) = F_{31}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{28}\! \left(x \right)+F_{53}\! \left(x \right)+F_{58}\! \left(x \right)+F_{60}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{48}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = 2 F_{28}\! \left(x \right)+F_{53}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{31}\! \left(x \right)+F_{55}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{57}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{27}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{67}\! \left(x \right)
F_{64}\! \left(x \right) = F_{22}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{28}\! \left(x \right)+F_{41}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{28}\! \left(x \right)+F_{34}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{4}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)+F_{74}\! \left(x \right)
F_{71}\! \left(x \right) = F_{11}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{28}\! \left(x \right)+F_{45}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{74}\! \left(x \right) = F_{31}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{28}\! \left(x \right)+F_{53}\! \left(x \right)+F_{58}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \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_21(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))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_10(x) + F_17(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_1(x) + F_18(x))
Eq(F_21(x), F_22(x) + F_40(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_26(x))
Eq(F_25(x), F_1(x) + F_14(x))
Eq(F_26(x), F_18(x) + F_27(x))
Eq(F_27(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_11(x) + F_31(x))
Eq(F_31(x), F_28(x) + F_29(x) + F_32(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_18(x) + F_31(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_33(x) + F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_18(x) + F_37(x))
Eq(F_40(x), F_28(x) + F_41(x) + F_62(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_51(x))
Eq(F_43(x), F_11(x) + F_44(x))
Eq(F_44(x), F_28(x) + F_45(x) + F_50(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_11(x) + F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_4(x)*F_47(x))
Eq(F_50(x), F_13(x)*F_4(x))
Eq(F_51(x), F_31(x) + F_52(x))
Eq(F_52(x), F_28(x) + F_53(x) + F_58(x) + F_60(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_48(x) + F_55(x))
Eq(F_55(x), 2*F_28(x) + F_53(x) + F_56(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_31(x) + F_55(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_57(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_27(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_64(x) + F_67(x))
Eq(F_64(x), F_22(x) + F_65(x))
Eq(F_65(x), F_28(x) + F_41(x) + F_66(x))
Eq(F_66(x), F_4(x)*F_64(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_28(x) + F_34(x) + F_69(x))
Eq(F_69(x), F_4(x)*F_70(x))
Eq(F_70(x), F_71(x) + F_74(x))
Eq(F_71(x), F_11(x) + F_72(x))
Eq(F_72(x), F_28(x) + F_45(x) + F_73(x))
Eq(F_73(x), F_17(x)*F_4(x))
Eq(F_74(x), F_31(x) + F_75(x))
Eq(F_75(x), F_28(x) + F_53(x) + F_58(x) + F_76(x))
Eq(F_76(x), F_36(x)*F_4(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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