0123_0132_1203_2301
Counting sequence:
1, 1, 2, 6, 20, 63, 181, 478, 1179, 2757, 6188, 13462, 28606, 59731, 123127, 251454, 510125, 1030073, 2073286, 4163878, 8350232, 16729287, 33495113, 67036062, 134129071, 268328269, 536742192, 1073588214, 2147301410, 4294752283, 8589682219, 17179574398, 34359395617, 68719079937, 137438495978, 274877381510, 549755212636, 1099510942159, 2199022476317, 4398045628254, 8796092024963, 17592184921173, 35184370827124, 70368742764118, 140737486775622, 281474974949475, 562949951462303, 1125899904668350, 2251799811277141, 4503599624708809, 9007199251804750, 18014398506248934, 36028797015410528, 72057594034029143, 144115188071585329, 288230376147041566, 576460752298324247, 1152921504601287645, 2305843009207641848, 4611686018420808630, 9223372036847633194, 18446744073701807659, 36893488147410718035, 73786976294829138174, 147573952589666617673, 295147905179342257681, 590295810358694262514, 1180591620717399042886, 2361183241434809422372, 4722366482869631050335, 9444732965739275227781, 18889465931478564559070, 37778931862957144255339, 75557863725914304741349, 151115727451828626869180, 302231454903657272345622, 604462909807314564586958, 1208925819614629150428531, 2417851639229258323543879, 4835703278458516671283006, 9671406556917033368348925, 19342813113834066764150745, 38685626227668133557509846, 77371252455336267146072230, 154742504910672534325133224, 309485009821345068685286887, 618970019642690137407724825, 1237940039285380274854833822, 2475880078570760549751391103, 4951760157141521099546954861, 9903520314283042199140645312, 19807040628566084398330706806, 39614081257132168796713632050, 79228162514264337593482410555, 158456325028528675187023025531, 316912650057057350374107447678, 633825300114114700748279622769, 1267650600228229401496627446817, 2535301200456458802993326716410, 5070602400912917605986729029382, 10141204801825835211973537586156
Generating function in Maple syntax:
-(4*x^6-16*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{4 x^{6}-16 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:
(-4*x**6 + 16*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)+4*x^6-16*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)+4 x^{6}-16 x^{5}+31 x^{4}-33 x^{3}+21 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
-7+8*2^n-119/20*n-13/24*n^3-35/24*n^2-1/24*n^4-1/120*n^5
Explicit closed form in latex syntax:
-7+8 \,2^{n}-\frac{119 n}{20}-\frac{13 n^{3}}{24}-\frac{35 n^{2}}{24}-\frac{n^{4}}{24}-\frac{n^{5}}{120}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 63
a(6) = 181
a(n+1) = 2*a(n)+1/120*(n-1)*(n^4+n^3+36*n^2-24*n+120), n >= 7
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) = 63
a \! \left(6\right) = 181
a \! \left(n +1\right) = 2 a \! \left(n \right)+\frac{\left(n -1\right) \left(n^{4}+n^{3}+36 n^{2}-24 n +120\right)}{120}, \quad n \geq 7
Specification 1
Strategy pack name: point_and_row_placements
Tree: http://permpal.com/tree/21138/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[14,x]*F[3,x]
F[3,x] = F[0,x]+F[4,x]
F[4,x] = F[5,x]
F[5,x] = F[14,x]*F[6,x]
F[6,x] = F[67,x]+F[7,x]
F[7,x] = F[32,x]+F[8,x]
F[8,x] = F[26,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[14,x]
F[12,x] = F[13,x]+F[15,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = x
F[15,x] = F[10,x]+F[16,x]
F[16,x] = F[17,x]+F[18,x]+F[22,x]
F[17,x] = 0
F[18,x] = F[14,x]*F[19,x]
F[19,x] = F[20,x]+F[21,x]
F[20,x] = F[14,x]
F[21,x] = F[16,x]
F[22,x] = F[14,x]*F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[14,x]*F[25,x]
F[25,x] = F[1,x]+F[23,x]
F[26,x] = F[23,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[14,x]*F[29,x]
F[29,x] = F[30,x]+F[31,x]
F[30,x] = F[23,x]
F[31,x] = F[27,x]
F[32,x] = F[33,x]+F[53,x]
F[33,x] = F[23,x]+F[34,x]
F[34,x] = F[17,x]+F[35,x]+F[37,x]
F[35,x] = F[14,x]*F[36,x]
F[36,x] = F[10,x]+F[34,x]
F[37,x] = F[14,x]*F[38,x]
F[38,x] = F[39,x]+F[43,x]
F[39,x] = F[23,x]+F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[14,x]*F[42,x]
F[42,x] = F[14,x]+F[40,x]
F[43,x] = F[34,x]+F[44,x]
F[44,x] = 2*F[17,x]+F[45,x]+F[49,x]
F[45,x] = F[14,x]*F[46,x]
F[46,x] = F[47,x]+F[48,x]
F[47,x] = F[22,x]
F[48,x] = F[45,x]
F[49,x] = F[14,x]*F[50,x]
F[50,x] = F[51,x]+F[52,x]
F[51,x] = F[40,x]
F[52,x] = F[44,x]
F[53,x] = F[54,x]+F[60,x]
F[54,x] = F[17,x]+F[55,x]+F[57,x]
F[55,x] = F[14,x]*F[56,x]
F[56,x] = F[23,x]+F[54,x]
F[57,x] = F[14,x]*F[58,x]
F[58,x] = F[23,x]+F[59,x]
F[59,x] = F[57,x]
F[60,x] = 2*F[17,x]+F[61,x]+F[63,x]
F[61,x] = F[14,x]*F[62,x]
F[62,x] = F[27,x]+F[60,x]
F[63,x] = F[14,x]*F[64,x]
F[64,x] = F[65,x]+F[66,x]
F[65,x] = F[54,x]
F[66,x] = F[60,x]
F[67,x] = F[14,x]*F[68,x]
F[68,x] = F[6,x]+F[69,x]
F[69,x] = F[70,x]
F[70,x] = F[23,x]*F[71,x]*F[77,x]
F[71,x] = F[25,x]+F[72,x]
F[72,x] = F[23,x]+F[73,x]
F[73,x] = F[17,x]+F[74,x]+F[76,x]
F[74,x] = F[14,x]*F[75,x]
F[75,x] = F[23,x]+F[73,x]
F[76,x] = F[14,x]*F[72,x]
F[77,x] = F[78,x]+F[9,x]
F[78,x] = F[23,x]*F[25,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_{14}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{0}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{14}\! \left(x \right) F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{67}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{32}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{26}\! \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_{14}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{15}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = x
F_{15}\! \left(x \right) = F_{10}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)+F_{22}\! \left(x \right)
F_{17}\! \left(x \right) = 0
F_{18}\! \left(x \right) = F_{14}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)
F_{20}\! \left(x \right) = F_{14}\! \left(x \right)
F_{21}\! \left(x \right) = F_{16}\! \left(x \right)
F_{22}\! \left(x \right) = F_{14}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{14}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{1}\! \left(x \right)+F_{23}\! \left(x \right)
F_{26}\! \left(x \right) = F_{23}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{14}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{31}\! \left(x \right)
F_{30}\! \left(x \right) = F_{23}\! \left(x \right)
F_{31}\! \left(x \right) = F_{27}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{53}\! \left(x \right)
F_{33}\! \left(x \right) = F_{23}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{17}\! \left(x \right)+F_{35}\! \left(x \right)+F_{37}\! \left(x \right)
F_{35}\! \left(x \right) = F_{14}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{10}\! \left(x \right)+F_{34}\! \left(x \right)
F_{37}\! \left(x \right) = F_{14}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{43}\! \left(x \right)
F_{39}\! \left(x \right) = F_{23}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{14}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{14}\! \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) = 2 F_{17}\! \left(x \right)+F_{45}\! \left(x \right)+F_{49}\! \left(x \right)
F_{45}\! \left(x \right) = F_{14}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{48}\! \left(x \right)
F_{47}\! \left(x \right) = F_{22}\! \left(x \right)
F_{48}\! \left(x \right) = F_{45}\! \left(x \right)
F_{49}\! \left(x \right) = F_{14}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{52}\! \left(x \right)
F_{51}\! \left(x \right) = F_{40}\! \left(x \right)
F_{52}\! \left(x \right) = F_{44}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{60}\! \left(x \right)
F_{54}\! \left(x \right) = F_{17}\! \left(x \right)+F_{55}\! \left(x \right)+F_{57}\! \left(x \right)
F_{55}\! \left(x \right) = F_{14}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{23}\! \left(x \right)+F_{54}\! \left(x \right)
F_{57}\! \left(x \right) = F_{14}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{23}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{57}\! \left(x \right)
F_{60}\! \left(x \right) = 2 F_{17}\! \left(x \right)+F_{61}\! \left(x \right)+F_{63}\! \left(x \right)
F_{61}\! \left(x \right) = F_{14}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{27}\! \left(x \right)+F_{60}\! \left(x \right)
F_{63}\! \left(x \right) = F_{14}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{66}\! \left(x \right)
F_{65}\! \left(x \right) = F_{54}\! \left(x \right)
F_{66}\! \left(x \right) = F_{60}\! \left(x \right)
F_{67}\! \left(x \right) = F_{14}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{6}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{23}\! \left(x \right) F_{71}\! \left(x \right) F_{77}\! \left(x \right)
F_{71}\! \left(x \right) = F_{25}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{23}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{17}\! \left(x \right)+F_{74}\! \left(x \right)+F_{76}\! \left(x \right)
F_{74}\! \left(x \right) = F_{14}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{23}\! \left(x \right)+F_{73}\! \left(x \right)
F_{76}\! \left(x \right) = F_{14}\! \left(x \right) F_{72}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right)+F_{9}\! \left(x \right)
F_{78}\! \left(x \right) = F_{23}\! \left(x \right) F_{25}\! \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_14(x)*F_3(x))
Eq(F_3(x), F_0(x) + F_4(x))
Eq(F_4(x), F_5(x))
Eq(F_5(x), F_14(x)*F_6(x))
Eq(F_6(x), F_67(x) + F_7(x))
Eq(F_7(x), F_32(x) + F_8(x))
Eq(F_8(x), F_26(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_14(x))
Eq(F_12(x), F_13(x) + F_15(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), x)
Eq(F_15(x), F_10(x) + F_16(x))
Eq(F_16(x), F_17(x) + F_18(x) + F_22(x))
Eq(F_17(x), 0)
Eq(F_18(x), F_14(x)*F_19(x))
Eq(F_19(x), F_20(x) + F_21(x))
Eq(F_20(x), F_14(x))
Eq(F_21(x), F_16(x))
Eq(F_22(x), F_14(x)*F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_14(x)*F_25(x))
Eq(F_25(x), F_1(x) + F_23(x))
Eq(F_26(x), F_23(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_14(x)*F_29(x))
Eq(F_29(x), F_30(x) + F_31(x))
Eq(F_30(x), F_23(x))
Eq(F_31(x), F_27(x))
Eq(F_32(x), F_33(x) + F_53(x))
Eq(F_33(x), F_23(x) + F_34(x))
Eq(F_34(x), F_17(x) + F_35(x) + F_37(x))
Eq(F_35(x), F_14(x)*F_36(x))
Eq(F_36(x), F_10(x) + F_34(x))
Eq(F_37(x), F_14(x)*F_38(x))
Eq(F_38(x), F_39(x) + F_43(x))
Eq(F_39(x), F_23(x) + F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_14(x)*F_42(x))
Eq(F_42(x), F_14(x) + F_40(x))
Eq(F_43(x), F_34(x) + F_44(x))
Eq(F_44(x), 2*F_17(x) + F_45(x) + F_49(x))
Eq(F_45(x), F_14(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_48(x))
Eq(F_47(x), F_22(x))
Eq(F_48(x), F_45(x))
Eq(F_49(x), F_14(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_52(x))
Eq(F_51(x), F_40(x))
Eq(F_52(x), F_44(x))
Eq(F_53(x), F_54(x) + F_60(x))
Eq(F_54(x), F_17(x) + F_55(x) + F_57(x))
Eq(F_55(x), F_14(x)*F_56(x))
Eq(F_56(x), F_23(x) + F_54(x))
Eq(F_57(x), F_14(x)*F_58(x))
Eq(F_58(x), F_23(x) + F_59(x))
Eq(F_59(x), F_57(x))
Eq(F_60(x), 2*F_17(x) + F_61(x) + F_63(x))
Eq(F_61(x), F_14(x)*F_62(x))
Eq(F_62(x), F_27(x) + F_60(x))
Eq(F_63(x), F_14(x)*F_64(x))
Eq(F_64(x), F_65(x) + F_66(x))
Eq(F_65(x), F_54(x))
Eq(F_66(x), F_60(x))
Eq(F_67(x), F_14(x)*F_68(x))
Eq(F_68(x), F_6(x) + F_69(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_23(x)*F_71(x)*F_77(x))
Eq(F_71(x), F_25(x) + F_72(x))
Eq(F_72(x), F_23(x) + F_73(x))
Eq(F_73(x), F_17(x) + F_74(x) + F_76(x))
Eq(F_74(x), F_14(x)*F_75(x))
Eq(F_75(x), F_23(x) + F_73(x))
Eq(F_76(x), F_14(x)*F_72(x))
Eq(F_77(x), F_78(x) + F_9(x))
Eq(F_78(x), F_23(x)*F_25(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"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": false, "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": "point_and_row_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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