01432_02431_10432_12430_20431_21430
Counting sequence:
1, 1, 2, 6, 24, 114, 600, 3372, 19824, 120426, 749976, 4762644, 30723792, 200778612, 1326360048, 8842981848, 59425117152, 402092408346, 2737156004376, 18732169337604, 128806616999184, 889479590046108, 6165939982059600, 42891532191557736, 299307319060137504, 2094684530280579204, 14698465904618271600, 103391859250744921032, 728921724455874437664, 5149722130698960310056, 36452793427943459541216, 258502230381854035397040, 1836252048586705964035008, 13064351285339412097452666, 93086856911367488874965784, 664194212869807647276675684, 4745398038984383168404384656, 33946006185024309584419138188, 243115989817433184083028079248, 1743090214757554905537513678792, 12510727228039899824443540005792, 89883165861304275497726457433068, 646376826183172836194296157755344, 4652483757356774570221586375075544, 33516379424491662872615797471726176, 241649128820923962435750703974420696, 1743625446363709768112039800174408992, 12590587146701777302368196690836722640, 90980758358285830188814991290082362944, 657886175476583519079435444907500602916, 4760332390677084581539663483009695394416, 34466527331862222321009111355303033993704, 249701196164851760773803778587492940113312, 1810073921029906072648056552378995252750232, 13128494645874546222193180072880140681328928, 95272456877355714342609853591191556566613008, 691743357403024365898560652397653035432617280, 5025041838744985930198913490786502026008913864, 36521087915854745438144359455987962173587937248, 265552112645559448851680853554026352702410113552, 1931749968200099720230997561300183368071609394752, 14058551723907651912165393734636130130689394847568, 102355813573075724181920262024018940996976042096064, 745522509073007171819006030616952269866027535330144, 5432248876291767535417303240917078903727163448878976, 39597139780188594027887721077937548406504234465920826, 288740758261776114113591863136074446498150993436396824, 2106238667249529533145237813129911396887982507996495652, 15369424291399086420224255180854306174116062386126061712, 112189877493227004366787381247069322374101025549117293548, 819202983092945268740956268198143438435477004733131278608, 5983662108096416715710643097119543226662604624951321514376, 43719618601413050434332076859839028433379696143458071186080, 319532974296699297658710470196447473692600968728108158485852, 2336046138209812933413279553857712698504403276606980913997328, 17083235073803093254657140104076787602754800342071165042006328, 124962145807890569452204984929148398043340726099667127219579872, 914330713946740475830977448781342411234561461256800363560575736, 6691778650190852452520229915907829747595406073683194154981715616, 48988068201702302028116506075656270284077180496372752740777400464, 358712678169587444454001061676846928264036056472802730078070211904, 2627290461054707576155659785771971746278985898400466333060858171660, 19247387378294848589695694969232071469321676318063976988177134457552, 141037727877177986557302915461969645785597561035837241696815163517304, 1033704031504405972128420222783734291349718956308257939157913296227296, 7577957582865503261430847647947457941146201206998083018267317411076296, 55564949917890717879737463479480435966152406924771892378338499640428384, 407511909616078816324505626662909519496028933493284135880140422982002992, 2989291069022834817912299550670398676444328526237735552906831188971334592, 21932220119253805641420220181801439175804835258736322686503175442662301368, 160946493884649536759913742492476295262545344585276908430924414391656987168, 1181307834098128307364496589571181291252244681070397898602126448169407400432, 8672123553482013528939449678852902731914189344531729930015792780477279551936, 63674694171543957799243486756545228676305151945433813232924749538230884359216, 467611953272222067082021832488215724812540463011033639910701411408743943142464, 3434630390883003500018846030445714607692604971489648782097174846920792298070944, 25231814037645356813978029435976117261295301350381020100226828064192998609269888, 185391342781785546347503953440334779327891653869728144258659718473701793892975460, 1362389936155974260282858570726688157899610208749783903669389047028573227393276016, 10013431841380955319656278717282408589325512782493229375657935217204610712941616424, 73609283846396475084679593013345762091317673234189918121253980559198224613893361824
Generating function in Maple syntax:
-2*x^2+2*x+1-x*(4*x^2-8*x+1)^(1/2)
Generating function in latex syntax:
-2 x^{2}+2 x +1-x \sqrt{4 x^{2}-8 x +1}
Generating function in sympy syntax:
-2*x**2 - x*sqrt(4*x**2 - 8*x + 1) + 2*x + 1
Implicit equation for the generating function in Maple syntax:
F(x)^2+(4*x^2-4*x-2)*F(x)-x^2+4*x+1 = 0
Implicit equation for the generating function in latex syntax:
F \! \left(x \right)^{2}+\left(4 x^{2}-4 x -2\right) F \! \left(x \right)-x^{2}+4 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(n+2) = -4*(n-2)/(1+n)*a(n)+4*(2*n-1)/(1+n)*a(1+n), n >= 3
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(n +2\right) = -\frac{4 \left(n -2\right) a \! \left(n \right)}{1+n}+\frac{4 \left(2 n -1\right) a \! \left(1+n \right)}{1+n}, \quad n \geq 3
Specification 1
Strategy pack name: row_placements_tracked_fusion
Tree: http://permpal.com/tree/24104/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[14,x]*F[3,x]
F[3,x] = F[1,x]+F[23,x]+F[4,x]
F[4,x] = F[14,x]*F[5,x]
F[5,x] = F[1,x]+F[20,x]+F[22,x]+F[6,x]
F[6,x] = F[14,x]*F[7,x]
F[7,x] = F[1,x]+F[19,x]+F[20,x]+F[21,x]+F[8,x]
F[8,x] = F[14,x]*F[9,x]
F[9,x] = F[10,x,1]
F[10,x,k[0]] = 1/(-1+k[0])*(F[11,x,k[0]]*k[0]-F[11,x,1])
F[11,x,k[0]] = F[1,x]+F[12,x,k[0]]+F[15,x,k[0]]+F[16,x,k[0]]+F[18,x,k[0]]
F[12,x,k[0]] = F[13,x,k[0]]*F[14,x]
F[13,x,k[0]] = 1/(-1+k[0])*(F[11,x,k[0]]*k[0]-F[11,x,1])
F[14,x] = x
F[15,x,k[0]] = F[13,x,k[0]]*F[14,x]
F[16,x,k[0]] = F[11,x,k[0]]*F[17,x,k[0]]
F[17,x,k[0]] = k[0]*x
F[18,x,k[0]] = F[10,x,k[0]]*F[14,x]
F[19,x] = F[14,x]*F[9,x]
F[20,x] = F[14,x]*F[7,x]
F[21,x] = F[14,x]*F[9,x]
F[22,x] = F[14,x]*F[7,x]
F[23,x] = F[14,x]*F[5,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_{14}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{23}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{14}\! \left(x \right) F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{20}\! \left(x \right)+F_{22}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{14}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{14}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x , 1\right)
F_{10}\! \left(x , y\right) = \frac{y F_{11}\! \left(x , y\right)-F_{11}\! \left(x , 1\right)}{-1+y}
F_{11}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{12}\! \left(x , y\right)+F_{15}\! \left(x , y\right)+F_{16}\! \left(x , y\right)+F_{18}\! \left(x , y\right)
F_{12}\! \left(x , y\right) = F_{13}\! \left(x , y\right) F_{14}\! \left(x \right)
F_{13}\! \left(x , y\right) = \frac{y F_{11}\! \left(x , y\right)-F_{11}\! \left(x , 1\right)}{-1+y}
F_{14}\! \left(x \right) = x
F_{15}\! \left(x , y\right) = F_{13}\! \left(x , y\right) F_{14}\! \left(x \right)
F_{16}\! \left(x , y\right) = F_{11}\! \left(x , y\right) F_{17}\! \left(x , y\right)
F_{17}\! \left(x , y\right) = y x
F_{18}\! \left(x , y\right) = F_{10}\! \left(x , y\right) F_{14}\! \left(x \right)
F_{19}\! \left(x \right) = F_{14}\! \left(x \right) F_{9}\! \left(x \right)
F_{20}\! \left(x \right) = F_{14}\! \left(x \right) F_{7}\! \left(x \right)
F_{21}\! \left(x \right) = F_{14}\! \left(x \right) F_{9}\! \left(x \right)
F_{22}\! \left(x \right) = F_{14}\! \left(x \right) F_{7}\! \left(x \right)
F_{23}\! \left(x \right) = F_{14}\! \left(x \right) F_{5}\! \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_14(x)*F_3(x))
Eq(F_3(x), F_1(x) + F_23(x) + F_4(x))
Eq(F_4(x), F_14(x)*F_5(x))
Eq(F_5(x), F_1(x) + F_20(x) + F_22(x) + F_6(x))
Eq(F_6(x), F_14(x)*F_7(x))
Eq(F_7(x), F_1(x) + F_19(x) + F_20(x) + F_21(x) + F_8(x))
Eq(F_8(x), F_14(x)*F_9(x))
Eq(F_9(x), F_10(x, 1))
Eq(F_10(x, k_0), (-k_0*F_11(x, k_0) + F_11(x, 1))/(1 - k_0))
Eq(F_11(x, k_0), F_1(x) + F_12(x, k_0) + F_15(x, k_0) + F_16(x, k_0) + F_18(x, k_0))
Eq(F_12(x, k_0), F_13(x, k_0)*F_14(x))
Eq(F_13(x, k_0), (k_0*F_11(x, k_0) - F_11(x, 1))/(k_0 - 1))
Eq(F_14(x), x)
Eq(F_15(x, k_0), F_13(x, k_0)*F_14(x))
Eq(F_16(x, k_0), F_11(x, k_0)*F_17(x, k_0))
Eq(F_17(x, k_0), k_0*x)
Eq(F_18(x, k_0), F_10(x, k_0)*F_14(x))
Eq(F_19(x), F_14(x)*F_9(x))
Eq(F_20(x), F_14(x)*F_7(x))
Eq(F_21(x), F_14(x)*F_9(x))
Eq(F_22(x), F_14(x)*F_7(x))
Eq(F_23(x), F_14(x)*F_5(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": false, "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.rearrange_assumption", "strategy_class": "RearrangeAssumptionFactory"}, {"class_module": "tilings.strategies.assumption_insertion", "strategy_class": "AddAssumptionFactory"}, {"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}, {"class_module": "tilings.strategies.fusion.fusion", "strategy_class": "FusionFactory", "tracked": true}], "iterative": false, "name": "row_placements_tracked_fusion", "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, 4, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 4, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 4, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 4, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 4, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 4, 3, 0], "pos": [[0, 0], [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, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[2, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [1, 3, 2, 0], "pos": [[2, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 4, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 4, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [1, 0, 4, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 4, 3, 0], "pos": [[0, 1], [0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [2, 0, 4, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [2, 1, 4, 3, 0], "pos": [[0, 1], [0, 1], [0, 1], [2, 1], [2, 1]]}], "requirements": [[{"patt": [0], "pos": [[1, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 4, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 4, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 4, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 4, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 4, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 4, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_placement", "direction": 3, "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "include_empty": true, "indices": [0], "own_col": true, "own_row": true, "strategy_class": "RequirementPlacementStrategy"}}, {"children": [{"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], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 4, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 4, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 4, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 4, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 4, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 4, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [1, 0], [1, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 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