0132_0231_1032_1230_2310
Counting sequence:
1, 1, 2, 6, 19, 55, 145, 358, 846, 1941, 4363, 9664, 21172, 45991, 99225, 212874, 454522, 966505, 2047831, 4325188, 9109296, 19136283, 40107781, 83885806, 175111894, 364904125, 759168675, 1577057928, 3271556716, 6777994831, 14025752113, 28991028754, 59861106162, 123480309201, 254476811695, 523986009484, 1078036790632, 2216203124035, 4552665333021, 9345848835318, 19172734008526, 39307540692133, 80539226733691, 164926744165456, 337550069726244, 690493302242295, 1411772930063305, 2885118511283098, 5893382324878186, 12033055254379321, 24558691718003463, 50102545854495444, 102175416545966752, 208291482765884011, 424464264879667765, 864691128455133694, 1760907454301862342, 3584865303386913165, 7295831396340201811, 14843864371813153048, 30192131901891803356, 61393070120314599583, 124803752873691183201, 253642731013506332706, 515355912559260596194, 1046852726183017052065, 2125987254495025821535, 4316538113248035075868, 8762203435012037015256, 17782661287056007755411, 36081831408175882958413, 73196680484479500809734, 148459396305214471402942, 301050863282939882370421, 610365867910901643867435, 1237260018511847045985504, 2507576602403781608469652, 5081266335567738249933895, 10294758932655826565854201, 20853970388352353263678378, 42236845822786106791293786, 85531501737735014110458633, 173178623659795629276656311, 350588487688242460664787556, 709639456113787325552521744, 1436203873702179459550933435, 2906257670353568535993643365, 5880215186605556305770836238, 11895830065007951079108767926, 24062459513609579093351723101, 48666517794406512056971816963, 98416233123187731854480371624, 198998861315124879190034214732, 402330512767748589342215368431, 813326605810494840608724610705, 1643984372170985005066036964914, 3322631065441960657829249412562, 6714586773083902611052849786225, 13567822830567767812894401490191, 27412944229935460807366206811308, 55380485597470771977887221279816
Generating function in Maple syntax:
-(x^7-x^6-2*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{x^{7}-x^{6}-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:
(-x**7 + x**6 + 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)+x^7-x^6-2*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)+x^{7}-x^{6}-2 x^{5}+6 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 = 2,2,1/16*(7*n-1)*2^n-1/2*n^2-1/2*n+2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(7 n -1\right) 2^{n}}{16}-\frac{n^{2}}{2}-\frac{n}{2}+2 & \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(6) = 145
a(7) = 358
a(n+2) = -1/2*n^2-4*a(n)+4*a(n+1)+3/2*n+3, 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) = 55
a \! \left(6\right) = 145
a \! \left(7\right) = 358
a \! \left(n +2\right) = -\frac{n^{2}}{2}-4 a \! \left(n \right)+4 a \! \left(n +1\right)+\frac{3 n}{2}+3, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18868/
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[35,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[14,x]+F[27,x]
F[27,x] = F[18,x]+F[28,x]+F[32,x]
F[28,x] = F[12,x]*F[29,x]
F[29,x] = F[12,x]+F[30,x]
F[30,x] = F[18,x]+F[28,x]+F[31,x]
F[31,x] = F[12,x]*F[14,x]
F[32,x] = F[12,x]*F[33,x]
F[33,x] = F[14,x]+F[34,x]
F[34,x] = F[32,x]
F[35,x] = F[18,x]+F[36,x]+F[53,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[20,x]+F[38,x]
F[38,x] = F[17,x]+F[39,x]
F[39,x] = F[18,x]+F[40,x]+F[46,x]+F[51,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[42,x]+F[44,x]
F[42,x] = F[43,x]
F[43,x] = F[10,x]*F[12,x]
F[44,x] = 2*F[18,x]+F[40,x]+F[45,x]
F[45,x] = F[12,x]*F[17,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[17,x]+F[48,x]
F[48,x] = 2*F[18,x]+F[46,x]+F[49,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[34,x]+F[48,x]
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[27,x]+F[39,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[55,x]+F[58,x]
F[55,x] = F[23,x]+F[56,x]
F[56,x] = F[18,x]+F[36,x]+F[57,x]
F[57,x] = F[12,x]*F[55,x]
F[58,x] = F[59,x]+F[67,x]
F[59,x] = F[18,x]+F[32,x]+F[60,x]
F[60,x] = F[12,x]*F[61,x]
F[61,x] = F[62,x]+F[64,x]
F[62,x] = F[12,x]+F[63,x]
F[63,x] = F[31,x]
F[64,x] = F[30,x]+F[65,x]
F[65,x] = F[66,x]
F[66,x] = F[12,x]*F[34,x]
F[67,x] = F[18,x]+F[46,x]+F[68,x]+F[75,x]
F[68,x] = F[12,x]*F[69,x]
F[69,x] = F[70,x]+F[72,x]
F[70,x] = F[42,x]+F[71,x]
F[71,x] = F[45,x]
F[72,x] = F[44,x]+F[73,x]
F[73,x] = F[74,x]
F[74,x] = F[12,x]*F[48,x]
F[75,x] = F[12,x]*F[58,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_{35}\! \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_{14}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{18}\! \left(x \right)+F_{28}\! \left(x \right)+F_{32}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{18}\! \left(x \right)+F_{28}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{14}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{18}\! \left(x \right)+F_{36}\! \left(x \right)+F_{53}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{20}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{17}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{18}\! \left(x \right)+F_{40}\! \left(x \right)+F_{46}\! \left(x \right)+F_{51}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{44}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{44}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{40}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{17}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{46}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{34}\! \left(x \right)+F_{48}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{27}\! \left(x \right)+F_{39}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{58}\! \left(x \right)
F_{55}\! \left(x \right) = F_{23}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{18}\! \left(x \right)+F_{36}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{67}\! \left(x \right)
F_{59}\! \left(x \right) = F_{18}\! \left(x \right)+F_{32}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{64}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{31}\! \left(x \right)
F_{64}\! \left(x \right) = F_{30}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{67}\! \left(x \right) = F_{18}\! \left(x \right)+F_{46}\! \left(x \right)+F_{68}\! \left(x \right)+F_{75}\! \left(x \right)
F_{68}\! \left(x \right) = F_{12}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)+F_{72}\! \left(x \right)
F_{70}\! \left(x \right) = F_{42}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{45}\! \left(x \right)
F_{72}\! \left(x \right) = F_{44}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{75}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \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_35(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_14(x) + F_27(x))
Eq(F_27(x), F_18(x) + F_28(x) + F_32(x))
Eq(F_28(x), F_12(x)*F_29(x))
Eq(F_29(x), F_12(x) + F_30(x))
Eq(F_30(x), F_18(x) + F_28(x) + F_31(x))
Eq(F_31(x), F_12(x)*F_14(x))
Eq(F_32(x), F_12(x)*F_33(x))
Eq(F_33(x), F_14(x) + F_34(x))
Eq(F_34(x), F_32(x))
Eq(F_35(x), F_18(x) + F_36(x) + F_53(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_20(x) + F_38(x))
Eq(F_38(x), F_17(x) + F_39(x))
Eq(F_39(x), F_18(x) + F_40(x) + F_46(x) + F_51(x))
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_44(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_10(x)*F_12(x))
Eq(F_44(x), 2*F_18(x) + F_40(x) + F_45(x))
Eq(F_45(x), F_12(x)*F_17(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_17(x) + F_48(x))
Eq(F_48(x), 2*F_18(x) + F_46(x) + F_49(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_34(x) + F_48(x))
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_27(x) + F_39(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_58(x))
Eq(F_55(x), F_23(x) + F_56(x))
Eq(F_56(x), F_18(x) + F_36(x) + F_57(x))
Eq(F_57(x), F_12(x)*F_55(x))
Eq(F_58(x), F_59(x) + F_67(x))
Eq(F_59(x), F_18(x) + F_32(x) + F_60(x))
Eq(F_60(x), F_12(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_64(x))
Eq(F_62(x), F_12(x) + F_63(x))
Eq(F_63(x), F_31(x))
Eq(F_64(x), F_30(x) + F_65(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_12(x)*F_34(x))
Eq(F_67(x), F_18(x) + F_46(x) + F_68(x) + F_75(x))
Eq(F_68(x), F_12(x)*F_69(x))
Eq(F_69(x), F_70(x) + F_72(x))
Eq(F_70(x), F_42(x) + F_71(x))
Eq(F_71(x), F_45(x))
Eq(F_72(x), F_44(x) + F_73(x))
Eq(F_73(x), F_74(x))
Eq(F_74(x), F_12(x)*F_48(x))
Eq(F_75(x), F_12(x)*F_58(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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