0132_0231_1032_1230_2301
Counting sequence:
1, 1, 2, 6, 19, 55, 146, 364, 869, 2013, 4564, 10186, 22463, 49075, 106406, 229272, 491401, 1048441, 2228072, 4718422, 9961283, 20971311, 44039962, 92274436, 192937709, 402652885, 838860476, 1744830114, 3623878279, 7516192363, 15569256014, 32212254256, 66571992593, 137438952945, 283467840976, 584115551662, 1202590842251, 2473901161831, 5085241277762, 10445360463132, 21440476740853, 43980465110221, 90159953476772, 184717953465466, 378231999953999, 774056185953315, 1583296743996406, 3236962232171464, 6614661952699289, 13510798882110313, 27584547717643064, 56294995342129926, 114841790497946323, 234187180623264415, 477381560501271146, 972777519512025652, 1981583836043016701, 4035225266123962821, 8214565720323783052, 16717361816799279442, 34011184385901984023, 69175290276410816731, 140656423562035329182, 285924533142498048096, 581072438321850873889, 1180591620717411301345, 2398076729582241707936, 4869940435459321624414, 9887454823508319663899, 20070057552195992155863, 40730410914750689965714, 82641413450218791237196, 167644010141872405083653, 340010386766614455383485, 689465506498968201196916, 1397820478929414983251242, 2833419889721787128214751, 5742397643169488579851411, 11635911013790805806543942, 23574053482485268906767352, 47752569874777852400890793, 96714065569170333976490841, 195845982777569926302397192, 396527668833598369303622326, 802726744224113772004897379, 1624796301562061610805096975, 3288278229351791355200795066, 6653927711158918977582788964, 13462597927228510489527972109, 27234680864278366047780729013, 55088331748199422233011023964, 111414603535684224740921176066, 225305087149939210031640604583, 455561934457019941162877710155, 921027389228322924524948418286, 1861861819085211933448282828432, 3763337719427556035693337636401, 7605903601369376408980219227601, 15370263527767281493147526360432, 31057439705591620336669228526862, 62748704711297355374086808661163
Generating function in Maple syntax:
(2*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{2 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:
(2*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)-2*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)-2 x^{5}+6 x^{4}-14 x^{3}+14 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,-1/2*n^2+1/2*n+1+2^(-1+n)*n-2^(-1+n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ -\frac{n^{2}}{2}+\frac{n}{2}+1+2^{-1+n} n -2^{-1+n} & \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) = 55
a(n+2) = -4*a(n)+4*a(n+1)-1/2*n*(n-5), n >= 6
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) = 55
a \! \left(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{n \left(n -5\right)}{2}, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18867/
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[22,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[17,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[18,x]+F[19,x]+F[21,x]
F[18,x] = 0
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[10,x]+F[17,x]
F[21,x] = F[12,x]*F[13,x]
F[22,x] = F[23,x]+F[46,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[16,x]+F[26,x]
F[26,x] = F[23,x]+F[27,x]
F[27,x] = F[18,x]+F[28,x]+F[40,x]
F[28,x] = F[12,x]*F[29,x]
F[29,x] = F[30,x]+F[33,x]
F[30,x] = F[12,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[12,x]*F[14,x]
F[33,x] = F[34,x]+F[35,x]
F[34,x] = F[18,x]+F[28,x]+F[32,x]
F[35,x] = F[36,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[12,x]*F[39,x]
F[39,x] = F[14,x]+F[37,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[14,x]+F[42,x]
F[42,x] = F[18,x]+F[40,x]+F[43,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[12,x]+F[45,x]
F[45,x] = F[18,x]+F[32,x]+F[43,x]
F[46,x] = F[18,x]+F[47,x]+F[74,x]
F[47,x] = F[12,x]*F[48,x]
F[48,x] = F[20,x]+F[49,x]
F[49,x] = F[50,x]+F[51,x]
F[50,x] = F[18,x]+F[21,x]+F[47,x]
F[51,x] = F[18,x]+F[52,x]+F[66,x]+F[68,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[54,x]+F[59,x]
F[54,x] = F[55,x]+F[57,x]
F[55,x] = F[56,x]
F[56,x] = F[10,x]*F[12,x]
F[57,x] = F[58,x]
F[58,x] = F[12,x]*F[17,x]
F[59,x] = F[60,x]+F[61,x]
F[60,x] = 2*F[18,x]+F[52,x]+F[58,x]
F[61,x] = F[62,x]
F[62,x] = F[12,x]*F[63,x]
F[63,x] = 2*F[18,x]+F[64,x]+F[66,x]
F[64,x] = F[12,x]*F[65,x]
F[65,x] = F[17,x]+F[63,x]
F[66,x] = F[12,x]*F[67,x]
F[67,x] = F[37,x]+F[63,x]
F[68,x] = F[12,x]*F[69,x]
F[69,x] = F[17,x]+F[70,x]
F[70,x] = F[18,x]+F[66,x]+F[68,x]+F[71,x]
F[71,x] = F[12,x]*F[72,x]
F[72,x] = F[55,x]+F[73,x]
F[73,x] = 2*F[18,x]+F[58,x]+F[71,x]
F[74,x] = F[12,x]*F[75,x]
F[75,x] = F[13,x]+F[76,x]
F[76,x] = F[42,x]+F[70,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_{22}\! \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_{17}\! \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_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{46}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{16}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{23}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{18}\! \left(x \right)+F_{28}\! \left(x \right)+F_{40}\! \left(x \right)
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_{33}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{35}\! \left(x \right)
F_{34}\! \left(x \right) = F_{18}\! \left(x \right)+F_{28}\! \left(x \right)+F_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{14}\! \left(x \right)+F_{37}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \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_{18}\! \left(x \right)+F_{40}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{18}\! \left(x \right)+F_{32}\! \left(x \right)+F_{43}\! \left(x \right)
F_{46}\! \left(x \right) = F_{18}\! \left(x \right)+F_{47}\! \left(x \right)+F_{74}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{20}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{51}\! \left(x \right)
F_{50}\! \left(x \right) = F_{18}\! \left(x \right)+F_{21}\! \left(x \right)+F_{47}\! \left(x \right)
F_{51}\! \left(x \right) = F_{18}\! \left(x \right)+F_{52}\! \left(x \right)+F_{66}\! \left(x \right)+F_{68}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{59}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{57}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{61}\! \left(x \right)
F_{60}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{52}\! \left(x \right)+F_{58}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{64}\! \left(x \right)+F_{66}\! \left(x \right)
F_{64}\! \left(x \right) = F_{12}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{17}\! \left(x \right)+F_{63}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{37}\! \left(x \right)+F_{63}\! \left(x \right)
F_{68}\! \left(x \right) = F_{12}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{17}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{18}\! \left(x \right)+F_{66}\! \left(x \right)+F_{68}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{55}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{58}\! \left(x \right)+F_{71}\! \left(x \right)
F_{74}\! \left(x \right) = F_{12}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{13}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{42}\! \left(x \right)+F_{70}\! \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_22(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_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_21(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_10(x) + F_17(x))
Eq(F_21(x), F_12(x)*F_13(x))
Eq(F_22(x), F_23(x) + F_46(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_16(x) + F_26(x))
Eq(F_26(x), F_23(x) + F_27(x))
Eq(F_27(x), F_18(x) + F_28(x) + F_40(x))
Eq(F_28(x), F_12(x)*F_29(x))
Eq(F_29(x), F_30(x) + F_33(x))
Eq(F_30(x), F_12(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_12(x)*F_14(x))
Eq(F_33(x), F_34(x) + F_35(x))
Eq(F_34(x), F_18(x) + F_28(x) + F_32(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_12(x)*F_39(x))
Eq(F_39(x), F_14(x) + F_37(x))
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_14(x) + F_42(x))
Eq(F_42(x), F_18(x) + F_40(x) + F_43(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_12(x) + F_45(x))
Eq(F_45(x), F_18(x) + F_32(x) + F_43(x))
Eq(F_46(x), F_18(x) + F_47(x) + F_74(x))
Eq(F_47(x), F_12(x)*F_48(x))
Eq(F_48(x), F_20(x) + F_49(x))
Eq(F_49(x), F_50(x) + F_51(x))
Eq(F_50(x), F_18(x) + F_21(x) + F_47(x))
Eq(F_51(x), F_18(x) + F_52(x) + F_66(x) + F_68(x))
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_59(x))
Eq(F_54(x), F_55(x) + F_57(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_10(x)*F_12(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_12(x)*F_17(x))
Eq(F_59(x), F_60(x) + F_61(x))
Eq(F_60(x), 2*F_18(x) + F_52(x) + F_58(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_12(x)*F_63(x))
Eq(F_63(x), 2*F_18(x) + F_64(x) + F_66(x))
Eq(F_64(x), F_12(x)*F_65(x))
Eq(F_65(x), F_17(x) + F_63(x))
Eq(F_66(x), F_12(x)*F_67(x))
Eq(F_67(x), F_37(x) + F_63(x))
Eq(F_68(x), F_12(x)*F_69(x))
Eq(F_69(x), F_17(x) + F_70(x))
Eq(F_70(x), F_18(x) + F_66(x) + F_68(x) + F_71(x))
Eq(F_71(x), F_12(x)*F_72(x))
Eq(F_72(x), F_55(x) + F_73(x))
Eq(F_73(x), 2*F_18(x) + F_58(x) + F_71(x))
Eq(F_74(x), F_12(x)*F_75(x))
Eq(F_75(x), F_13(x) + F_76(x))
Eq(F_76(x), F_42(x) + F_70(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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