1302_03142_20314_31420
Counting sequence:
1, 1, 2, 6, 23, 100, 464, 2236, 11048, 55588, 283648, 1463868, 7626296, 40049188, 211768752, 1126513788, 6024316488, 32368294756, 174645900768, 945893037948, 5140619493464, 28024941448420, 153219244012432, 839884005112636, 4615012956649832, 25415266785076900, 140253540192296384, 775473907195044796, 4295344555193978488, 23831707655380474020, 132431845025074893680, 737000643339676165372, 4107175916108980677768, 22918379326224020445924, 128043570482815609144224, 716202221256392443648764, 4010427349728623669850264, 22480084192631712064863588, 126134700383203543731760464, 708401332986749629287831996, 3982104163434810194968594344, 22403507845877736710370367524, 126145731739032424937399115648, 710830973739691898936980118844, 4008506006061304299475846991544, 22620764023981729811671278287652, 127740183581307558927809588443440, 721823711011904499304923500886396, 4081376299717901733777768619422408, 23091026756695782240638548847681124, 130716622138902142984985867813513568, 740388711731842760154489468487207548, 4195861374168847453277015774019440856, 23790650240781794678440638721953092068, 134960681036377286788175574128959155984, 765978070091112012208351352972980085820, 4349369391939400037718320201960546080232, 24707583023302272582921088311406344875428, 140417656221977816832481013631239536400704, 798352081403911117584499661027893929345724, 4540906831762628610829330185097823012445944, 25838095129348214869208315452367239110836644, 147076340925755953470398212318474863986820848, 837499317387333181095110196328847729303157244, 4770683013729964729415918990303013240573417736, 27184820319906665181835609164662540113459267044, 154959333025754709264878374985254403957275322144, 883588651685441822720659542117238110149282719228, 5039872814519536282452981006158294385409924604184, 28755583896502874939224741790206243195667767069284, 164117254865086303974402120510636099995274114627792, 936940829301301554165490083839773358782021494037180, 5350477301520602142169043586186258460699811512411176, 30562722575297004919139666525847041322906506496698148, 174625453393501776906251777900219780400589570117094144, 998012577117880352855173869575193280253468470545323068, 5705247978208263597221940985098819629774390064670506808, 32622728056479067373088707114234103745405922700538178596, 186582352950188387518241282467995657306392351375987162288, 1067389180843877264175944046889474514398055605642752222844, 6107654554972821907079608437694897076999703350762951614280, 34956115164297243513411834067121299913818221925560294527332, 200108971033425777605131183678300300847461050810783747882208, 1145783107773222790185317325416193445552121957748683476594044, 6561884272730204224144040071839885759484853393252934044344664, 37587455322514403933910485227188467892921938030909323660044004, 215349303988323545840364173682328064324578875036903474266196624, 1234037225847664685134982847965206100987573081018214232005573948, 7072865614982689221933651256536086151028183729813872421198819944, 40545539994396942801477362101101785762864544402347126035008996516, 232471408266233750917840981198387360189130218116607520990034097344, 1333131761562650935440968749143477066503339538895368375680539546044, 7646312203682652174438659728181127472199280555651024541752765067128, 43863653548810608199508206778997092060438679732805336796912854366884, 251669077432496981808387090094110977847964228594723757666189069015664, 1444194514669535852057279862120695693114127925399514815153024255557372, 8288784570547430254748895926792345964091997301701878239864921277416840, 47579944619583654675831423176206561293372126875961550295695292533014756, 273164063898087571119207807492479378126085446783821187651891922363427488, 1568514096293372565929352199929309432312988984663245908789110138824696060, 9007768766633886173496619707181351503548239711628843931319566149000319384
Generating function in Maple syntax:
1/2*((-x^4+7*x^3-13*x^2+6*x-1)*(x^2-6*x+1)^(1/2)-x^5+10*x^4-30*x^3+29*x^2-7*x+1)/x
Generating function in latex syntax:
\frac{\left(-x^{4}+7 x^{3}-13 x^{2}+6 x -1\right) \sqrt{x^{2}-6 x +1}-x^{5}+10 x^{4}-30 x^{3}+29 x^{2}-7 x +1}{2 x}
Generating function in sympy syntax:
(-x**5/2 + 5*x**4 - 15*x**3 + 29*x**2/2 - 7*x/2 + sqrt(x**2 - 6*x + 1)*(-x**4 + 7*x**3 - 13*x**2 + 6*x - 1)/2 + 1/2)/x
Implicit equation for the generating function in Maple syntax:
x*F(x)^2+(x^5-10*x^4+30*x^3-29*x^2+7*x-1)*F(x)+3*x^4-14*x^3+22*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
x F \! \left(x \right)^{2}+\left(x^{5}-10 x^{4}+30 x^{3}-29 x^{2}+7 x -1\right) F \! \left(x \right)+3 x^{4}-14 x^{3}+22 x^{2}-7 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 23
a(5) = 100
a(n+6) = -(n-4)/(7+n)*a(n)+(-29+13*n)/(7+n)*a(1+n)-2*(-11+28*n)/(7+n)*a(n+2)+(136+91*n)/(7+n)*a(n+3)-(169+50*n)/(7+n)*a(n+4)+3*(21+4*n)/(7+n)*a(n+5), n >= 6
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) = 23
a \! \left(5\right) = 100
a \! \left(n +6\right) = -\frac{\left(n -4\right) a \! \left(n \right)}{7+n}+\frac{\left(-29+13 n \right) a \! \left(1+n \right)}{7+n}-\frac{2 \left(-11+28 n \right) a \! \left(n +2\right)}{7+n}+\frac{\left(136+91 n \right) a \! \left(n +3\right)}{7+n}-\frac{\left(169+50 n \right) a \! \left(n +4\right)}{7+n}+\frac{3 \left(21+4 n \right) a \! \left(n +5\right)}{7+n}, \quad n \geq 6
Specification 1
Strategy pack name: row_and_col_placements_req_corrob_expand_verified
Tree: http://permpal.com/tree/24053/
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[8,x]
F[3,x] = F[1,x]+F[29,x]+F[4,x]
F[4,x] = F[5,x]*F[8,x]
F[5,x] = F[3,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[3,x]*F[8,x]*F[9,x]
F[8,x] = x
F[9,x] = F[10,x]+F[27,x]
F[10,x] = F[11,x]+F[19,x]
F[11,x] = F[1,x]+F[12,x]
F[12,x] = F[13,x]*F[8,x]
F[13,x] = F[11,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[11,x]*F[16,x]*F[8,x]
F[16,x] = F[13,x]+F[17,x]
F[17,x] = F[18,x]
F[18,x] = F[13,x]*F[8,x]*F[9,x]
F[19,x] = F[20,x]
F[20,x] = F[11,x]*F[21,x]*F[8,x]
F[21,x] = F[10,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[10,x]*F[24,x]*F[8,x]
F[24,x] = F[13,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[13,x]*F[21,x]*F[8,x]
F[27,x] = F[28,x]
F[28,x] = F[10,x]*F[16,x]*F[8,x]
F[29,x] = F[30,x]*F[8,x]
F[30,x] = F[31,x]+F[43,x]
F[31,x] = F[1,x]+F[32,x]+F[36,x]
F[32,x] = F[33,x]*F[8,x]
F[33,x] = F[31,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[31,x]*F[8,x]*F[9,x]
F[36,x] = F[37,x]*F[8,x]
F[37,x] = F[38,x]+F[41,x]
F[38,x] = F[1,x]+F[39,x]+F[40,x]
F[39,x] = F[37,x]*F[8,x]
F[40,x] = F[33,x]*F[8,x]
F[41,x] = F[42,x]
F[42,x] = F[21,x]*F[38,x]*F[8,x]
F[43,x] = F[44,x]
F[44,x] = F[31,x]*F[45,x]*F[8,x]
F[45,x] = F[16,x]+F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[11,x]*F[13,x]*F[48,x]*F[8,x]
F[48,x] = 1/F[11,x]/F[8,x]*F[49,x]
F[49,x] = F[50,x]
F[50,x] = -F[38,x]+F[37,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_{8}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{29}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{5}\! \left(x \right) F_{8}\! \left(x \right)
F_{5}\! \left(x \right) = F_{3}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{3}\! \left(x \right) F_{8}\! \left(x \right) F_{9}\! \left(x \right)
F_{8}\! \left(x \right) = x
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{27}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{19}\! \left(x \right)
F_{11}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{8}\! \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_{15}\! \left(x \right) = F_{11}\! \left(x \right) F_{16}\! \left(x \right) F_{8}\! \left(x \right)
F_{16}\! \left(x \right) = F_{13}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{13}\! \left(x \right) F_{8}\! \left(x \right) F_{9}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{11}\! \left(x \right) F_{21}\! \left(x \right) F_{8}\! \left(x \right)
F_{21}\! \left(x \right) = F_{10}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{10}\! \left(x \right) F_{24}\! \left(x \right) F_{8}\! \left(x \right)
F_{24}\! \left(x \right) = F_{13}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{13}\! \left(x \right) F_{21}\! \left(x \right) F_{8}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{10}\! \left(x \right) F_{16}\! \left(x \right) F_{8}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{8}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{43}\! \left(x \right)
F_{31}\! \left(x \right) = F_{1}\! \left(x \right)+F_{32}\! \left(x \right)+F_{36}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{8}\! \left(x \right)
F_{33}\! \left(x \right) = F_{31}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{31}\! \left(x \right) F_{8}\! \left(x \right) F_{9}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{8}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{41}\! \left(x \right)
F_{38}\! \left(x \right) = F_{1}\! \left(x \right)+F_{39}\! \left(x \right)+F_{40}\! \left(x \right)
F_{39}\! \left(x \right) = F_{37}\! \left(x \right) F_{8}\! \left(x \right)
F_{40}\! \left(x \right) = F_{33}\! \left(x \right) F_{8}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{21}\! \left(x \right) F_{38}\! \left(x \right) F_{8}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{31}\! \left(x \right) F_{45}\! \left(x \right) F_{8}\! \left(x \right)
F_{45}\! \left(x \right) = F_{16}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{11}\! \left(x \right) F_{13}\! \left(x \right) F_{48}\! \left(x \right) F_{8}\! \left(x \right)
F_{48}\! \left(x \right) = \frac{F_{49}\! \left(x \right)}{F_{11}\! \left(x \right) F_{8}\! \left(x \right)}
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = -F_{38}\! \left(x \right)+F_{37}\! \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_8(x))
Eq(F_3(x), F_1(x) + F_29(x) + F_4(x))
Eq(F_4(x), F_5(x)*F_8(x))
Eq(F_5(x), F_3(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_3(x)*F_8(x)*F_9(x))
Eq(F_8(x), x)
Eq(F_9(x), F_10(x) + F_27(x))
Eq(F_10(x), F_11(x) + F_19(x))
Eq(F_11(x), F_1(x) + F_12(x))
Eq(F_12(x), F_13(x)*F_8(x))
Eq(F_13(x), F_11(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_11(x)*F_16(x)*F_8(x))
Eq(F_16(x), F_13(x) + F_17(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_13(x)*F_8(x)*F_9(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_11(x)*F_21(x)*F_8(x))
Eq(F_21(x), F_10(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_10(x)*F_24(x)*F_8(x))
Eq(F_24(x), F_13(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_13(x)*F_21(x)*F_8(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_10(x)*F_16(x)*F_8(x))
Eq(F_29(x), F_30(x)*F_8(x))
Eq(F_30(x), F_31(x) + F_43(x))
Eq(F_31(x), F_1(x) + F_32(x) + F_36(x))
Eq(F_32(x), F_33(x)*F_8(x))
Eq(F_33(x), F_31(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_31(x)*F_8(x)*F_9(x))
Eq(F_36(x), F_37(x)*F_8(x))
Eq(F_37(x), F_38(x) + F_41(x))
Eq(F_38(x), F_1(x) + F_39(x) + F_40(x))
Eq(F_39(x), F_37(x)*F_8(x))
Eq(F_40(x), F_33(x)*F_8(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_21(x)*F_38(x)*F_8(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_31(x)*F_45(x)*F_8(x))
Eq(F_45(x), F_16(x) + F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_11(x)*F_13(x)*F_48(x)*F_8(x))
Eq(F_48(x), F_49(x)/(F_11(x)*F_8(x)))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_37(x) - F_38(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}, {"class_module": "tilings.strategies.requirement_insertion", "ignore_parent": true, "strategy_class": "RequirementCorroborationFactory"}], "iterative": false, "name": "row_and_col_placements_req_corrob_expand_verified", "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": [1, 3, 0, 2], "pos": [[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": [2, 0, 3, 1, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 4, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"class_module": "comb_spec_searcher.strategies.rule", "idx": 1, "original_rule": {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 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], "pos": [[2, 1]]}, {"patt": [0], "pos": [[3, 0]]}, {"patt": [0], "pos": [[3, 1]]}, {"patt": [0, 1], "pos": [[0, 0], [2, 0]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[0, 2], [3, 2]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[2, 2], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 2], [2, 2]]}, {"patt": [0, 2, 1], "pos": [[0, 2], [0, 2], [2, 2]]}, {"patt": [1, 0, 2], "pos": [[0, 2], [0, 0], [0, 2]]}, {"patt": [1, 0, 2], "pos": [[2, 2], [3, 2], [3, 2]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 2], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 2], [2, 2], [2, 2]]}, {"patt": [2, 0, 1], "pos": [[2, 2], [2, 2], [3, 2]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 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