0132_0231_0321_1203_3021
Counting sequence:
1, 1, 2, 6, 19, 56, 157, 429, 1157, 3097, 8248, 21883, 57886, 152764, 402410, 1058508, 2781241, 7301522, 19156003, 50231919, 131670707, 345042211, 903980062, 2367946381, 6201956044, 16242115846, 42532779872, 111373000734, 291619776487, 763553437292, 1999174752793, 5234239256193, 13704079886321, 35879074144189, 93935290029460, 245931090911023, 643866572637706, 1685685806870752, 4413225207712358, 11554058535742464, 30249087838467877, 79193479857567446, 207331901490047647, 542803324124202531, 1421080269905814719, 3720441883639751911, 9740254177106462362, 25500338239865678689, 66760795726862661592, 174782119309466482762, 457585702939025140988, 1197975270982585649778, 3136340672958685228531, 8211047873793376877264, 21496805200221259087285, 56279372230470027753813, 147341320498388078913821, 385744607279092718468257, 1009892537367687095453488, 2643933076881562605818659, 6921906837392188797856822, 18121787723525379939461956, 47443456909644703323950882, 124208584158330234639235956, 325182297871189009807449169, 851338314066922813210497626, 2228832653552951466678817627, 5835159665038675660535504919, 15276646378456563662346798347, 39994779544117991621343094507, 104707692401471363791358895958, 274128297955444004932086417013, 717677202055156461363606004516, 1878903309390616999876142897614, 4919032728477877779699645292760, 12878194880765382822092438191878, 33715551923263003652316959707711, 88268460907913094066337021783412, 231089830838255210409651267349393, 605001031682410400888531103681129, 1583913264360091719707770690529417, 4146738761700096213138438261580741, 10856303021344659829514858681562892, 28422170303542809095020766957811031, 74410207891701619094776700541279394, 194808453376397751467767851364848616, 510015152247163041865443886950912542, 1335237003384434187242397876283184424, 3495695857944825146089417875489234877, 9151850570527412503481192017365711902, 23959855853792154869264830710970287703, 62727716991158537114134645184269928523, 164223295120302426492781794979289056151, 429942168370986682403496120028496360239, 1125603209995133500796277325655998269010, 2946867461619365580142477378039094939593, 7714999174872866759945437850660479539376, 20198130063019041740322402258340729661842, 52879391014223872542278901093158481416948, 138440042979731804049028565358728258534874, 362440737925129995929835323658213382083787
Generating function in Maple syntax:
(3*x^6-5*x^5+14*x^4-24*x^3+19*x^2-7*x+1)/(2*x-1)/(x^2-3*x+1)/(x-1)^3
Generating function in latex syntax:
\frac{3 x^{6}-5 x^{5}+14 x^{4}-24 x^{3}+19 x^{2}-7 x +1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(3*x**6 - 5*x**5 + 14*x**4 - 24*x**3 + 19*x**2 - 7*x + 1)/((x - 1)**3*(2*x - 1)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^2-3*x+1)*(x-1)^3*F(x)-3*x^6+5*x^5-14*x^4+24*x^3-19*x^2+7*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{3} F \! \left(x \right)-3 x^{6}+5 x^{5}-14 x^{4}+24 x^{3}-19 x^{2}+7 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/10*(16*5^(1/2)-30)*(3/2-1/2*5^(1/2))^(-n)+1/10*(-16*5^(1/2)-30)*(3/2+1/2*5^(1/2))^(-n)+1/2*n^2-7/2*n-2^(n-1)+6)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(16 \sqrt{5}-30\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{10}+\frac{\left(-16 \sqrt{5}-30\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{10}+\frac{n^{2}}{2}-\frac{7 n}{2}-\\2^{n -1}+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) = 19
a(5) = 56
a(6) = 157
a(n+3) = 1/2*n^2+2*a(n)-7*a(n+1)+5*a(n+2)-7/2*n+4, 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) = 19
a \! \left(5\right) = 56
a \! \left(6\right) = 157
a \! \left(n +3\right) = \frac{n^{2}}{2}+2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)-\frac{7 n}{2}+4, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18669/
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[18,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[12,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[17,x]
F[15,x] = 0
F[16,x] = F[10,x]*F[12,x]
F[17,x] = F[12,x]*F[13,x]
F[18,x] = F[19,x]+F[2,x]
F[19,x] = F[15,x]+F[20,x]+F[49,x]
F[20,x] = F[12,x]*F[21,x]
F[21,x] = F[22,x]+F[29,x]
F[22,x] = F[23,x]+F[6,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[12,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[12,x]*F[26,x]
F[29,x] = F[19,x]+F[30,x]
F[30,x] = 2*F[15,x]+F[31,x]+F[35,x]
F[31,x] = F[12,x]*F[32,x]
F[32,x] = F[33,x]+F[34,x]
F[33,x] = F[23,x]
F[34,x] = F[30,x]
F[35,x] = F[12,x]*F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[38,x]+F[43,x]
F[38,x] = F[39,x]
F[39,x] = F[12,x]*F[40,x]
F[40,x] = F[41,x]+F[42,x]
F[41,x] = F[12,x]
F[42,x] = F[38,x]
F[43,x] = 2*F[15,x]+F[44,x]+F[48,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[46,x]+F[47,x]
F[46,x] = F[27,x]
F[47,x] = F[43,x]
F[48,x] = F[12,x]*F[37,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[51,x]+F[64,x]
F[51,x] = F[2,x]+F[52,x]
F[52,x] = F[15,x]+F[53,x]+F[63,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[55,x]+F[57,x]
F[55,x] = F[10,x]+F[56,x]
F[56,x] = F[28,x]
F[57,x] = F[52,x]+F[58,x]
F[58,x] = 2*F[15,x]+F[48,x]+F[59,x]
F[59,x] = F[12,x]*F[60,x]
F[60,x] = F[61,x]+F[62,x]
F[61,x] = F[56,x]
F[62,x] = F[58,x]
F[63,x] = F[12,x]*F[51,x]
F[64,x] = F[38,x]+F[65,x]
F[65,x] = 2*F[15,x]+F[44,x]+F[66,x]
F[66,x] = F[12,x]*F[64,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_{18}\! \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_{12}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{2}\! \left(x \right)
F_{19}\! \left(x \right) = F_{15}\! \left(x \right)+F_{20}\! \left(x \right)+F_{49}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{29}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{6}\! \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_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{26}\! \left(x \right)
F_{29}\! \left(x \right) = F_{19}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{31}\! \left(x \right)+F_{35}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{34}\! \left(x \right)
F_{33}\! \left(x \right) = F_{23}\! \left(x \right)
F_{34}\! \left(x \right) = F_{30}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \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_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{42}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right)
F_{42}\! \left(x \right) = F_{38}\! \left(x \right)
F_{43}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{44}\! \left(x \right)+F_{48}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{47}\! \left(x \right)
F_{46}\! \left(x \right) = F_{27}\! \left(x \right)
F_{47}\! \left(x \right) = F_{43}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{64}\! \left(x \right)
F_{51}\! \left(x \right) = F_{2}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{15}\! \left(x \right)+F_{53}\! \left(x \right)+F_{63}\! \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_{57}\! \left(x \right)
F_{55}\! \left(x \right) = F_{10}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{28}\! \left(x \right)
F_{57}\! \left(x \right) = F_{52}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{48}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{12}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{62}\! \left(x \right)
F_{61}\! \left(x \right) = F_{56}\! \left(x \right)
F_{62}\! \left(x \right) = F_{58}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{64}\! \left(x \right) = F_{38}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{44}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \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_18(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_12(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_17(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_10(x)*F_12(x))
Eq(F_17(x), F_12(x)*F_13(x))
Eq(F_18(x), F_19(x) + F_2(x))
Eq(F_19(x), F_15(x) + F_20(x) + F_49(x))
Eq(F_20(x), F_12(x)*F_21(x))
Eq(F_21(x), F_22(x) + F_29(x))
Eq(F_22(x), F_23(x) + F_6(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_12(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_12(x)*F_26(x))
Eq(F_29(x), F_19(x) + F_30(x))
Eq(F_30(x), 2*F_15(x) + F_31(x) + F_35(x))
Eq(F_31(x), F_12(x)*F_32(x))
Eq(F_32(x), F_33(x) + F_34(x))
Eq(F_33(x), F_23(x))
Eq(F_34(x), F_30(x))
Eq(F_35(x), F_12(x)*F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_38(x) + F_43(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_12(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_42(x))
Eq(F_41(x), F_12(x))
Eq(F_42(x), F_38(x))
Eq(F_43(x), 2*F_15(x) + F_44(x) + F_48(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_47(x))
Eq(F_46(x), F_27(x))
Eq(F_47(x), F_43(x))
Eq(F_48(x), F_12(x)*F_37(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_64(x))
Eq(F_51(x), F_2(x) + F_52(x))
Eq(F_52(x), F_15(x) + F_53(x) + F_63(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_57(x))
Eq(F_55(x), F_10(x) + F_56(x))
Eq(F_56(x), F_28(x))
Eq(F_57(x), F_52(x) + F_58(x))
Eq(F_58(x), 2*F_15(x) + F_48(x) + F_59(x))
Eq(F_59(x), F_12(x)*F_60(x))
Eq(F_60(x), F_61(x) + F_62(x))
Eq(F_61(x), F_56(x))
Eq(F_62(x), F_58(x))
Eq(F_63(x), F_12(x)*F_51(x))
Eq(F_64(x), F_38(x) + F_65(x))
Eq(F_65(x), 2*F_15(x) + F_44(x) + F_66(x))
Eq(F_66(x), F_12(x)*F_64(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:
{"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": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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, 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": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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