0132_0231_0321_1032_1320_2031_3120
Counting sequence:
1, 1, 2, 6, 17, 45, 114, 278, 657, 1515, 3428, 7644, 16851, 36809, 79806, 171954, 368549, 786327, 1671048, 3538808, 7470951, 15728469, 33029954, 69205806, 144703257, 301989635, 629145324, 1308622548, 2717908667, 5637144225, 11676941958, 24159190634, 49928994381, 103079214639, 212600880656, 438086663664, 901943131599, 1855425871277, 3813930958218, 7834020347238, 16080357555521, 32985348832539, 67619965107444, 138538465098956, 283673999965347, 580542139464825, 1187472557997134, 2427721674128418, 4960996464524277, 10133099161582535, 20688410788232088, 42221246506597224, 86131342873459511, 175640385467448069, 358036170375953106, 729583139634018974, 1486187877032262249, 3026418949592971827, 6160924290242836988, 12538021362599459268, 25508388289426487691, 51881467707308112209, 105492317671526496534, 214443399856873535706, 435804328741388155037, 885443715538058475615, 1798557547186681280544, 3652455326594491217888, 7415591117631239747487, 15052543164146994116445, 30547808186063017473818, 61981060087664093427414, 125733007606404303812241, 255007790074960841537099, 517099129874226150897156, 1048365359197061237437884, 2125064917291340346160499, 4306798232377116434887977, 8726933260343104354907358, 17680540111863951680074898, 35814427406083389300667461, 72535549176877750482367479, 146884487083177444726797224, 297395751625198776977716056, 602045058168085329003672327, 1218597226171546208103822005, 2466208672013843516400595554, 4990445783369189233187090958, 10096948445421382867145978297, 20426010648208774535835545955, 41316248811149566674758267148, 83560952651763168555690881204, 168978815362454407523730452571, 341671450842764955872158281729, 690770541921242193393711312806, 1396396364313908950086212120394, 2822503289570667026770003226349, 5704427701027032306735164419727, 11527697645825461119860644769328, 23293079779193715252501921394128, 47061528533473016530565106494831
Generating function in Maple syntax:
(2*x^7-2*x^5-4*x^4+14*x^3-14*x^2+6*x-1)/(2*x-1)^2/(x-1)^3
Generating function in latex syntax:
\frac{2 x^{7}-2 x^{5}-4 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:
(2*x**7 - 2*x**5 - 4*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)+2*x^7-2*x^5-4*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)+2 x^{7}-2 x^{5}-4 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 = 2,2,1/8*(3*n-3)*2^n-1/2*n^2+5/2*n-3)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(3 n -3\right) 2^{n}}{8}-\frac{n^{2}}{2}+\frac{5 n}{2}-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) = 17
a(5) = 45
a(6) = 114
a(7) = 278
a(n+2) = -1/2*n^2-4*a(n)+4*a(n+1)+9/2*n-8, 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) = 17
a \! \left(5\right) = 45
a \! \left(6\right) = 114
a \! \left(7\right) = 278
a \! \left(n +2\right) = -\frac{n^{2}}{2}-4 a \! \left(n \right)+4 a \! \left(n +1\right)+\frac{9 n}{2}-8, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/22988/
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[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[12,x]*F[7,x]
F[7,x] = F[25,x]+F[8,x]
F[8,x] = F[16,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[13,x]*F[9,x]
F[12,x] = x
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[13,x]
F[16,x] = F[17,x]+F[20,x]
F[17,x] = F[1,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[16,x]
F[20,x] = F[21,x]+F[23,x]
F[21,x] = F[14,x]*F[22,x]
F[22,x] = F[1,x]+F[12,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[13,x]*F[20,x]
F[25,x] = F[12,x]+F[26,x]
F[26,x] = F[27,x]+F[28,x]+F[29,x]
F[27,x] = 0
F[28,x] = F[12,x]*F[14,x]
F[29,x] = F[12,x]*F[30,x]
F[30,x] = F[31,x]+F[44,x]
F[31,x] = F[12,x]+F[32,x]
F[32,x] = F[27,x]+F[28,x]+F[33,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[35,x]+F[38,x]
F[35,x] = F[12,x]+F[36,x]
F[36,x] = F[27,x]+F[28,x]+F[37,x]
F[37,x] = F[12,x]*F[35,x]
F[38,x] = F[39,x]+F[41,x]
F[39,x] = F[40,x]
F[40,x] = x^2
F[41,x] = 2*F[27,x]+F[42,x]+F[43,x]
F[42,x] = F[12,x]*F[36,x]
F[43,x] = F[12,x]*F[38,x]
F[44,x] = F[45,x]+F[48,x]
F[45,x] = F[46,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[12,x]+F[45,x]
F[48,x] = 2*F[27,x]+F[49,x]+F[54,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[36,x]+F[51,x]
F[51,x] = 2*F[27,x]+F[49,x]+F[52,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[45,x]+F[51,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[53,x]+F[56,x]
F[56,x] = F[57,x]+F[59,x]
F[57,x] = F[58,x]
F[58,x] = F[12,x]*F[45,x]
F[59,x] = 3*F[27,x]+F[60,x]+F[61,x]
F[60,x] = F[12,x]*F[51,x]
F[61,x] = F[12,x]*F[56,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_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{12}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{25}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{16}\! \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_{13}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{20}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{23}\! \left(x \right)
F_{21}\! \left(x \right) = F_{14}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right) F_{20}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{29}\! \left(x \right)
F_{27}\! \left(x \right) = 0
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{44}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{38}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{41}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = x^{2}
F_{41}\! \left(x \right) = 2 F_{27}\! \left(x \right)+F_{42}\! \left(x \right)+F_{43}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{48}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right)+F_{45}\! \left(x \right)
F_{48}\! \left(x \right) = 2 F_{27}\! \left(x \right)+F_{49}\! \left(x \right)+F_{54}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{36}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = 2 F_{27}\! \left(x \right)+F_{49}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{45}\! \left(x \right)+F_{51}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{53}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{59}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{59}\! \left(x \right) = 3 F_{27}\! \left(x \right)+F_{60}\! \left(x \right)+F_{61}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \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_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_12(x)*F_7(x))
Eq(F_7(x), F_25(x) + F_8(x))
Eq(F_8(x), F_16(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_13(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_13(x))
Eq(F_16(x), F_17(x) + F_20(x))
Eq(F_17(x), F_1(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_16(x))
Eq(F_20(x), F_21(x) + F_23(x))
Eq(F_21(x), F_14(x)*F_22(x))
Eq(F_22(x), F_1(x) + F_12(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_13(x)*F_20(x))
Eq(F_25(x), F_12(x) + F_26(x))
Eq(F_26(x), F_27(x) + F_28(x) + F_29(x))
Eq(F_27(x), 0)
Eq(F_28(x), F_12(x)*F_14(x))
Eq(F_29(x), F_12(x)*F_30(x))
Eq(F_30(x), F_31(x) + F_44(x))
Eq(F_31(x), F_12(x) + F_32(x))
Eq(F_32(x), F_27(x) + F_28(x) + F_33(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_38(x))
Eq(F_35(x), F_12(x) + F_36(x))
Eq(F_36(x), F_27(x) + F_28(x) + F_37(x))
Eq(F_37(x), F_12(x)*F_35(x))
Eq(F_38(x), F_39(x) + F_41(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), x**2)
Eq(F_41(x), 2*F_27(x) + F_42(x) + F_43(x))
Eq(F_42(x), F_12(x)*F_36(x))
Eq(F_43(x), F_12(x)*F_38(x))
Eq(F_44(x), F_45(x) + F_48(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_12(x) + F_45(x))
Eq(F_48(x), 2*F_27(x) + F_49(x) + F_54(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_36(x) + F_51(x))
Eq(F_51(x), 2*F_27(x) + F_49(x) + F_52(x))
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_45(x) + F_51(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_53(x) + F_56(x))
Eq(F_56(x), F_57(x) + F_59(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_12(x)*F_45(x))
Eq(F_59(x), 3*F_27(x) + F_60(x) + F_61(x))
Eq(F_60(x), F_12(x)*F_51(x))
Eq(F_61(x), F_12(x)*F_56(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": true, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
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