01324_01342_03124_03142_10324_10342_13024_13042_13204_13240_13402_13420_30124_30142_31024_31042_31204_31240_31402_31420
Counting sequence:
1, 1, 2, 6, 24, 100, 408, 1624, 6336, 24336, 92320, 346720, 1291392, 4776512, 17562496, 64245120, 233969664, 848748800, 3068269056, 11057710592, 39740405760, 142466343936, 509572929536, 1818872207360, 6480018948096, 23046008934400, 81831356112896, 290135484555264, 1027272821342208, 3632590541701120, 12830110235197440, 45264894250811392, 159529306684391424, 561687970055258112, 1975835493826428928, 6944333879453286400, 24386851453244276736, 85574750057890316288, 300066971494183862272, 1051453058529759854592, 3681933633467669544960, 12885180440821539799040, 45065751907718349717504, 157527532346679675060224, 550340373138344364736512, 1921683646933368852971520, 6706839222616064931135488, 23396376683087933525196800, 81579805870762693767462912, 284333606278466443122049024, 990587402851420346305740800, 3449718484441885274183565312, 12009018295479087582236639232, 41789840695503854942625464320, 145371505794113570897985011712, 505520651232729328550422773760, 1757336471925708539196742828032, 6107063480674883457162745479168, 21216622803432111846581931606016, 73687013759839644942346737418240, 255847323813170223764952223580160, 888076026240783146989342964056064, 3081787462606566572306919313113088, 10691588302008234639138343851393024, 37082784288534395123738043079458816, 128586803564852400776090804761395200, 445777005729836791670951400871821312, 1545038169950163701269050425612435456, 5353822964888074608655876091327545344, 18547905197519301907496859413275213824, 64244284978676195140478354685962485760, 222477190637432061907307464980986593280, 770283016826689124534218872375736270848, 2666437260733613672745267916708457218048, 9228495659619335101232771907104867876864, 31933839568959705099875089020757632614400, 110482667463722001040132685201796073783296, 382174729841556813252302980046059995136000, 1321771939922892426408840473036261135745024, 4570668243695716342467286790514595833315328, 15802772158717903219437801803321319578664960, 54628464515228950475688914264674578154389504, 188815877086913431190596504807058722451881984, 652519407955432026152129434854060946452643840, 2254682065513979180138376803712821094739083264, 7789608540332892015211214152475591075472343040, 26908274031322811212593862083985046962431852544, 92938856860326588174132124742357447920022716416, 320960382639426836664005675483916816940145836032, 1108279874248716341170059845758043964217732628480, 3826419954841128162016103803927794353809580359680, 13209372848544221506655195825034714825484232818688, 45595220148952962924984425351914867687405394067456, 157363904001202458494348907794437178793640851406848, 543051114787903440913521473302044336393528333565952, 1873816869268248878194323652858110138971959014195200, 6464954241811349691493804580127329773853737488482304, 22302658769727465188702911525511396078512609555382272, 76931050770733095967202293154066158029003357566271488, 265339231163224063514683693392906178487546633181462528, 915075557239524633026695050149599285481092717058457600
Generating function in Maple syntax:
(4*x^4-6*x^3+14*x^2-7*x+1)/(2*x^2-4*x+1)^2
Generating function in latex syntax:
\frac{4 x^{4}-6 x^{3}+14 x^{2}-7 x +1}{\left(2 x^{2}-4 x +1\right)^{2}}
Generating function in sympy syntax:
(4*x**4 - 6*x**3 + 14*x**2 - 7*x + 1)/(2*x**2 - 4*x + 1)**2
Implicit equation for the generating function in Maple syntax:
(2*x^2-4*x+1)^2*F(x)-4*x^4+6*x^3-14*x^2+7*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x^{2}-4 x +1\right)^{2} F \! \left(x \right)-4 x^{4}+6 x^{3}-14 x^{2}+7 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,-1/4*n*(2*(1-1/2*2^(1/2))^(-n)*2^(1/2)-2*(1+1/2*2^(1/2))^(-n)*2^(1/2)-3*(1-1/2*2^(1/2))^(-n)-3*(1+1/2*2^(1/2))^(-n)))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ -\frac{n \left(2 \left(1-\frac{\sqrt{2}}{2}\right)^{-n} \sqrt{2}-2 \left(1+\frac{\sqrt{2}}{2}\right)^{-n} \sqrt{2}-3 \left(1-\frac{\sqrt{2}}{2}\right)^{-n}-3 \left(1+\frac{\sqrt{2}}{2}\right)^{-n}\right)}{4} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 24
a(n+4) = -4*a(n)+16*a(n+1)-20*a(n+2)+8*a(n+3), n >= 5
Recurrence in latex format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 24
a{\left(n + 4 \right)} = - 4 a{\left(n \right)} + 16 a{\left(n + 1 \right)} - 20 a{\left(n + 2 \right)} + 8 a{\left(n + 3 \right)}, \quad n \geq 5
Specification 1
Strategy pack name: regular_insertion_encoding_left
Tree: http://permpal.com/tree/25686/
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[15,x]*F[4,x]
F[4,x] = F[24,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[15,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[16,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[15,x]
F[12,x] = F[13,x]+F[14,x]
F[13,x] = F[1,x]+F[10,x]
F[14,x] = F[10,x]
F[15,x] = x
F[16,x] = F[17,x]+F[18,x]+F[22,x]
F[17,x] = 0
F[18,x] = F[15,x]*F[19,x]
F[19,x] = F[20,x]+F[21,x]
F[20,x] = F[16,x]+F[6,x]
F[21,x] = F[16,x]
F[22,x] = F[15,x]*F[23,x]
F[23,x] = F[9,x]
F[24,x] = F[25,x]+F[38,x]
F[25,x] = F[26,x]
F[26,x] = F[15,x]*F[27,x]
F[27,x] = F[13,x]+F[28,x]
F[28,x] = F[25,x]+F[29,x]
F[29,x] = F[17,x]+F[30,x]+F[32,x]
F[30,x] = F[15,x]*F[31,x]
F[31,x] = F[10,x]
F[32,x] = F[15,x]*F[33,x]
F[33,x] = F[34,x]+F[37,x]
F[34,x] = F[15,x]+F[35,x]
F[35,x] = F[17,x]+F[32,x]+F[36,x]
F[36,x] = F[10,x]*F[15,x]
F[37,x] = F[35,x]
F[38,x] = F[17,x]+F[39,x]+F[62,x]
F[39,x] = F[15,x]*F[40,x]
F[40,x] = F[20,x]+F[41,x]
F[41,x] = F[38,x]+F[42,x]
F[42,x] = F[17,x]+F[43,x]+F[45,x]+F[59,x]
F[43,x] = F[15,x]*F[44,x]
F[44,x] = F[16,x]
F[45,x] = F[15,x]*F[46,x]
F[46,x] = F[47,x]+F[58,x]
F[47,x] = F[48,x]+F[54,x]
F[48,x] = F[17,x]+F[49,x]+F[50,x]
F[49,x] = F[15,x]*F[6,x]
F[50,x] = F[15,x]*F[51,x]
F[51,x] = F[52,x]+F[53,x]
F[52,x] = F[15,x]+F[48,x]
F[53,x] = F[35,x]+F[54,x]
F[54,x] = F[17,x]+F[45,x]+F[55,x]+F[56,x]
F[55,x] = F[15,x]*F[16,x]
F[56,x] = F[15,x]*F[57,x]
F[57,x] = F[53,x]
F[58,x] = F[54,x]
F[59,x] = F[15,x]*F[60,x]
F[60,x] = F[61,x]
F[61,x] = F[29,x]+F[42,x]
F[62,x] = F[15,x]*F[63,x]
F[63,x] = F[24,x]+F[61,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_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{24}\! \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_{15}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{16}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{15}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{14}\! \left(x \right) = F_{10}\! \left(x \right)
F_{15}\! \left(x \right) = x
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_{15}\! \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_{16}\! \left(x \right)+F_{6}\! \left(x \right)
F_{21}\! \left(x \right) = F_{16}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{9}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{38}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{15}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{13}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{25}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{17}\! \left(x \right)+F_{30}\! \left(x \right)+F_{32}\! \left(x \right)
F_{30}\! \left(x \right) = F_{15}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{10}\! \left(x \right)
F_{32}\! \left(x \right) = F_{15}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{37}\! \left(x \right)
F_{34}\! \left(x \right) = F_{15}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{17}\! \left(x \right)+F_{32}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{10}\! \left(x \right) F_{15}\! \left(x \right)
F_{37}\! \left(x \right) = F_{35}\! \left(x \right)
F_{38}\! \left(x \right) = F_{17}\! \left(x \right)+F_{39}\! \left(x \right)+F_{62}\! \left(x \right)
F_{39}\! \left(x \right) = F_{15}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{20}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{38}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{17}\! \left(x \right)+F_{43}\! \left(x \right)+F_{45}\! \left(x \right)+F_{59}\! \left(x \right)
F_{43}\! \left(x \right) = F_{15}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{16}\! \left(x \right)
F_{45}\! \left(x \right) = F_{15}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{58}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{54}\! \left(x \right)
F_{48}\! \left(x \right) = F_{17}\! \left(x \right)+F_{49}\! \left(x \right)+F_{50}\! \left(x \right)
F_{49}\! \left(x \right) = F_{15}\! \left(x \right) F_{6}\! \left(x \right)
F_{50}\! \left(x \right) = F_{15}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{53}\! \left(x \right)
F_{52}\! \left(x \right) = F_{15}\! \left(x \right)+F_{48}\! \left(x \right)
F_{53}\! \left(x \right) = F_{35}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{17}\! \left(x \right)+F_{45}\! \left(x \right)+F_{55}\! \left(x \right)+F_{56}\! \left(x \right)
F_{55}\! \left(x \right) = F_{15}\! \left(x \right) F_{16}\! \left(x \right)
F_{56}\! \left(x \right) = F_{15}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{53}\! \left(x \right)
F_{58}\! \left(x \right) = F_{54}\! \left(x \right)
F_{59}\! \left(x \right) = F_{15}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{29}\! \left(x \right)+F_{42}\! \left(x \right)
F_{62}\! \left(x \right) = F_{15}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{24}\! \left(x \right)+F_{61}\! \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_15(x)*F_4(x))
Eq(F_4(x), F_24(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_15(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_16(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_15(x))
Eq(F_12(x), F_13(x) + F_14(x))
Eq(F_13(x), F_1(x) + F_10(x))
Eq(F_14(x), F_10(x))
Eq(F_15(x), x)
Eq(F_16(x), F_17(x) + F_18(x) + F_22(x))
Eq(F_17(x), 0)
Eq(F_18(x), F_15(x)*F_19(x))
Eq(F_19(x), F_20(x) + F_21(x))
Eq(F_20(x), F_16(x) + F_6(x))
Eq(F_21(x), F_16(x))
Eq(F_22(x), F_15(x)*F_23(x))
Eq(F_23(x), F_9(x))
Eq(F_24(x), F_25(x) + F_38(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_15(x)*F_27(x))
Eq(F_27(x), F_13(x) + F_28(x))
Eq(F_28(x), F_25(x) + F_29(x))
Eq(F_29(x), F_17(x) + F_30(x) + F_32(x))
Eq(F_30(x), F_15(x)*F_31(x))
Eq(F_31(x), F_10(x))
Eq(F_32(x), F_15(x)*F_33(x))
Eq(F_33(x), F_34(x) + F_37(x))
Eq(F_34(x), F_15(x) + F_35(x))
Eq(F_35(x), F_17(x) + F_32(x) + F_36(x))
Eq(F_36(x), F_10(x)*F_15(x))
Eq(F_37(x), F_35(x))
Eq(F_38(x), F_17(x) + F_39(x) + F_62(x))
Eq(F_39(x), F_15(x)*F_40(x))
Eq(F_40(x), F_20(x) + F_41(x))
Eq(F_41(x), F_38(x) + F_42(x))
Eq(F_42(x), F_17(x) + F_43(x) + F_45(x) + F_59(x))
Eq(F_43(x), F_15(x)*F_44(x))
Eq(F_44(x), F_16(x))
Eq(F_45(x), F_15(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_58(x))
Eq(F_47(x), F_48(x) + F_54(x))
Eq(F_48(x), F_17(x) + F_49(x) + F_50(x))
Eq(F_49(x), F_15(x)*F_6(x))
Eq(F_50(x), F_15(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_53(x))
Eq(F_52(x), F_15(x) + F_48(x))
Eq(F_53(x), F_35(x) + F_54(x))
Eq(F_54(x), F_17(x) + F_45(x) + F_55(x) + F_56(x))
Eq(F_55(x), F_15(x)*F_16(x))
Eq(F_56(x), F_15(x)*F_57(x))
Eq(F_57(x), F_53(x))
Eq(F_58(x), F_54(x))
Eq(F_59(x), F_15(x)*F_60(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_29(x) + F_42(x))
Eq(F_62(x), F_15(x)*F_63(x))
Eq(F_63(x), F_24(x) + F_61(x))
Pack JSON:
{"name": "regular_insertion_encoding_left", "initial_strats": [{"class_module": "tilings.strategies.factor", "strategy_class": "FactorFactory", "interleaving": null, "unions": false, "ignore_parent": true, "workable": true, "tracked": false}, {"class_module": "tilings.strategies.requirement_insertion", "strategy_class": "CellInsertionFactory", "ignore_parent": true, "maxreqlen": 1, "extra_basis": [], "one_cell_only": true}], "inferral_strats": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}], "expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "strategy_class": "RowAndColumnPlacementFactory", "partial": false, "ignore_parent": false, "dirs": [0], "place_row": false, "place_col": true}]], "symmetries": [], "iterative": false}
Specification JSON:
{"root": {"class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 4, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 4, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 4, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, 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