0132_0231_1203_2013_2130
Counting sequence:
1, 1, 2, 6, 19, 55, 148, 379, 938, 2268, 5400, 12732, 29840, 69688, 162415, 378087, 879589, 2045584, 4756358, 11058329, 25708843, 59767389, 138944161, 323008110, 750905191, 1745645562, 4058136739, 9434026910, 21931455766, 50984453521, 118524479624, 275535991686, 640543473857, 1489082922815, 3461697817906, 8047471087963, 18708100557472, 43491057321572, 101104441945542, 235039311759428, 546400108717950, 1270226144590456, 2952917528105597, 6864700403865351, 15958492299987699, 37098993620351280, 86244696664958028, 200494595054174449, 466093385452974833, 1083535662915550973, 2518914812894896175, 5855766498306581062, 13613005532045522685, 31646398412418324258, 73568950671470531945, 171027060721620494862, 397589679234338771244, 924283866931245883729, 2148699303046680632878, 4995119854511888933334, 11612244824373551450543, 26995194067143557152119, 62756212407195457524656, 145890493911672809758155, 339154250987771071417046, 788437977547163052301212, 1832895924577619823871444, 4260966069626304438474780, 9905544337270836720030636, 23027596797137521107063976, 53532667786497194319657899, 124448354102487377464931191, 289307323531605264862598977, 672557930176338233977652160, 1563507497568291549672751986, 3634713955883803445925615513, 8449684802691165472409062327, 19643133993874181075048778317, 45664746332124015726253899037, 106157655811314850501073278958, 246786608763570701125760896275, 573709261000206418101390112186, 1333712221284792702553721908367, 3100504750617535972584146485646, 7207799070283228930746385844354, 16756099930899407549624586565505, 38953206402749300759965134592308, 90555218416732316111392616592694, 210515342375597754363872167265053, 489388796696077931628796403311903, 1137690923753770602270037492111950, 2644810520244753697916391837094667, 6148438509922797820837896930493580, 14293385413032656668950944609529028, 33228089739497128063093431804824450, 77245936902348868672216303126043012, 179575016641085006559412990378147674, 417461265858054410396899796687324424, 970479701194342086744089712431868473, 2256091588508002445997882534299256439, 5244776628993377574902367974721513743
Generating function in Maple syntax:
(x^7-4*x^6+8*x^5-15*x^4+19*x^3-15*x^2+6*x-1)/(x^3-2*x^2+3*x-1)/(x-1)^4
Generating function in latex syntax:
\frac{x^{7}-4 x^{6}+8 x^{5}-15 x^{4}+19 x^{3}-15 x^{2}+6 x -1}{\left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(x**7 - 4*x**6 + 8*x**5 - 15*x**4 + 19*x**3 - 15*x**2 + 6*x - 1)/((x - 1)**4*(x**3 - 2*x**2 + 3*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^3-2*x^2+3*x-1)*(x-1)^4*F(x)-x^7+4*x^6-8*x^5+15*x^4-19*x^3+15*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{4} F \! \left(x \right)-x^{7}+4 x^{6}-8 x^{5}+15 x^{4}-19 x^{3}+15 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/13800*(-92*2^(1/3)*((17/46*23^(1/2)+I)*3^(1/2)+51/46*I*23^(1/2)+1)*(11+3*23^(1/2)*3^(1/2))^(2/3)+9200-805*((13/161*23^(1/2)+I)*3^(1/2)-39/161*I*23^(1/2)-1)*2^(2/3)*(11+3*23^(1/2)*3^(1/2))^(1/3))*(11/600*2^(1/3)*((I-3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)+1)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/13800*(805*((-13/161*23^(1/2)+I)*3^(1/2)-39/161*I*23^(1/2)+1)*2^(2/3)*(11+3*23^(1/2)*3^(1/2))^(1/3)+9200+92*2^(1/3)*((-17/46*23^(1/2)+I)*3^(1/2)+51/46*I*23^(1/2)-1)*(11+3*23^(1/2)*3^(1/2))^(2/3))*(-11/600*2^(1/3)*((I+3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)-1)*(11+3*23^(1/2)*3^(1/2))^(2/3)+1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/13800*((130*2^(2/3)*23^(1/2)*3^(1/2)-1610*2^(2/3))*(11+3*23^(1/2)*3^(1/2))^(1/3)+9200+(68*2^(1/3)*3^(1/2)*23^(1/2)+184*2^(1/3))*(11+3*23^(1/2)*3^(1/2))^(2/3))*(1/300*2^(1/3)*(3*23^(1/2)*3^(1/2)-11)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/6*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)-1/6*n^3-5/6*n-2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(-92 \,2^{\frac{1}{3}} \left(\left(\frac{17 \sqrt{23}}{46}+\mathrm{I}\right) \sqrt{3}+\frac{51 \,\mathrm{I} \sqrt{23}}{46}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+9200-805 \left(\left(\frac{13 \sqrt{23}}{161}+\mathrm{I}\right) \sqrt{3}-\frac{39 \,\mathrm{I} \sqrt{23}}{161}-1\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{11 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}-\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{13800}\\+\\\frac{\left(805 \left(\left(-\frac{13 \sqrt{23}}{161}+\mathrm{I}\right) \sqrt{3}-\frac{39 \,\mathrm{I} \sqrt{23}}{161}+1\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+9200+92 \,2^{\frac{1}{3}} \left(\left(-\frac{17 \sqrt{23}}{46}+\mathrm{I}\right) \sqrt{3}+\frac{51 \,\mathrm{I} \sqrt{23}}{46}-1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{11 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}+\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}-1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}+\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{13800}\\+\\\frac{\left(\left(130 \,2^{\frac{2}{3}} \sqrt{23}\, \sqrt{3}-1610 \,2^{\frac{2}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+9200+\left(68 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{23}+184 \,2^{\frac{1}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{2^{\frac{1}{3}} \left(3 \sqrt{23}\, \sqrt{3}-11\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{300}-\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{2}{3}\right)^{-n}}{13800}\\-\frac{n^{3}}{6}-\frac{5 n}{6}-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) = 148
a(7) = 379
a(n+3) = a(n)-2*a(n+1)+3*a(n+2)+1/6*(n+2)*(n^2+n+6), 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) = 148
a \! \left(7\right) = 379
a \! \left(n +3\right) = a \! \left(n \right)-2 a \! \left(n +1\right)+3 a \! \left(n +2\right)+\frac{\left(n +2\right) \left(n^{2}+n +6\right)}{6}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18969/
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[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[19,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[7,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[11,x]
F[18,x] = F[14,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[21,x]+F[22,x]+F[45,x]
F[21,x] = 0
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[30,x]
F[24,x] = F[25,x]+F[7,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[28,x]+F[29,x]
F[28,x] = F[7,x]
F[29,x] = F[25,x]
F[30,x] = F[31,x]+F[36,x]
F[31,x] = F[32,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]+F[35,x]
F[34,x] = F[7,x]
F[35,x] = F[31,x]
F[36,x] = F[37,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]+F[40,x]
F[39,x] = F[25,x]
F[40,x] = F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[43,x]+F[44,x]
F[43,x] = F[25,x]
F[44,x] = F[41,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[50,x]
F[47,x] = F[2,x]+F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[4,x]*F[47,x]
F[50,x] = F[20,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_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{19}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{45}\! \left(x \right)
F_{21}\! \left(x \right) = 0
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{30}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{7}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)
F_{28}\! \left(x \right) = F_{7}\! \left(x \right)
F_{29}\! \left(x \right) = F_{25}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{36}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{35}\! \left(x \right)
F_{34}\! \left(x \right) = F_{7}\! \left(x \right)
F_{35}\! \left(x \right) = F_{31}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{40}\! \left(x \right)
F_{39}\! \left(x \right) = F_{25}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{44}\! \left(x \right)
F_{43}\! \left(x \right) = F_{25}\! \left(x \right)
F_{44}\! \left(x \right) = F_{41}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{50}\! \left(x \right)
F_{47}\! \left(x \right) = F_{2}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{50}\! \left(x \right) = F_{20}\! \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_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_19(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_7(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_11(x))
Eq(F_18(x), F_14(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_21(x) + F_22(x) + F_45(x))
Eq(F_21(x), 0)
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_30(x))
Eq(F_24(x), F_25(x) + F_7(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_28(x) + F_29(x))
Eq(F_28(x), F_7(x))
Eq(F_29(x), F_25(x))
Eq(F_30(x), F_31(x) + F_36(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_35(x))
Eq(F_34(x), F_7(x))
Eq(F_35(x), F_31(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x) + F_40(x))
Eq(F_39(x), F_25(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_44(x))
Eq(F_43(x), F_25(x))
Eq(F_44(x), F_41(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_50(x))
Eq(F_47(x), F_2(x) + F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_4(x)*F_47(x))
Eq(F_50(x), F_20(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": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "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": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 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