0132_1203_1230_2130_3102
Counting sequence:
1, 1, 2, 6, 19, 51, 118, 247, 488, 935, 1769, 3342, 6344, 12134, 23398, 45458, 88879, 174665, 344622, 682037, 1352960, 2688645, 5350233, 10657796, 21247904, 42387916, 84603198, 168929352, 337413103, 674108295, 1347058534, 2692247347, 5381474024, 10758065699, 21508237433, 43003708698, 85986768584, 171940134770, 343826232262, 687565040222, 1374988635775, 2749748421125, 5499126567438, 10997654031761, 21994338709664, 43987108988481, 87971680220553, 175939254284432, 351871864688768, 703732979376280, 1407448564909374, 2814868986015252, 5629692434427679, 11259311187495939, 22518503156079718, 45036813411941263, 90073314704809064, 180146124390386975, 360291431642533577, 720581541127663782, 1441160942959756040, 2882318424466613630, 5764631248184838118, 11529253434168474026, 23058492205387446799, 46116960685624285121, 92233882983148560558, 184467703853046608237, 368935307204742804608, 737870451794884800765, 1475740640473818503289, 2951480855217085227932, 5902961021587667713184, 11805920928598281045604, 23611840053773005114878, 47223677189545399026528, 94447349657666630034319, 188894691675908481014655, 377789370990968015131942, 755578721981662304495467, 1511157411602201936515880, 3022314770843007474987035, 6044629456963495879661273, 12089258776843076291180466, 24178517331879718044130184, 48357034304869086082120586, 96714068029041387620112838, 193428135118495640690209142, 386856268716707362286534623, 773712534973543670929440125, 1547425065966932369121648654, 3094850125493838711862728137, 6189700240567496424607957664, 12379400464274785823718160569, 24758800901269183622821391913, 49517601758397772195530435368, 99035203445374561316333940224, 198070406775187544507828960560, 396140813363392527816092441982, 792281626424240916307780868652, 1584563252358955132091592639295
Generating function in Maple syntax:
-(2*x^9-4*x^8-5*x^7+2*x^6-2*x^4+13*x^3-14*x^2+6*x-1)/(2*x-1)/(x^2+x-1)/(x-1)^4
Generating function in latex syntax:
-\frac{2 x^{9}-4 x^{8}-5 x^{7}+2 x^{6}-2 x^{4}+13 x^{3}-14 x^{2}+6 x -1}{\left(2 x -1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(-2*x**9 + 4*x**8 + 5*x**7 - 2*x**6 + 2*x**4 - 13*x**3 + 14*x**2 - 6*x + 1)/((x - 1)**4*(2*x - 1)*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^2+x-1)*(x-1)^4*F(x)+2*x^9-4*x^8-5*x^7+2*x^6-2*x^4+13*x^3-14*x^2+6*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+2 x^{9}-4 x^{8}-5 x^{7}+2 x^{6}-2 x^{4}+13 x^{3}-14 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/4*(-2*5^(1/2)+6)*(-1/2*5^(1/2)-1/2)^(-n)+1/4*(2*5^(1/2)+6)*(1/2*5^(1/2)-1/2)^(-n)+1/2*n^3-7/2*n^2+2*n+5/4*2^n-3)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(-2 \sqrt{5}+6\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{4}+\frac{\left(2 \sqrt{5}+6\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{4}+\frac{n^{3}}{2}-\frac{7 n^{2}}{2}+\\2 n +\frac{5 \,2^{n}}{4}-3 & \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) = 51
a(6) = 118
a(7) = 247
a(8) = 488
a(9) = 935
a(n+3) = 1/2*n^3-13/2*n^2+3*a(n+2)-2*a(n)-a(n+1)+13*n+2, n >= 10
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) = 51
a \! \left(6\right) = 118
a \! \left(7\right) = 247
a \! \left(8\right) = 488
a \! \left(9\right) = 935
a \! \left(n +3\right) = \frac{n^{3}}{2}-\frac{13 n^{2}}{2}+3 a \! \left(n +2\right)-2 a \! \left(n \right)-a \! \left(n +1\right)+13 n +2, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19861/
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[29,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[21,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[10,x]+F[17,x]
F[17,x] = F[14,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[14,x]
F[21,x] = F[19,x]+F[22,x]+F[23,x]
F[22,x] = 0
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[28,x]
F[25,x] = F[11,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[25,x]*F[4,x]
F[28,x] = F[23,x]
F[29,x] = F[30,x]+F[40,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[34,x]
F[33,x] = F[1,x]+F[14,x]
F[34,x] = F[30,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[39,x]
F[38,x] = F[30,x]
F[39,x] = F[35,x]
F[40,x] = F[22,x]+F[41,x]+F[60,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[43,x]+F[54,x]
F[43,x] = F[44,x]+F[7,x]
F[44,x] = F[22,x]+F[23,x]+F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[20,x]+F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[22,x]+F[49,x]+F[51,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[25,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[20,x]+F[53,x]
F[53,x] = F[48,x]
F[54,x] = F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[57,x]+F[59,x]
F[57,x] = F[58,x]+F[7,x]
F[58,x] = F[23,x]
F[59,x] = F[55,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[38,x]+F[62,x]
F[62,x] = F[63,x]
F[63,x] = F[22,x]+F[64,x]+F[67,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[66,x]
F[66,x] = F[14,x]+F[48,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[38,x]+F[69,x]
F[69,x] = F[63,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_{29}\! \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_{21}\! \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_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{14}\! \left(x \right)
F_{21}\! \left(x \right) = F_{19}\! \left(x \right)+F_{22}\! \left(x \right)+F_{23}\! \left(x \right)
F_{22}\! \left(x \right) = 0
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = F_{11}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{23}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{40}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{34}\! \left(x \right)
F_{33}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{34}\! \left(x \right) = F_{30}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{39}\! \left(x \right)
F_{38}\! \left(x \right) = F_{30}\! \left(x \right)
F_{39}\! \left(x \right) = F_{35}\! \left(x \right)
F_{40}\! \left(x \right) = F_{22}\! \left(x \right)+F_{41}\! \left(x \right)+F_{60}\! \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_{54}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{7}\! \left(x \right)
F_{44}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{20}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{22}\! \left(x \right)+F_{49}\! \left(x \right)+F_{51}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{25}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{20}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{48}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{59}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{7}\! \left(x \right)
F_{58}\! \left(x \right) = F_{23}\! \left(x \right)
F_{59}\! \left(x \right) = F_{55}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{38}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{22}\! \left(x \right)+F_{64}\! \left(x \right)+F_{67}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{14}\! \left(x \right)+F_{48}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{38}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{63}\! \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_29(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_21(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_10(x) + F_17(x))
Eq(F_17(x), F_14(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_14(x))
Eq(F_21(x), F_19(x) + F_22(x) + F_23(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_28(x))
Eq(F_25(x), F_11(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_25(x)*F_4(x))
Eq(F_28(x), F_23(x))
Eq(F_29(x), F_30(x) + F_40(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_34(x))
Eq(F_33(x), F_1(x) + F_14(x))
Eq(F_34(x), F_30(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_39(x))
Eq(F_38(x), F_30(x))
Eq(F_39(x), F_35(x))
Eq(F_40(x), F_22(x) + F_41(x) + F_60(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_54(x))
Eq(F_43(x), F_44(x) + F_7(x))
Eq(F_44(x), F_22(x) + F_23(x) + F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_20(x) + F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_22(x) + F_49(x) + F_51(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_25(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_20(x) + F_53(x))
Eq(F_53(x), F_48(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_57(x) + F_59(x))
Eq(F_57(x), F_58(x) + F_7(x))
Eq(F_58(x), F_23(x))
Eq(F_59(x), F_55(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_38(x) + F_62(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_22(x) + F_64(x) + F_67(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_14(x) + F_48(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_38(x) + F_69(x))
Eq(F_69(x), F_63(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": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "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": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 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