0132_0231_0321_1203_1320
Counting sequence:
1, 1, 2, 6, 19, 57, 164, 460, 1269, 3459, 9342, 25050, 66795, 177337, 469256, 1238556, 3262717, 8582271, 22549526, 59197258, 155304255, 407239641, 1067463072, 2797246536, 7328470629, 19196553727, 50277967514, 131670902970, 344801849959, 902868864309, 2364073178072, 6189887540440, 16206663184665, 42432249496767, 111094380272466, 290859481254726, 761501243360367, 1993678608564181, 5219603301808316, 13665268735813608, 35776477783538785, 93664714370615931, 245218764839936042, 641993779172446966, 1680766970723915139, 4400315929092319797, 11520198408739087764, 30160314481497031380, 78960815404496183049, 206722272469479872059, 541206283478920142702, 1416897140917233976230, 3709486265172688627435, 9711563906400645590097, 25425209957628875512076, 66564074973685235685796, 174267032977825341025917, 456237059988587806354491, 1194444219045532115964006, 3127095741263196617391858, 8186843292974433887921715, 21433434714120857349795121, 56113462002309642768308912, 146906953598651080168823796, 384607403405329616165548549, 1006915265840709805182595767, 2636138412563543873091788414, 6901500008743409961511870690, 18068361687453662306282028039, 47303585201191529547010624209, 123842394211268831514102668232, 324223598022910775354003029920, 848828401038055115265317722605, 2222261607452437811876772742327, 5817956426041624803234645715586, 15231607680117169563566454829266, 39876866633199349818943299624367, 104398992257259811756220605750701, 273320110214137949175632841044096, 715561338536269763222506564217008, 1873363905696902795395544145280545, 4904530379158901532771440458974711, 12840227232988727622533406406346682, 33616151322225132974058037109474526, 88008226738522374578099221620898359, 230408528903013397317156661150866637, 603217359989860630487204828626996964, 1579243551105254120372125958320718400, 4134513293403272983084509313516349929, 10824296329259307333792074516590718259, 28338375694684134028113059304980582162, 74190830755412064770189793535800586510, 194234116572790000321741701577320297675, 508511518965433816273606071745958550957, 1331300440328463208656218034760151847996, 3485389802029859330009331074733689982636, 9124868965780921822000341273839304083217, 23889217095352520217248824915580994237811, 62542782320355866992260397810497222576086, 163739129865873537084560897191097761386558, 428674607277581656911479351113170237380275
Generating function in Maple syntax:
(2*x^6-4*x^5+14*x^4-24*x^3+19*x^2-7*x+1)/(2*x-1)/(x^2-3*x+1)/(-1+x)^3
Generating function in latex syntax:
\frac{2 x^{6}-4 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(-1+x \right)^{3}}
Generating function in sympy syntax:
(2*x**6 - 4*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)*(-1+x)^3*F(x)-2*x^6+4*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(-1+x \right)^{3} F \! \left(x \right)-2 x^{6}+4 x^{5}-14 x^{4}+24 x^{3}-19 x^{2}+7 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/10*(12*5^(1/2)-20)*(3/2-1/2*5^(1/2))^(-n)+1/10*(-12*5^(1/2)-20)*(3/2+1/2*5^(1/2))^(-n)+1/2*n^2-5/2*n-2^n+5)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(12 \sqrt{5}-20\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{10}+\frac{\left(-12 \sqrt{5}-20\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{10}+\frac{n^{2}}{2}-\frac{5 n}{2}-2^{n}\\+5 & \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) = 57
a(6) = 164
a(n+3) = 2*a(n)-7*a(n+1)+5*a(n+2)+1/2*(n-3)*(n-2), 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) = 19
a \! \left(5\right) = 57
a \! \left(6\right) = 164
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 -3\right) \left(n -2\right)}{2}, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18661/
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[12,x]*F[4,x]
F[4,x] = F[18,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[12,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[17,x]
F[15,x] = 0
F[16,x] = F[10,x]*F[12,x]
F[17,x] = F[12,x]*F[13,x]
F[18,x] = F[19,x]+F[29,x]
F[19,x] = F[20,x]
F[20,x] = F[12,x]*F[21,x]
F[21,x] = F[22,x]+F[9,x]
F[22,x] = F[19,x]+F[23,x]
F[23,x] = F[15,x]+F[24,x]+F[28,x]
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[26,x]+F[27,x]
F[26,x] = F[10,x]
F[27,x] = F[23,x]
F[28,x] = F[12,x]*F[22,x]
F[29,x] = F[15,x]+F[30,x]+F[60,x]
F[30,x] = F[12,x]*F[31,x]
F[31,x] = F[32,x]+F[39,x]
F[32,x] = F[33,x]+F[6,x]
F[33,x] = F[34,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[12,x]+F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[12,x]*F[36,x]
F[39,x] = F[29,x]+F[40,x]
F[40,x] = 2*F[15,x]+F[41,x]+F[45,x]
F[41,x] = F[12,x]*F[42,x]
F[42,x] = F[43,x]+F[44,x]
F[43,x] = F[33,x]
F[44,x] = F[40,x]
F[45,x] = F[12,x]*F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[48,x]+F[54,x]
F[48,x] = F[15,x]+F[49,x]+F[53,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[51,x]+F[52,x]
F[51,x] = F[12,x]
F[52,x] = F[48,x]
F[53,x] = F[12,x]*F[19,x]
F[54,x] = 2*F[15,x]+F[55,x]+F[59,x]
F[55,x] = F[12,x]*F[56,x]
F[56,x] = F[57,x]+F[58,x]
F[57,x] = F[37,x]
F[58,x] = F[54,x]
F[59,x] = F[12,x]*F[47,x]
F[60,x] = F[12,x]*F[61,x]
F[61,x] = F[62,x]+F[75,x]
F[62,x] = F[19,x]+F[63,x]
F[63,x] = F[15,x]+F[64,x]+F[74,x]
F[64,x] = F[12,x]*F[65,x]
F[65,x] = F[66,x]+F[68,x]
F[66,x] = F[10,x]+F[67,x]
F[67,x] = F[38,x]
F[68,x] = F[63,x]+F[69,x]
F[69,x] = 2*F[15,x]+F[59,x]+F[70,x]
F[70,x] = F[12,x]*F[71,x]
F[71,x] = F[72,x]+F[73,x]
F[72,x] = F[67,x]
F[73,x] = F[69,x]
F[74,x] = F[12,x]*F[62,x]
F[75,x] = F[48,x]+F[76,x]
F[76,x] = F[15,x]+F[55,x]+F[77,x]+F[78,x]
F[77,x] = F[12,x]*F[63,x]
F[78,x] = F[12,x]*F[75,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{18}\! \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_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{12}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{29}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{9}\! \left(x \right)
F_{22}\! \left(x \right) = F_{19}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{15}\! \left(x \right)+F_{24}\! \left(x \right)+F_{28}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)
F_{26}\! \left(x \right) = F_{10}\! \left(x \right)
F_{27}\! \left(x \right) = F_{23}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{22}\! \left(x \right)
F_{29}\! \left(x \right) = F_{15}\! \left(x \right)+F_{30}\! \left(x \right)+F_{60}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{39}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{6}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{39}\! \left(x \right) = F_{29}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{41}\! \left(x \right)+F_{45}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{44}\! \left(x \right)
F_{43}\! \left(x \right) = F_{33}\! \left(x \right)
F_{44}\! \left(x \right) = F_{40}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{54}\! \left(x \right)
F_{48}\! \left(x \right) = F_{15}\! \left(x \right)+F_{49}\! \left(x \right)+F_{53}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{52}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right)
F_{52}\! \left(x \right) = F_{48}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{19}\! \left(x \right)
F_{54}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{55}\! \left(x \right)+F_{59}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{58}\! \left(x \right)
F_{57}\! \left(x \right) = F_{37}\! \left(x \right)
F_{58}\! \left(x \right) = F_{54}\! \left(x \right)
F_{59}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{75}\! \left(x \right)
F_{62}\! \left(x \right) = F_{19}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{15}\! \left(x \right)+F_{64}\! \left(x \right)+F_{74}\! \left(x \right)
F_{64}\! \left(x \right) = F_{12}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{68}\! \left(x \right)
F_{66}\! \left(x \right) = F_{10}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{38}\! \left(x \right)
F_{68}\! \left(x \right) = F_{63}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{59}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{12}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)+F_{73}\! \left(x \right)
F_{72}\! \left(x \right) = F_{67}\! \left(x \right)
F_{73}\! \left(x \right) = F_{69}\! \left(x \right)
F_{74}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \left(x \right)
F_{75}\! \left(x \right) = F_{48}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{15}\! \left(x \right)+F_{55}\! \left(x \right)+F_{77}\! \left(x \right)+F_{78}\! \left(x \right)
F_{77}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \left(x \right)
F_{78}\! \left(x \right) = F_{12}\! \left(x \right) F_{75}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_18(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_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_12(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_17(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_10(x)*F_12(x))
Eq(F_17(x), F_12(x)*F_13(x))
Eq(F_18(x), F_19(x) + F_29(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_12(x)*F_21(x))
Eq(F_21(x), F_22(x) + F_9(x))
Eq(F_22(x), F_19(x) + F_23(x))
Eq(F_23(x), F_15(x) + F_24(x) + F_28(x))
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_26(x) + F_27(x))
Eq(F_26(x), F_10(x))
Eq(F_27(x), F_23(x))
Eq(F_28(x), F_12(x)*F_22(x))
Eq(F_29(x), F_15(x) + F_30(x) + F_60(x))
Eq(F_30(x), F_12(x)*F_31(x))
Eq(F_31(x), F_32(x) + F_39(x))
Eq(F_32(x), F_33(x) + F_6(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_12(x) + F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_12(x)*F_36(x))
Eq(F_39(x), F_29(x) + F_40(x))
Eq(F_40(x), 2*F_15(x) + F_41(x) + F_45(x))
Eq(F_41(x), F_12(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_44(x))
Eq(F_43(x), F_33(x))
Eq(F_44(x), F_40(x))
Eq(F_45(x), F_12(x)*F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_48(x) + F_54(x))
Eq(F_48(x), F_15(x) + F_49(x) + F_53(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_52(x))
Eq(F_51(x), F_12(x))
Eq(F_52(x), F_48(x))
Eq(F_53(x), F_12(x)*F_19(x))
Eq(F_54(x), 2*F_15(x) + F_55(x) + F_59(x))
Eq(F_55(x), F_12(x)*F_56(x))
Eq(F_56(x), F_57(x) + F_58(x))
Eq(F_57(x), F_37(x))
Eq(F_58(x), F_54(x))
Eq(F_59(x), F_12(x)*F_47(x))
Eq(F_60(x), F_12(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_75(x))
Eq(F_62(x), F_19(x) + F_63(x))
Eq(F_63(x), F_15(x) + F_64(x) + F_74(x))
Eq(F_64(x), F_12(x)*F_65(x))
Eq(F_65(x), F_66(x) + F_68(x))
Eq(F_66(x), F_10(x) + F_67(x))
Eq(F_67(x), F_38(x))
Eq(F_68(x), F_63(x) + F_69(x))
Eq(F_69(x), 2*F_15(x) + F_59(x) + F_70(x))
Eq(F_70(x), F_12(x)*F_71(x))
Eq(F_71(x), F_72(x) + F_73(x))
Eq(F_72(x), F_67(x))
Eq(F_73(x), F_69(x))
Eq(F_74(x), F_12(x)*F_62(x))
Eq(F_75(x), F_48(x) + F_76(x))
Eq(F_76(x), F_15(x) + F_55(x) + F_77(x) + F_78(x))
Eq(F_77(x), F_12(x)*F_63(x))
Eq(F_78(x), F_12(x)*F_75(x))
Pack JSON:
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Specification JSON:
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