0132_0213_2130_3120
Counting sequence:
1, 1, 2, 6, 20, 62, 170, 415, 922, 1907, 3746, 7108, 13216, 24364, 44946, 83509, 156854, 298233, 573738, 1115074, 2185100, 4308906, 8535962, 16964411, 33789650, 67401487, 134578450, 268876400, 537405752, 1074385912, 2148254146, 4295883313, 8591017326, 17181142069, 34361227226, 68721209934, 137440962084, 274880224918, 549758478218, 1099514678679, 2199026736650, 4398050469611, 8796097509122, 17592191114716, 35184377801680, 70368750596612, 140737495548530, 281474984751085, 562949962386982, 1125899916816817, 2251799824756746, 4503599639633818, 9007199268296636, 18014398524436674, 36028797035430906, 72057594056027059, 144115188095714114, 288230376173463303, 576460752327210098, 1152921504632818184, 2305843009242007400, 4611686018458209616, 9223372036888280482, 18446744073745922921, 36893488147458534110, 73786976294880899437, 147573952589722580410, 295147905179402690470, 590295810358759446612, 1180591620717469272622, 2361183241434885005546, 4722366482869712308623, 9444732965739362497146, 18889465931478658190179, 37778931862957244613986, 75557863725914412208884, 151115727451828741842944, 302231454903657395239388, 604462909807314695831378, 1208925819614629290471589, 2417851639229258472851350, 4835703278458516830338921, 9671406556917033537656042, 19342813113834066944231026, 38685626227668133748904940, 77371252455336267349343962, 154742504910672534540864090, 309485009821345068914080555, 618970019642690137650206642, 1237940039285380275111651327, 2475880078570760550023214546, 4951760157141521099834477728, 9903520314283042199444584856, 19807040628566084398651804584, 39614081257132168797052654466, 79228162514264337593840149409, 158456325028528675187400298574, 316912650057057350374505099173, 633825300114114700748698524058, 1267650600228229401497068496894, 2535301200456458802993790842500
Generating function in Maple syntax:
(2*x^7-x^6+17*x^5-31*x^4+33*x^3-21*x^2+7*x-1)/(2*x-1)/(x-1)^6
Generating function in latex syntax:
\frac{2 x^{7}-x^{6}+17 x^{5}-31 x^{4}+33 x^{3}-21 x^{2}+7 x -1}{\left(2 x -1\right) \left(x -1\right)^{6}}
Generating function in sympy syntax:
(2*x**7 - x**6 + 17*x**5 - 31*x**4 + 33*x**3 - 21*x**2 + 7*x - 1)/((x - 1)**6*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^6*F(x)-2*x^7+x^6-17*x^5+31*x^4-33*x^3+21*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{6} F \! \left(x \right)-2 x^{7}+x^{6}-17 x^{5}+31 x^{4}-33 x^{3}+21 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,-2+2^(n+1)+1/24*n^5-1/3*n^4+9/8*n^3-8/3*n^2+5/6*n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ -2+2^{n +1}+\frac{n^{5}}{24}-\frac{n^{4}}{3}+\frac{9 n^{3}}{8}-\frac{8 n^{2}}{3}+\frac{5 n}{6} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 62
a(6) = 170
a(7) = 415
a(n+1) = 2*a(n)-1/24*(n-1)*(n^4-12*n^3+37*n^2-70*n+24), 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) = 20
a \! \left(5\right) = 62
a \! \left(6\right) = 170
a \! \left(7\right) = 415
a \! \left(n +1\right) = 2 a \! \left(n \right)-\frac{\left(n -1\right) \left(n^{4}-12 n^{3}+37 n^{2}-70 n +24\right)}{24}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15302/
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[14,x]+F[18,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[9,x]
F[17,x] = F[14,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[13,x]
F[20,x] = F[21,x]+F[26,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[24,x]+F[25,x]
F[24,x] = F[1,x]+F[14,x]
F[25,x] = F[21,x]
F[26,x] = F[27,x]+F[28,x]+F[50,x]
F[27,x] = 0
F[28,x] = F[12,x]*F[29,x]
F[29,x] = F[30,x]+F[49,x]
F[30,x] = F[31,x]+F[6,x]
F[31,x] = F[27,x]+F[32,x]+F[38,x]
F[32,x] = F[12,x]*F[33,x]
F[33,x] = F[34,x]+F[37,x]
F[34,x] = F[10,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[12,x]*F[34,x]
F[37,x] = F[18,x]
F[38,x] = F[12,x]*F[39,x]
F[39,x] = F[13,x]+F[40,x]
F[40,x] = F[41,x]+F[47,x]
F[41,x] = F[27,x]+F[42,x]+F[44,x]
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[34,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[17,x]+F[46,x]
F[46,x] = F[41,x]
F[47,x] = F[48,x]
F[48,x] = F[12,x]*F[40,x]
F[49,x] = F[26,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[52,x]+F[60,x]
F[52,x] = F[21,x]+F[53,x]
F[53,x] = F[27,x]+F[54,x]+F[59,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[56,x]+F[58,x]
F[56,x] = F[10,x]+F[57,x]
F[57,x] = F[19,x]+F[27,x]+F[32,x]
F[58,x] = F[53,x]
F[59,x] = F[12,x]*F[52,x]
F[60,x] = F[61,x]+F[69,x]
F[61,x] = F[27,x]+F[62,x]+F[65,x]
F[62,x] = F[12,x]*F[63,x]
F[63,x] = F[64,x]
F[64,x] = F[14,x]+F[41,x]
F[65,x] = F[12,x]*F[66,x]
F[66,x] = F[67,x]+F[68,x]
F[67,x] = F[21,x]
F[68,x] = F[61,x]
F[69,x] = 2*F[27,x]+F[70,x]+F[74,x]
F[70,x] = F[12,x]*F[71,x]
F[71,x] = F[72,x]
F[72,x] = F[18,x]+F[73,x]
F[73,x] = F[48,x]
F[74,x] = F[12,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_{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_{14}\! \left(x \right)+F_{18}\! \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_{9}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \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_{26}\! \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_{25}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{25}\! \left(x \right) = F_{21}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{50}\! \left(x \right)
F_{27}\! \left(x \right) = 0
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{49}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{6}\! \left(x \right)
F_{31}\! \left(x \right) = F_{27}\! \left(x \right)+F_{32}\! \left(x \right)+F_{38}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{37}\! \left(x \right)
F_{34}\! \left(x \right) = F_{10}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{37}\! \left(x \right) = F_{18}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{13}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{47}\! \left(x \right)
F_{41}\! \left(x \right) = F_{27}\! \left(x \right)+F_{42}\! \left(x \right)+F_{44}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{34}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{17}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{41}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{49}\! \left(x \right) = F_{26}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{60}\! \left(x \right)
F_{52}\! \left(x \right) = F_{21}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{27}\! \left(x \right)+F_{54}\! \left(x \right)+F_{59}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{58}\! \left(x \right)
F_{56}\! \left(x \right) = F_{10}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{19}\! \left(x \right)+F_{27}\! \left(x \right)+F_{32}\! \left(x \right)
F_{58}\! \left(x \right) = F_{53}\! \left(x \right)
F_{59}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{69}\! \left(x \right)
F_{61}\! \left(x \right) = F_{27}\! \left(x \right)+F_{62}\! \left(x \right)+F_{65}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{14}\! \left(x \right)+F_{41}\! \left(x \right)
F_{65}\! \left(x \right) = F_{12}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)+F_{68}\! \left(x \right)
F_{67}\! \left(x \right) = F_{21}\! \left(x \right)
F_{68}\! \left(x \right) = F_{61}\! \left(x \right)
F_{69}\! \left(x \right) = 2 F_{27}\! \left(x \right)+F_{70}\! \left(x \right)+F_{74}\! \left(x \right)
F_{70}\! \left(x \right) = F_{12}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{18}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{48}\! \left(x \right)
F_{74}\! \left(x \right) = F_{12}\! \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_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_14(x) + F_18(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_9(x))
Eq(F_17(x), F_14(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_13(x))
Eq(F_20(x), F_21(x) + F_26(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_25(x))
Eq(F_24(x), F_1(x) + F_14(x))
Eq(F_25(x), F_21(x))
Eq(F_26(x), F_27(x) + F_28(x) + F_50(x))
Eq(F_27(x), 0)
Eq(F_28(x), F_12(x)*F_29(x))
Eq(F_29(x), F_30(x) + F_49(x))
Eq(F_30(x), F_31(x) + F_6(x))
Eq(F_31(x), F_27(x) + F_32(x) + F_38(x))
Eq(F_32(x), F_12(x)*F_33(x))
Eq(F_33(x), F_34(x) + F_37(x))
Eq(F_34(x), F_10(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_12(x)*F_34(x))
Eq(F_37(x), F_18(x))
Eq(F_38(x), F_12(x)*F_39(x))
Eq(F_39(x), F_13(x) + F_40(x))
Eq(F_40(x), F_41(x) + F_47(x))
Eq(F_41(x), F_27(x) + F_42(x) + F_44(x))
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_34(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_17(x) + F_46(x))
Eq(F_46(x), F_41(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_12(x)*F_40(x))
Eq(F_49(x), F_26(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_60(x))
Eq(F_52(x), F_21(x) + F_53(x))
Eq(F_53(x), F_27(x) + F_54(x) + F_59(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_58(x))
Eq(F_56(x), F_10(x) + F_57(x))
Eq(F_57(x), F_19(x) + F_27(x) + F_32(x))
Eq(F_58(x), F_53(x))
Eq(F_59(x), F_12(x)*F_52(x))
Eq(F_60(x), F_61(x) + F_69(x))
Eq(F_61(x), F_27(x) + F_62(x) + F_65(x))
Eq(F_62(x), F_12(x)*F_63(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_14(x) + F_41(x))
Eq(F_65(x), F_12(x)*F_66(x))
Eq(F_66(x), F_67(x) + F_68(x))
Eq(F_67(x), F_21(x))
Eq(F_68(x), F_61(x))
Eq(F_69(x), 2*F_27(x) + F_70(x) + F_74(x))
Eq(F_70(x), F_12(x)*F_71(x))
Eq(F_71(x), F_72(x))
Eq(F_72(x), F_18(x) + F_73(x))
Eq(F_73(x), F_48(x))
Eq(F_74(x), F_12(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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