0132_0321_1302_2301_3021
Counting sequence:
1, 1, 2, 6, 19, 54, 142, 358, 886, 2183, 5396, 13429, 33687, 85161, 216779, 555054, 1428037, 3688248, 9554956, 24812968, 64556476, 168201329, 438737042, 1145388631, 2992186269, 7820684739, 20448896777, 53484062928, 139919406331, 366106384338, 958064202826, 2507415135994, 6562839028402, 17178417595211, 44967045048704, 117711980133289, 308147420515347, 806687331740493, 2111828675360951, 5528626895651298, 14473708414210081, 37891811152212444, 99200350652893432, 259706492027399356, 679913627871166744, 1780023396469824149, 4660134571305751262, 12200336336982319723, 31940786478710987001, 83621847177290198343, 218924403209438720981, 573150658763584189284, 1500526165706430294967, 3928425023605939590486, 10284743275611854264854, 26925793544230554779374, 70492614839081673222382, 184552005937018191184463, 483163312899980352922796, 1264937752618937772765853, 3311649584668862775736911, 8670010280811710175167409, 22698379816614386991208547, 59425126286727689281342806, 155576993278961157818587069, 407305842020940738105950784, 1066340509725430964362327972, 2791715641038491970707156368, 7308806321156324579211385396, 19134703137963041029831486073, 50095302723797917036092042986, 131151204295560947130062580799, 343358308687145398457331572757, 898923718814396196448403881611, 2353412841853085087300823557729, 6161314794938942858279953760184, 16130531519351911073190811657267, 42230279715893125532596029077658, 110560307533880135867204371304866, 289450642696852622754231276292258, 757791620178888413765917840479466, 1983924217084233981284379010958099, 5193981029562656255568932724015512, 13598018868581420236385846224326481, 35600075570136975355515460075436619, 93202207829740247634014242254925269, 244006547894905251154234683195219407, 638817435806618473044104640342489738, 1672445759428236102408908903855759641, 4378519842284661703044281403271805028, 11463113767038892744447253970053683216, 30010821458058304005744117835077296164, 78569350605588594223678374191554304272, 205697230355612628567077554052337810461, 538522340455059591281127386590963510454, 1409869790997186744883450802971561483219, 3691087032511741842583517416825738459377, 9663391306488521181295686236509688930703, 25299086886854786498160710870711398399661, 66233869353877767906900785531640646397388, 173402521174382376409970324036242821045871
Generating function in Maple syntax:
-(x^7-4*x^6+7*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{x^{7}-4 x^{6}+7 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:
(-x**7 + 4*x**6 - 7*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)+x^7-4*x^6+7*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)+x^{7}-4 x^{6}+7 x^{5}-14 x^{4}+24 x^{3}-19 x^{2}+7 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/20*(-2*5^(1/2)+10)*(3/2-1/2*5^(1/2))^(-n)+1/20*(2*5^(1/2)+10)*(3/2+1/2*5^(1/2))^(-n)-1/2*n^2-3/2*n+5/4*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(-2 \sqrt{5}+10\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\frac{\left(2 \sqrt{5}+10\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{20}-\frac{n^{2}}{2}-\frac{3 n}{2}+\frac{5 \,2^{n}}{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) = 19
a(5) = 54
a(6) = 142
a(7) = 358
a(n+3) = 2*a(n)-7*a(n+1)+5*a(n+2)-1/2*(n+4)*(-1+n), 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) = 54
a \! \left(6\right) = 142
a \! \left(7\right) = 358
a \! \left(n +3\right) = 2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)-\frac{\left(n +4\right) \left(-1+n \right)}{2}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21759/
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[23,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[11,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[19,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[11,x]+F[18,x]
F[18,x] = F[16,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[4,x]
F[21,x] = F[15,x]+F[19,x]+F[22,x]
F[22,x] = F[11,x]*F[4,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]+F[37,x]
F[26,x] = F[27,x]+F[5,x]
F[27,x] = F[28,x]*F[7,x]
F[28,x] = F[10,x]+F[29,x]
F[29,x] = F[30,x]+F[33,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[1,x]+F[30,x]
F[33,x] = F[15,x]+F[34,x]+F[36,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[11,x]+F[33,x]
F[36,x] = F[29,x]*F[4,x]
F[37,x] = F[28,x]*F[38,x]
F[38,x] = F[39,x]+F[40,x]
F[39,x] = F[10,x]*F[30,x]
F[40,x] = F[41,x]
F[41,x] = F[30,x]*F[4,x]*F[42,x]
F[42,x] = F[32,x]+F[43,x]+F[44,x]
F[43,x] = F[4,x]*F[42,x]
F[44,x] = F[45,x]
F[45,x] = F[32,x]*F[4,x]*F[42,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_{23}\! \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_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{19}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{15}\! \left(x \right)+F_{19}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{37}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{5}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{7}\! \left(x \right)
F_{28}\! \left(x \right) = F_{10}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{33}\! \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_{1}\! \left(x \right)+F_{30}\! \left(x \right)
F_{33}\! \left(x \right) = F_{15}\! \left(x \right)+F_{34}\! \left(x \right)+F_{36}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{11}\! \left(x \right)+F_{33}\! \left(x \right)
F_{36}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{28}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{40}\! \left(x \right)
F_{39}\! \left(x \right) = F_{10}\! \left(x \right) F_{30}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{32}\! \left(x \right)+F_{43}\! \left(x \right)+F_{44}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right) F_{42}\! \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_23(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_11(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_19(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_11(x) + F_18(x))
Eq(F_18(x), F_16(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_4(x))
Eq(F_21(x), F_15(x) + F_19(x) + F_22(x))
Eq(F_22(x), F_11(x)*F_4(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x) + F_37(x))
Eq(F_26(x), F_27(x) + F_5(x))
Eq(F_27(x), F_28(x)*F_7(x))
Eq(F_28(x), F_10(x) + F_29(x))
Eq(F_29(x), F_30(x) + F_33(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_1(x) + F_30(x))
Eq(F_33(x), F_15(x) + F_34(x) + F_36(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_11(x) + F_33(x))
Eq(F_36(x), F_29(x)*F_4(x))
Eq(F_37(x), F_28(x)*F_38(x))
Eq(F_38(x), F_39(x) + F_40(x))
Eq(F_39(x), F_10(x)*F_30(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_30(x)*F_4(x)*F_42(x))
Eq(F_42(x), F_32(x) + F_43(x) + F_44(x))
Eq(F_43(x), F_4(x)*F_42(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_32(x)*F_4(x)*F_42(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": true, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "strategy_class": "RequirementInsertionStrategy"}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], 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