0132_0213_1320_3021
Counting sequence:
1, 1, 2, 6, 20, 66, 214, 692, 2258, 7476, 25148, 85879, 297278, 1041433, 3686815, 13172332, 47444109, 172105201, 628258532, 2306241682, 8507943863, 31525760093, 117280815758, 437860095606, 1639987807815, 6160452320591, 23202682928094, 87602964189252, 331488181232233, 1256926975631257, 4775048367083136, 18172378823435863, 69272217329906085, 264467479952093398, 1011134001045988819, 3871060148173501514, 14838876420322592960, 56949549252469393224, 218810817887588230302, 841608296359775002567, 3240332723714875935118, 12487778082778348304963, 48169929745614146941523, 185970037805612396943711, 718568725470573546659381, 2778660592310138893835274, 10752988439938469598286463, 41642328058233700577778704, 161376230128390019361757968, 625792460294357124156810516, 2428261869456188586036251968, 9428098335678995859443945889, 36627366387322234144039683618, 142373942785779769843356489323, 553718899522166244182448907289, 2154633961217237784255121902939, 8388315498203861928851099917265, 32672723839852199141480873640084, 127320371595840194782656475043327, 496369910921428090587533442319814, 1935982260960077626426594862623674, 7554041927010453994349924390675106, 29487199316554583739786187568682082, 115148449269458106683862304710518916, 449829242177411707772583522316135866, 1757912041903790662033535779806487544, 6872291088393147316439115888793195988, 26875478637853069864958853097860796299, 105137256927377421047641769490598640738, 411432472701527346509123455771166120829, 1610563610064759533604116723385679198667, 6306532961898441953070760058541012983681, 24702060108242214050669929982528557068999, 96783605408069589569017560583902724162548, 379308947111256541993791144286015402239565, 1486975234152923640506201828507507103823452, 5830832479041888497632546233679362955512114, 22870229447553472317256271504517979253619476, 89726489155447941010811329782186040225274192, 352109873813077643940738273002038097031672477, 1382102953928391481299414016121966235576016498, 5426310466468938265534111902460069300428301013, 21309262467024911246510999607755025350354034633, 83700743120512300642606791070919338722254912771, 328840264566800078255755543974357697822569425431, 1292210282649536805427540377394183349975814517704, 5078924450250303800679726697420450650172260143483, 19966345087941774862082751164391171452001713878604, 78507587042454704233887082370326173987135557603258, 308751422255465244846180236817263770158375724773214, 1214475326783819251403185676511269511843639298722772, 4778031305886934442040939185942942154887216600842651, 18801308914217568989453196008908915741544631321912964, 73995322971798529427231937437865083999842401349755003, 291270091558217556021027353637350720481621525950694675, 1146730466196732162484633786268595645905828810490927747, 4515429808541802703776758478526297218163198783979540589, 17783106298295244644876119592932427356276579518295191296, 70046345496368471473116408411487154961835251140948797087, 275950545488494596963763299827175225492370233940077978922, 1087285461815761482684260341014701979707500712475833233158
Generating function in Maple syntax:
1/4*(-4*(x-1/2)^2*(x-1)^4*(1-4*x)^(1/2)-2*x^7+12*x^6-36*x^5+51*x^4-46*x^3+26*x^2-8*x+1)/(x-1/2)/x/(x-1)^3/(x^2-3*x+1)
Generating function in latex syntax:
\frac{-4 \left(x -\frac{1}{2}\right)^{2} \left(x -1\right)^{4} \sqrt{1-4 x}-2 x^{7}+12 x^{6}-36 x^{5}+51 x^{4}-46 x^{3}+26 x^{2}-8 x +1}{4 \left(x -\frac{1}{2}\right) x \left(x -1\right)^{3} \left(x^{2}-3 x +1\right)}
Generating function in sympy syntax:
(-x**7/2 + 3*x**6 - 9*x**5 + 51*x**4/4 - 23*x**3/2 + 13*x**2/2 - 2*x - sqrt(1 - 4*x)*(x - 1)**4*(x - 1/2)**2 + 1/4)/(x*(x - 1)**3*(x - 1/2)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
x*(2*x-1)^2*(x^2-3*x+1)^2*(x-1)^6*F(x)^2+(2*x-1)*(x^2-3*x+1)*(2*x^7-12*x^6+36*x^5-51*x^4+46*x^3-26*x^2+8*x-1)*(x-1)^3*F(x)+x^13+4*x^12-92*x^11+501*x^10-1498*x^9+2927*x^8-3982*x^7+3884*x^6-2741*x^5+1388*x^4-491*x^3+115*x^2-16*x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(2 x -1\right)^{2} \left(x^{2}-3 x +1\right)^{2} \left(x -1\right)^{6} F \! \left(x \right)^{2}+\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(2 x^{7}-12 x^{6}+36 x^{5}-51 x^{4}+46 x^{3}-26 x^{2}+8 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{13}+4 x^{12}-92 x^{11}+501 x^{10}-1498 x^{9}+2927 x^{8}-3982 x^{7}+3884 x^{6}-2741 x^{5}+1388 x^{4}-491 x^{3}+115 x^{2}-16 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 66
a(6) = 214
a(7) = 692
a(8) = 2258
a(9) = 7476
a(n+6) = -8*(2*n+1)/(n+7)*a(n)+4*(21*n+26)/(n+7)*a(n+1)-4*(38*n+89)/(n+7)*a(n+2)+3*(43*n+151)/(n+7)*a(n+3)-2*(28*n+131)/(n+7)*a(n+4)+2*(6*n+35)/(n+7)*a(n+5)-2*(3*n^2-9*n-10)/(n+7), 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) = 20
a \! \left(5\right) = 66
a \! \left(6\right) = 214
a \! \left(7\right) = 692
a \! \left(8\right) = 2258
a \! \left(9\right) = 7476
a \! \left(n +6\right) = -\frac{8 \left(2 n +1\right) a \! \left(n \right)}{n +7}+\frac{4 \left(21 n +26\right) a \! \left(n +1\right)}{n +7}-\frac{4 \left(38 n +89\right) a \! \left(n +2\right)}{n +7}+\frac{3 \left(43 n +151\right) a \! \left(n +3\right)}{n +7}-\frac{2 \left(28 n +131\right) a \! \left(n +4\right)}{n +7}+\frac{2 \left(6 n +35\right) a \! \left(n +5\right)}{n +7}-\frac{2 \left(3 n^{2}-9 n -10\right)}{n +7}, \quad n \geq 10
Specification 1
Strategy pack name: row_and_col_placements
Tree: http://permpal.com/tree/21199/
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[7,x]
F[3,x] = F[4,x]+F[8,x]
F[4,x] = F[1,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[4,x]^2*F[7,x]
F[7,x] = x
F[8,x] = F[9,x]
F[9,x] = F[10,x]*F[7,x]
F[10,x] = F[11,x]+F[28,x]
F[11,x] = F[12,x]+F[15,x]+F[17,x]
F[12,x] = F[1,x]+F[13,x]
F[13,x] = F[14,x]
F[14,x] = F[12,x]*F[7,x]
F[15,x] = F[16,x]
F[16,x] = F[11,x]*F[12,x]*F[4,x]*F[7,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[7,x]
F[19,x] = F[20,x]+F[21,x]+F[27,x]
F[20,x] = F[4,x]^2
F[21,x] = F[22,x]
F[22,x] = F[23,x]*F[4,x]*F[7,x]
F[23,x] = F[12,x]+F[24,x]+F[26,x]
F[24,x] = F[25,x]
F[25,x] = F[12,x]*F[23,x]*F[7,x]
F[26,x] = F[23,x]*F[7,x]
F[27,x] = F[19,x]*F[7,x]
F[28,x] = F[29,x]
F[29,x] = F[12,x]*F[30,x]*F[7,x]
F[30,x] = F[31,x]+F[35,x]+F[43,x]
F[31,x] = F[32,x]+F[5,x]
F[32,x] = F[1,x]+F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[12,x]*F[32,x]*F[7,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[7,x]
F[37,x] = F[38,x]+F[39,x]
F[38,x] = F[23,x]*F[4,x]
F[39,x] = F[40,x]*F[42,x]
F[40,x] = F[1,x]+F[41,x]
F[41,x] = F[40,x]*F[7,x]
F[42,x] = F[33,x]
F[43,x] = F[30,x]*F[7,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_{7}\! \left(x \right)
F_{3}\! \left(x \right) = F_{4}\! \left(x \right)+F_{8}\! \left(x \right)
F_{4}\! \left(x \right) = F_{1}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{4} \left(x \right)^{2} F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = x
F_{8}\! \left(x \right) = F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{7}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{28}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{15}\! \left(x \right)+F_{17}\! \left(x \right)
F_{12}\! \left(x \right) = F_{1}\! \left(x \right)+F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{12}\! \left(x \right) F_{7}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{11}\! \left(x \right) F_{12}\! \left(x \right) F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{7}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{27}\! \left(x \right)
F_{20}\! \left(x \right) = F_{4} \left(x \right)^{2}
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right)+F_{24}\! \left(x \right)+F_{26}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right) F_{7}\! \left(x \right)
F_{26}\! \left(x \right) = F_{23}\! \left(x \right) F_{7}\! \left(x \right)
F_{27}\! \left(x \right) = F_{19}\! \left(x \right) F_{7}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right) F_{30}\! \left(x \right) F_{7}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{35}\! \left(x \right)+F_{43}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{5}\! \left(x \right)
F_{32}\! \left(x \right) = F_{1}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{32}\! \left(x \right) F_{7}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{7}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{39}\! \left(x \right)
F_{38}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right) F_{42}\! \left(x \right)
F_{40}\! \left(x \right) = F_{1}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{40}\! \left(x \right) F_{7}\! \left(x \right)
F_{42}\! \left(x \right) = F_{33}\! \left(x \right)
F_{43}\! \left(x \right) = F_{30}\! \left(x \right) F_{7}\! \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)*F_7(x))
Eq(F_3(x), F_4(x) + F_8(x))
Eq(F_4(x), F_1(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_4(x)**2*F_7(x))
Eq(F_7(x), x)
Eq(F_8(x), F_9(x))
Eq(F_9(x), F_10(x)*F_7(x))
Eq(F_10(x), F_11(x) + F_28(x))
Eq(F_11(x), F_12(x) + F_15(x) + F_17(x))
Eq(F_12(x), F_1(x) + F_13(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_12(x)*F_7(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_11(x)*F_12(x)*F_4(x)*F_7(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x)*F_7(x))
Eq(F_19(x), F_20(x) + F_21(x) + F_27(x))
Eq(F_20(x), F_4(x)**2)
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_23(x)*F_4(x)*F_7(x))
Eq(F_23(x), F_12(x) + F_24(x) + F_26(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_12(x)*F_23(x)*F_7(x))
Eq(F_26(x), F_23(x)*F_7(x))
Eq(F_27(x), F_19(x)*F_7(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_12(x)*F_30(x)*F_7(x))
Eq(F_30(x), F_31(x) + F_35(x) + F_43(x))
Eq(F_31(x), F_32(x) + F_5(x))
Eq(F_32(x), F_1(x) + F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_12(x)*F_32(x)*F_7(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_7(x))
Eq(F_37(x), F_38(x) + F_39(x))
Eq(F_38(x), F_23(x)*F_4(x))
Eq(F_39(x), F_40(x)*F_42(x))
Eq(F_40(x), F_1(x) + F_41(x))
Eq(F_41(x), F_40(x)*F_7(x))
Eq(F_42(x), F_33(x))
Eq(F_43(x), F_30(x)*F_7(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": true, "place_row": true, "strategy_class": "RowAndColumnPlacementFactory"}]], "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": "row_and_col_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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "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], "pos": [[0, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[1, 2]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 2]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 2], [0, 2], [0, 2]]}, 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