0132_0231_1023_1230_2301
Counting sequence:
1, 1, 2, 6, 19, 53, 129, 285, 592, 1186, 2332, 4552, 8881, 17379, 34155, 67415, 133558, 265368, 528398, 1053738, 2103551, 4202145, 8398117, 16788641, 33568044, 67124958, 134236624, 268457500, 536896477, 1073771311, 2147517487, 4295005947, 8589978546, 17179918964, 34359794530, 68719539870, 137439024203, 274877985933, 549755901833, 1099511725413, 2199023363656, 4398046630490, 8796093153732, 17592186188976, 35184372247369, 70368744351163, 140737488544819, 281474976917215, 562949953646062, 1125899907086736, 2251799813949942, 4503599627657042, 9007199255050711, 18014398509816249, 36028797019324205, 72057594038315625, 144115188076272548, 288230376152158998, 576460752303902968, 1152921504607360388, 2305843009214243061, 4611686018427974535, 9223372036855401847, 18446744073710219011, 36893488147419813994, 73786976294838962668, 147573952589677216714, 295147905179353679430, 590295810358706557347, 1180591620717412263461, 2361183241434823623697, 4722366482869646289837, 9444732965739291565376, 18889465931478582057234, 37778931862957162979180, 75557863725914324758680, 151115727451828648250593, 302231454903657295164563, 604462909807314588919803, 1208925819614629176354663, 2417851639229258351145766, 4835703278458516700646280, 9671406556917033399562462, 19342813113834066797306746, 38685626227668133592703919, 77371252455336267183403473, 154742504910672534364704309, 309485009821345068727204145, 618970019642690137452098332, 1237940039285380274901777486, 2475880078570760549801022752, 4951760157141521099599396332, 9903520314283042199196022541, 19807040628566084398389149919, 39614081257132168796775275455, 79228162514264337593547393035, 158456325028528675187091490338, 316912650057057350374179542628, 633825300114114700748355500338, 1267650600228229401496707264238, 2535301200456458802993410635771
Generating function in Maple syntax:
(2*x^7-x^6-6*x^5+12*x^4-18*x^3+15*x^2-6*x+1)/(2*x-1)/(x-1)^5
Generating function in latex syntax:
\frac{2 x^{7}-x^{6}-6 x^{5}+12 x^{4}-18 x^{3}+15 x^{2}-6 x +1}{\left(2 x -1\right) \left(x -1\right)^{5}}
Generating function in sympy syntax:
(2*x**7 - x**6 - 6*x**5 + 12*x**4 - 18*x**3 + 15*x**2 - 6*x + 1)/((x - 1)**5*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
-(2*x-1)*(x-1)^5*F(x)+2*x^7-x^6-6*x^5+12*x^4-18*x^3+15*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right) \left(x -1\right)^{5} F \! \left(x \right)+2 x^{7}-x^{6}-6 x^{5}+12 x^{4}-18 x^{3}+15 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,-6+2^(n+1)+1/12*n^3-61/24*n^2+53/12*n+1/24*n^4)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ -6+2^{n +1}+\frac{n^{3}}{12}-\frac{61 n^{2}}{24}+\frac{53 n}{12}+\frac{n^{4}}{24} & \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) = 53
a(6) = 129
a(7) = 285
a(n+1) = -1/24*n^4+1/12*n^3+73/24*n^2+2*a(n)-109/12*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) = 19
a \! \left(5\right) = 53
a \! \left(6\right) = 129
a \! \left(7\right) = 285
a \! \left(n +1\right) = -\frac{n^{4}}{24}+\frac{n^{3}}{12}+\frac{73 n^{2}}{24}+2 a \! \left(n \right)-\frac{109 n}{12}+8, \quad n \geq 8
Specification 1
Strategy pack name: row_and_col_placements
Tree: http://permpal.com/tree/21692/
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[5,x]
F[3,x] = F[1,x]+F[4,x]+F[58,x]
F[4,x] = F[5,x]*F[6,x]
F[5,x] = x
F[6,x] = F[21,x]+F[7,x]
F[7,x] = F[11,x]+F[8,x]
F[8,x] = F[1,x]+F[9,x]
F[9,x] = F[10,x]
F[10,x] = F[5,x]*F[8,x]
F[11,x] = F[12,x]+F[19,x]
F[12,x] = F[13,x]
F[13,x] = F[14,x]*F[5,x]
F[14,x] = F[15,x]+F[18,x]
F[15,x] = F[1,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[15,x]*F[5,x]
F[18,x] = F[12,x]
F[19,x] = F[20,x]
F[20,x] = F[11,x]*F[5,x]
F[21,x] = F[22,x]+F[53,x]
F[22,x] = F[23,x]+F[34,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[5,x]
F[25,x] = F[26,x]+F[27,x]
F[26,x] = F[1,x]+F[12,x]
F[27,x] = F[23,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[5,x]
F[30,x] = F[31,x]+F[32,x]
F[31,x] = F[9,x]
F[32,x] = F[33,x]
F[33,x] = F[29,x]
F[34,x] = F[35,x]+F[36,x]+F[48,x]
F[35,x] = 0
F[36,x] = F[37,x]*F[5,x]
F[37,x] = F[38,x]+F[43,x]
F[38,x] = F[39,x]+F[9,x]
F[39,x] = F[40,x]
F[40,x] = F[41,x]*F[5,x]
F[41,x] = F[42,x]+F[9,x]
F[42,x] = F[40,x]
F[43,x] = F[34,x]+F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[46,x]*F[5,x]
F[46,x] = F[42,x]+F[47,x]
F[47,x] = F[45,x]
F[48,x] = F[49,x]*F[5,x]
F[49,x] = F[50,x]+F[9,x]
F[50,x] = F[35,x]+F[48,x]+F[51,x]
F[51,x] = F[5,x]*F[52,x]
F[52,x] = F[50,x]+F[9,x]
F[53,x] = F[28,x]+F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[5,x]*F[56,x]
F[56,x] = F[33,x]+F[57,x]
F[57,x] = F[55,x]
F[58,x] = F[5,x]*F[59,x]
F[59,x] = F[3,x]+F[60,x]
F[60,x] = F[61,x]
F[61,x] = F[5,x]*F[62,x]
F[62,x] = F[61,x]+F[63,x]+F[64,x]
F[63,x] = F[8,x]^2
F[64,x] = F[65,x]
F[65,x] = F[5,x]*F[66,x]*F[8,x]
F[66,x] = F[52,x]+F[67,x]
F[67,x] = F[68,x]*F[8,x]
F[68,x] = F[69,x]+F[8,x]
F[69,x] = F[70,x]+F[9,x]
F[70,x] = F[35,x]+F[71,x]+F[73,x]
F[71,x] = F[5,x]*F[72,x]
F[72,x] = F[70,x]+F[9,x]
F[73,x] = F[5,x]*F[69,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_{5}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{58}\! \left(x \right)
F_{4}\! \left(x \right) = F_{5}\! \left(x \right) F_{6}\! \left(x \right)
F_{5}\! \left(x \right) = x
F_{6}\! \left(x \right) = F_{21}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{11}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{1}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{5}\! \left(x \right) F_{8}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{19}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{5}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{18}\! \left(x \right)
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{15}\! \left(x \right) F_{5}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{11}\! \left(x \right) F_{5}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{53}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{34}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{5}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)
F_{26}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{27}\! \left(x \right) = F_{23}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{5}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)
F_{31}\! \left(x \right) = F_{9}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{29}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{48}\! \left(x \right)
F_{35}\! \left(x \right) = 0
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{5}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{43}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{9}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right) F_{5}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{9}\! \left(x \right)
F_{42}\! \left(x \right) = F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{34}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right) F_{5}\! \left(x \right)
F_{46}\! \left(x \right) = F_{42}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{45}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right) F_{5}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{9}\! \left(x \right)
F_{50}\! \left(x \right) = F_{35}\! \left(x \right)+F_{48}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{5}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{50}\! \left(x \right)+F_{9}\! \left(x \right)
F_{53}\! \left(x \right) = F_{28}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{5}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{33}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{55}\! \left(x \right)
F_{58}\! \left(x \right) = F_{5}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{3}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{5}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{61}\! \left(x \right)+F_{63}\! \left(x \right)+F_{64}\! \left(x \right)
F_{63}\! \left(x \right) = F_{8} \left(x \right)^{2}
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{5}\! \left(x \right) F_{66}\! \left(x \right) F_{8}\! \left(x \right)
F_{66}\! \left(x \right) = F_{52}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right) F_{8}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)+F_{8}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)+F_{9}\! \left(x \right)
F_{70}\! \left(x \right) = F_{35}\! \left(x \right)+F_{71}\! \left(x \right)+F_{73}\! \left(x \right)
F_{71}\! \left(x \right) = F_{5}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{70}\! \left(x \right)+F_{9}\! \left(x \right)
F_{73}\! \left(x \right) = F_{5}\! \left(x \right) F_{69}\! \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)*F_5(x))
Eq(F_3(x), F_1(x) + F_4(x) + F_58(x))
Eq(F_4(x), F_5(x)*F_6(x))
Eq(F_5(x), x)
Eq(F_6(x), F_21(x) + F_7(x))
Eq(F_7(x), F_11(x) + F_8(x))
Eq(F_8(x), F_1(x) + F_9(x))
Eq(F_9(x), F_10(x))
Eq(F_10(x), F_5(x)*F_8(x))
Eq(F_11(x), F_12(x) + F_19(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_14(x)*F_5(x))
Eq(F_14(x), F_15(x) + F_18(x))
Eq(F_15(x), F_1(x) + F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_15(x)*F_5(x))
Eq(F_18(x), F_12(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_11(x)*F_5(x))
Eq(F_21(x), F_22(x) + F_53(x))
Eq(F_22(x), F_23(x) + F_34(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_5(x))
Eq(F_25(x), F_26(x) + F_27(x))
Eq(F_26(x), F_1(x) + F_12(x))
Eq(F_27(x), F_23(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_5(x))
Eq(F_30(x), F_31(x) + F_32(x))
Eq(F_31(x), F_9(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_29(x))
Eq(F_34(x), F_35(x) + F_36(x) + F_48(x))
Eq(F_35(x), 0)
Eq(F_36(x), F_37(x)*F_5(x))
Eq(F_37(x), F_38(x) + F_43(x))
Eq(F_38(x), F_39(x) + F_9(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_41(x)*F_5(x))
Eq(F_41(x), F_42(x) + F_9(x))
Eq(F_42(x), F_40(x))
Eq(F_43(x), F_34(x) + F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_46(x)*F_5(x))
Eq(F_46(x), F_42(x) + F_47(x))
Eq(F_47(x), F_45(x))
Eq(F_48(x), F_49(x)*F_5(x))
Eq(F_49(x), F_50(x) + F_9(x))
Eq(F_50(x), F_35(x) + F_48(x) + F_51(x))
Eq(F_51(x), F_5(x)*F_52(x))
Eq(F_52(x), F_50(x) + F_9(x))
Eq(F_53(x), F_28(x) + F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_5(x)*F_56(x))
Eq(F_56(x), F_33(x) + F_57(x))
Eq(F_57(x), F_55(x))
Eq(F_58(x), F_5(x)*F_59(x))
Eq(F_59(x), F_3(x) + F_60(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_5(x)*F_62(x))
Eq(F_62(x), F_61(x) + F_63(x) + F_64(x))
Eq(F_63(x), F_8(x)**2)
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_5(x)*F_66(x)*F_8(x))
Eq(F_66(x), F_52(x) + F_67(x))
Eq(F_67(x), F_68(x)*F_8(x))
Eq(F_68(x), F_69(x) + F_8(x))
Eq(F_69(x), F_70(x) + F_9(x))
Eq(F_70(x), F_35(x) + F_71(x) + F_73(x))
Eq(F_71(x), F_5(x)*F_72(x))
Eq(F_72(x), F_70(x) + F_9(x))
Eq(F_73(x), F_5(x)*F_69(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": true, "place_row": true, "strategy_class": "RowAndColumnPlacementFactory"}]], "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": "row_and_col_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:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[1, 2]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 2]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 2], 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