0213_1032_1320_2301
Counting sequence:
1, 1, 2, 6, 20, 60, 163, 416, 1027, 2501, 6081, 14863, 36636, 91159, 228896, 579432, 1476962, 3786294, 9751273, 25205858, 65342545, 169773791, 441882327, 1151679601, 3004768650, 7845849985, 20499227798, 53584725546, 140120732192, 366509036736, 958869508351, 2509025747828, 6566060252911, 17184860045129, 44979929949501, 117737749935907, 308198960121672, 806790410954299, 2112034833789788, 5529039212510268, 14474533047929390, 37893460419652506, 99203649187775077, 259713089097164246, 679926822010698205, 1780049784748888835, 4660187347863882483, 12200441890098584101, 31940997584943517782, 83622269389755262021, 218925247634368850546, 573152347613444450718, 1500529543406150820236, 3928431779005380643524, 10284756786410736373531, 26925820565828318999432, 70492668882277201665307, 184552114023409248073229, 483163529072762466703353, 1264938184964502000330103, 3311650449359991230868660, 8670012010193967085434271, 22698383275378900811745752, 59425133204256716922420816, 155577007114019213100746810, 407305869691056848670274110, 1066340565065663185490978593, 2791715751718956412964461706, 7308806542517253463726000297, 19134703580684898798860720231, 50095303609241632574150515791, 131151206066448378206179531033, 343358312228920260609565477986, 898923725897945920752871696969, 2353412856020184535909759193486, 6161314823273141755497825036882, 16130531576020308867626554215992, 42230279829229921121467514200584, 110560307760553727044947341556343, 289450643150199805109717216800988, 757791621085582778476889721502855, 1983924218897622710706322773010961, 5193981033189433714412820248127477, 13598018875834975154073621272556811, 35600075584644085190891010171903840, 93202207858754467304765342447866435, 244006547952933690495736883581108628, 638817435922675351727109041114275236, 1672445759660349859774917705399337862, 4378519842748889217776299006358968866, 11463113767967347773911289176228018461, 30010821459915214064672188247425974398, 78569350609302414341534515016251668661, 205697230363040268802789835701732547339, 538522340469914871752551949889752992491, 1409869791026897305826299929569140455757, 3691087032571162964469215670020896413102, 9663391306607363425067082742900004846989, 25299086887092470985703503883492030241258, 66233869354353136881986371557201910089798, 173402521175333114360141496087365348440100
Generating function in Maple syntax:
(2*x^6-9*x^5+15*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{2 x^{6}-9 x^{5}+15 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:
(2*x**6 - 9*x**5 + 15*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)-2*x^6+9*x^5-15*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)-2 x^{6}+9 x^{5}-15 x^{4}+24 x^{3}-19 x^{2}+7 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/10*(5-5^(1/2))*(3/2-1/2*5^(1/2))^(-n)+1/10*(5+5^(1/2))*(3/2+1/2*5^(1/2))^(-n)-3/2*n^2+1/2*n+2^(n+1)-3)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(5-\sqrt{5}\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{10}+\frac{\left(5+\sqrt{5}\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{10}-\frac{3 n^{2}}{2}+\frac{n}{2}+2^{n +1}-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) = 20
a(5) = 60
a(6) = 163
a(n+3) = -3/2*n^2+2*a(n)-7*a(n+1)+5*a(n+2)+1/2*n+3, n >= 7
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) = 60
a \! \left(6\right) = 163
a \! \left(n +3\right) = -\frac{3 n^{2}}{2}+2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)+\frac{n}{2}+3, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21382/
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[9,x]
F[4,x] = F[15,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[10,x]*F[9,x]
F[9,x] = x
F[10,x] = F[11,x]+F[5,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[13,x]*F[9,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[13,x]*F[9,x]
F[15,x] = F[16,x]+F[2,x]
F[16,x] = F[17,x]
F[17,x] = F[18,x]*F[9,x]
F[18,x] = F[19,x]+F[26,x]
F[19,x] = F[20,x]
F[20,x] = F[13,x]*F[21,x]
F[21,x] = F[22,x]+F[7,x]
F[22,x] = F[23,x]^2
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[9,x]
F[25,x] = F[1,x]+F[23,x]
F[26,x] = F[27,x]
F[27,x] = F[25,x]*F[28,x]*F[9,x]
F[28,x] = F[29,x]+F[42,x]
F[29,x] = F[30,x]+F[40,x]
F[30,x] = F[13,x]*F[31,x]
F[31,x] = F[25,x]+F[32,x]
F[32,x] = F[33,x]+F[35,x]
F[33,x] = F[34,x]
F[34,x] = F[13,x]*F[9,x]
F[35,x] = F[36,x]+F[37,x]+F[39,x]
F[36,x] = 0
F[37,x] = F[38,x]*F[9,x]
F[38,x] = F[23,x]+F[35,x]
F[39,x] = F[32,x]*F[9,x]
F[40,x] = F[41,x]
F[41,x] = F[13,x]^2*F[23,x]*F[9,x]
F[42,x] = F[13,x]*F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[45,x]*F[9,x]
F[45,x] = F[25,x]+F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[25,x]*F[48,x]*F[9,x]
F[48,x] = F[45,x]+F[49,x]
F[49,x] = F[13,x]*F[23,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_{9}\! \left(x \right)
F_{4}\! \left(x \right) = F_{15}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{10}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = x
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{5}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{13}\! \left(x \right) F_{9}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{13}\! \left(x \right) F_{9}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{2}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{9}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{26}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{7}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23} \left(x \right)^{2}
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{9}\! \left(x \right)
F_{25}\! \left(x \right) = F_{1}\! \left(x \right)+F_{23}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{25}\! \left(x \right) F_{28}\! \left(x \right) F_{9}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{42}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{40}\! \left(x \right)
F_{30}\! \left(x \right) = F_{13}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{25}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{35}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{13}\! \left(x \right) F_{9}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{37}\! \left(x \right)+F_{39}\! \left(x \right)
F_{36}\! \left(x \right) = 0
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{9}\! \left(x \right)
F_{38}\! \left(x \right) = F_{23}\! \left(x \right)+F_{35}\! \left(x \right)
F_{39}\! \left(x \right) = F_{32}\! \left(x \right) F_{9}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{13} \left(x \right)^{2} F_{23}\! \left(x \right) F_{9}\! \left(x \right)
F_{42}\! \left(x \right) = F_{13}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right) F_{9}\! \left(x \right)
F_{45}\! \left(x \right) = F_{25}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{25}\! \left(x \right) F_{48}\! \left(x \right) F_{9}\! \left(x \right)
F_{48}\! \left(x \right) = F_{45}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{13}\! \left(x \right) F_{23}\! \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_9(x))
Eq(F_4(x), F_15(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_10(x)*F_9(x))
Eq(F_9(x), x)
Eq(F_10(x), F_11(x) + F_5(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_13(x)*F_9(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_13(x)*F_9(x))
Eq(F_15(x), F_16(x) + F_2(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_18(x)*F_9(x))
Eq(F_18(x), F_19(x) + F_26(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_13(x)*F_21(x))
Eq(F_21(x), F_22(x) + F_7(x))
Eq(F_22(x), F_23(x)**2)
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_9(x))
Eq(F_25(x), F_1(x) + F_23(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_25(x)*F_28(x)*F_9(x))
Eq(F_28(x), F_29(x) + F_42(x))
Eq(F_29(x), F_30(x) + F_40(x))
Eq(F_30(x), F_13(x)*F_31(x))
Eq(F_31(x), F_25(x) + F_32(x))
Eq(F_32(x), F_33(x) + F_35(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_13(x)*F_9(x))
Eq(F_35(x), F_36(x) + F_37(x) + F_39(x))
Eq(F_36(x), 0)
Eq(F_37(x), F_38(x)*F_9(x))
Eq(F_38(x), F_23(x) + F_35(x))
Eq(F_39(x), F_32(x)*F_9(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_13(x)**2*F_23(x)*F_9(x))
Eq(F_42(x), F_13(x)*F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_45(x)*F_9(x))
Eq(F_45(x), F_25(x) + F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_25(x)*F_48(x)*F_9(x))
Eq(F_48(x), F_45(x) + F_49(x))
Eq(F_49(x), F_13(x)*F_23(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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 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"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 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