0123_0213_0231_1032_3012
Counting sequence:
1, 1, 2, 6, 19, 54, 148, 399, 1066, 2832, 7495, 19782, 52108, 137055, 360082, 945240, 2479735, 6502158, 17043124, 44659983, 117002362, 306478176, 802694311, 2102129046, 5504741404, 14414192319, 37742029858, 98820285864, 258735604951, 677420083422, 1773591754180, 4643489396847, 12157144871818, 31828482089520, 83329375138567, 218161790809830, 571160292258220, 1495327675899423, 3914839915309234, 10249226429766648, 26832908093467447, 70249635289589166, 183916272653206996, 481499732425845711, 1260584024135957914, 3300254539005283584, 8640183990926403943, 22620306229866950454, 59220752290860491836, 155041985827086613887, 405905275559143527490, 1062673981587832323912, 2782116950679330154903, 7283677433400111562110, 19068916475420911374052, 49923074244662436245295, 130700310762166024732330, 342177867049034892692688, 895833308399337162827719, 2345322094177773614754438, 6140133046191577719363532, 16075077188512147619192031, 42085098807575241289924306, 110180219810674328554004376, 288455561777369248978935799, 755186467827276427596496974, 1977103846316146052237943028, 5176125080344533765972107919, 13551271413164199319387932346, 35477689196041552339610792352, 92881796248747433994282651175, 243167699697774702232913574102, 636621303139724577883810896988, 1666696210311694841777224768575, 4363467328975951568165274712162, 11423705778977343104153421974760, 29907650012678444227164636425815, 78299244268502722543079777730078, 204990082811719189333553277619204, 536671004204433777320537216837103, 1405022929877140006353972696311242, 3678397785578101969193209518934896, 9630170427159397356129313154169991, 25212113496504553009002044530928166, 66006170063563187490491449613320684, 172806396696602861101701562658446239, 452413020031081099093071755060842738, 1184432663406311842734430735921731384, 3100884970207197242224054519499650231, 8118222247253965510165400956167816942, 21253781771632070540727484616184995860, 55643123067796988616927725426749561167, 145675587432068380319877036732788468698, 381383639229027122362346074909065407040, 998475330256250926806446568269306876647, 2614042351542201538135564390448653471350, 6843651724375305447757388124176250034300, 17916912821593618325450883024279289625343, 46907086740425356569223827033060004829314, 122804347399722065463477730243697496837768, 321505955458820067983723628035626029634327
Generating function in Maple syntax:
(2*x^6-4*x^5-3*x^3+8*x^2-5*x+1)/(x-1)/(2*x-1)/(x^2-3*x+1)
Generating function in latex syntax:
\frac{2 x^{6}-4 x^{5}-3 x^{3}+8 x^{2}-5 x +1}{\left(x -1\right) \left(2 x -1\right) \left(x^{2}-3 x +1\right)}
Generating function in sympy syntax:
(2*x**6 - 4*x**5 - 3*x**3 + 8*x**2 - 5*x + 1)/((x - 1)*(2*x - 1)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
-(x-1)*(2*x-1)*(x^2-3*x+1)*F(x)+2*x^6-4*x^5-3*x^3+8*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right) \left(2 x -1\right) \left(x^{2}-3 x +1\right) F \! \left(x \right)+2 x^{6}-4 x^{5}-3 x^{3}+8 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/20*(18*5^(1/2)-30)*(3/2-1/2*5^(1/2))^(-n)+1/20*(-18*5^(1/2)-30)*(3/2+1/2*5^(1/2))^(-n)-1/4*2^n-1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(18 \sqrt{5}-30\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\frac{\left(-18 \sqrt{5}-30\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{20}-\frac{2^{n}}{4}-1 & \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) = 54
a(6) = 148
a(n+3) = 2*a(n)-7*a(n+1)+5*a(n+2)-1, 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) = 54
a \! \left(6\right) = 148
a \! \left(n +3\right) = 2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)-1, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16713/
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[5,x]
F[4,x] = x
F[5,x] = F[14,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[7,x]
F[14,x] = F[15,x]+F[32,x]
F[15,x] = F[16,x]
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[10,x]+F[18,x]
F[18,x] = F[15,x]+F[19,x]
F[19,x] = F[20,x]+F[21,x]+F[31,x]
F[20,x] = 0
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[26,x]
F[23,x] = F[24,x]+F[4,x]
F[24,x] = F[25,x]
F[25,x] = F[11,x]*F[4,x]
F[26,x] = F[27,x]+F[29,x]
F[27,x] = F[20,x]+F[21,x]+F[28,x]
F[28,x] = F[15,x]*F[4,x]
F[29,x] = F[30,x]
F[30,x] = F[19,x]*F[4,x]
F[31,x] = F[18,x]*F[4,x]
F[32,x] = F[20,x]+F[33,x]+F[72,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]+F[44,x]
F[35,x] = F[36,x]+F[39,x]
F[36,x] = F[37,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[1,x]+F[4,x]
F[39,x] = F[20,x]+F[40,x]+F[42,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[4,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[11,x]
F[44,x] = F[45,x]+F[64,x]
F[45,x] = F[20,x]+F[33,x]+F[46,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[15,x]+F[48,x]
F[48,x] = F[20,x]+F[28,x]+F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[53,x]
F[51,x] = F[4,x]+F[52,x]
F[52,x] = F[20,x]+F[25,x]+F[40,x]
F[53,x] = F[48,x]+F[54,x]
F[54,x] = F[20,x]+F[30,x]+F[55,x]+F[62,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[57,x]+F[59,x]
F[57,x] = F[58,x]
F[58,x] = x^2
F[59,x] = F[60,x]
F[60,x] = 2*F[20,x]+F[55,x]+F[61,x]
F[61,x] = F[4,x]*F[48,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[48,x]
F[64,x] = F[20,x]+F[62,x]+F[65,x]+F[70,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[67,x]+F[68,x]
F[67,x] = F[58,x]
F[68,x] = F[69,x]
F[69,x] = 2*F[20,x]+F[61,x]+F[65,x]
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[19,x]
F[72,x] = F[4,x]*F[73,x]
F[73,x] = F[74,x]+F[76,x]
F[74,x] = F[15,x]+F[75,x]
F[75,x] = F[20,x]+F[31,x]+F[49,x]
F[76,x] = F[77,x]
F[77,x] = F[20,x]+F[21,x]+F[78,x]
F[78,x] = F[4,x]*F[79,x]
F[79,x] = F[18,x]+F[80,x]
F[80,x] = F[77,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_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{14}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{32}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{10}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{15}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{31}\! \left(x \right)
F_{20}\! \left(x \right) = 0
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{26}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{29}\! \left(x \right)
F_{27}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{20}\! \left(x \right)+F_{33}\! \left(x \right)+F_{72}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{44}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{39}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{20}\! \left(x \right)+F_{40}\! \left(x \right)+F_{42}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{11}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{64}\! \left(x \right)
F_{45}\! \left(x \right) = F_{20}\! \left(x \right)+F_{33}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{15}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{20}\! \left(x \right)+F_{28}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{53}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{20}\! \left(x \right)+F_{25}\! \left(x \right)+F_{40}\! \left(x \right)
F_{53}\! \left(x \right) = F_{48}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{20}\! \left(x \right)+F_{30}\! \left(x \right)+F_{55}\! \left(x \right)+F_{62}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{59}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = x^{2}
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{55}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{48}\! \left(x \right)
F_{64}\! \left(x \right) = F_{20}\! \left(x \right)+F_{62}\! \left(x \right)+F_{65}\! \left(x \right)+F_{70}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)+F_{68}\! \left(x \right)
F_{67}\! \left(x \right) = F_{58}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{61}\! \left(x \right)+F_{65}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{19}\! \left(x \right)
F_{72}\! \left(x \right) = F_{4}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)+F_{76}\! \left(x \right)
F_{74}\! \left(x \right) = F_{15}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{20}\! \left(x \right)+F_{31}\! \left(x \right)+F_{49}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{4}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{18}\! \left(x \right)+F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{77}\! \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_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_14(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_7(x))
Eq(F_14(x), F_15(x) + F_32(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_10(x) + F_18(x))
Eq(F_18(x), F_15(x) + F_19(x))
Eq(F_19(x), F_20(x) + F_21(x) + F_31(x))
Eq(F_20(x), 0)
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_26(x))
Eq(F_23(x), F_24(x) + F_4(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_11(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_29(x))
Eq(F_27(x), F_20(x) + F_21(x) + F_28(x))
Eq(F_28(x), F_15(x)*F_4(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_19(x)*F_4(x))
Eq(F_31(x), F_18(x)*F_4(x))
Eq(F_32(x), F_20(x) + F_33(x) + F_72(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x) + F_44(x))
Eq(F_35(x), F_36(x) + F_39(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_1(x) + F_4(x))
Eq(F_39(x), F_20(x) + F_40(x) + F_42(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_4(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_11(x))
Eq(F_44(x), F_45(x) + F_64(x))
Eq(F_45(x), F_20(x) + F_33(x) + F_46(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_15(x) + F_48(x))
Eq(F_48(x), F_20(x) + F_28(x) + F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_53(x))
Eq(F_51(x), F_4(x) + F_52(x))
Eq(F_52(x), F_20(x) + F_25(x) + F_40(x))
Eq(F_53(x), F_48(x) + F_54(x))
Eq(F_54(x), F_20(x) + F_30(x) + F_55(x) + F_62(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_57(x) + F_59(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), x**2)
Eq(F_59(x), F_60(x))
Eq(F_60(x), 2*F_20(x) + F_55(x) + F_61(x))
Eq(F_61(x), F_4(x)*F_48(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_48(x))
Eq(F_64(x), F_20(x) + F_62(x) + F_65(x) + F_70(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_67(x) + F_68(x))
Eq(F_67(x), F_58(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), 2*F_20(x) + F_61(x) + F_65(x))
Eq(F_70(x), F_4(x)*F_71(x))
Eq(F_71(x), F_19(x))
Eq(F_72(x), F_4(x)*F_73(x))
Eq(F_73(x), F_74(x) + F_76(x))
Eq(F_74(x), F_15(x) + F_75(x))
Eq(F_75(x), F_20(x) + F_31(x) + F_49(x))
Eq(F_76(x), F_77(x))
Eq(F_77(x), F_20(x) + F_21(x) + F_78(x))
Eq(F_78(x), F_4(x)*F_79(x))
Eq(F_79(x), F_18(x) + F_80(x))
Eq(F_80(x), F_77(x))
Pack JSON:
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Specification JSON:
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