0132_0321_1302_2130_3021
Counting sequence:
1, 1, 2, 6, 19, 54, 141, 353, 874, 2177, 5490, 14016, 36127, 93726, 244139, 637429, 1666446, 4359663, 11409634, 29865532, 78182305, 204675606, 535837427, 1402828071, 3672636434, 9615068879, 25172555576, 65902580648, 172535166269, 451702894812, 1182573491195, 3096017547771, 8105479116652, 21220419761791, 55555780122904, 145446920555154, 380784981484281, 996908023832308, 2609939089939529, 6832909245904767, 17888788647684160, 46833456697047261, 122611581443346552, 321001287632869886, 840392281455128299, 2200175556732367004, 5760134388741810561, 15080227609492887393, 39480548439736658164, 103361417709716876397, 270603704689413741950, 708449696358524100826, 1854745384386158291127, 4855786456799950481106, 12712613986013692837369, 33282055501241127691429, 87133552517709689871166, 228118602051887941528653, 597222253637954134292174, 1563548158861974460894452, 4093422222947969247905315, 10716718509981933282301466, 28056733306997830598443127, 73453481411011558512434201, 192303710926036844938226114, 503457651367098976301569179, 1318069243175260083965762846, 3450750078158681275594955088, 9034180991300783742818290309, 23651792895743669952859053682, 61921197695930226115757956255, 162111800192047008394413845931, 424414202880210799067482555302, 1111130808448585388808032734171, 2908978222465545367356614499284, 7615803858948050713261809551004, 19938433354378606772428812873601, 52199496204187769604024627719448, 136660055258184702039645068861319, 357780669570366336514910577365087, 936681953452914307505086661655520, 2452265190788376586000349405940971, 6420113618912215450495961554421652, 16808075665948269765487535255489766, 44004113378932593845966644210121629, 115204264470849511772412397372853904, 301608680033615941471270547906320181, 789621775629998312641399246343884483, 2067256646856378996452927191123005204, 5412148164939138676717382327022693417, 14169187847961037033699219789942523860, 37095415378943972424380277042802209586, 97117058288870880239441611338461314927, 254255759487668668293944556972578819736, 665650220174135124642392059579272099149, 1742694901034736705633231621765234298629, 4562434482930074992257302805716427479336, 11944608547755488271138676795384044679193, 31271391160336389821158727580435702950714, 81869564933253681192337505945923060413422, 214337303639424653755853790257333474373275
Generating function in Maple syntax:
(x^7-5*x^6+13*x^5-23*x^4+29*x^3-20*x^2+7*x-1)/(x^2-3*x+1)/(x-1)^5
Generating function in latex syntax:
\frac{x^{7}-5 x^{6}+13 x^{5}-23 x^{4}+29 x^{3}-20 x^{2}+7 x -1}{\left(x^{2}-3 x +1\right) \left(x -1\right)^{5}}
Generating function in sympy syntax:
(x**7 - 5*x**6 + 13*x**5 - 23*x**4 + 29*x**3 - 20*x**2 + 7*x - 1)/((x - 1)**5*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x-1)^5*F(x)-x^7+5*x^6-13*x^5+23*x^4-29*x^3+20*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x -1\right)^{5} F \! \left(x \right)-x^{7}+5 x^{6}-13 x^{5}+23 x^{4}-29 x^{3}+20 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/120*(72*5^(1/2)-120)*(3/2-1/2*5^(1/2))^(-n)+1/120*(-72*5^(1/2)-120)*(3/2+1/2*5^(1/2))^(-n)+1/24*n^4-1/12*n^3-1/24*n^2-11/12*n+2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(72 \sqrt{5}-120\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{120}+\frac{\left(-72 \sqrt{5}-120\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{120}+\frac{n^{4}}{24}-\frac{n^{3}}{12}-\\\frac{n^{2}}{24}-\frac{11 n}{12}+2 & \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) = 141
a(7) = 353
a(n+2) = -1/24*n^4-1/12*n^3+13/24*n^2-a(n)+3*a(n+1)+19/12*n-1, n >= 8
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) = 141
a \! \left(7\right) = 353
a \! \left(n +2\right) = -\frac{n^{4}}{24}-\frac{n^{3}}{12}+\frac{13 n^{2}}{24}-a \! \left(n \right)+3 a \! \left(n +1\right)+\frac{19 n}{12}-1, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19536/
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[20,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[10,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[19,x]
F[15,x] = 0
F[16,x] = F[12,x]*F[17,x]
F[17,x] = F[10,x]+F[18,x]
F[18,x] = F[16,x]
F[19,x] = F[12,x]*F[13,x]
F[20,x] = F[21,x]+F[38,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[24,x]+F[31,x]
F[24,x] = F[1,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[12,x]*F[27,x]
F[27,x] = F[28,x]+F[9,x]
F[28,x] = F[10,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[12,x]*F[28,x]
F[31,x] = F[21,x]+F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[21,x]+F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[12,x]*F[35,x]
F[38,x] = F[15,x]+F[39,x]+F[64,x]
F[39,x] = F[12,x]*F[40,x]
F[40,x] = F[41,x]+F[48,x]
F[41,x] = F[42,x]+F[6,x]
F[42,x] = F[43,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[17,x]+F[45,x]
F[45,x] = F[18,x]+F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[12,x]*F[45,x]
F[48,x] = F[38,x]+F[49,x]
F[49,x] = F[50,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[53,x]+F[62,x]
F[53,x] = F[15,x]+F[54,x]+F[60,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[56,x]+F[58,x]
F[56,x] = F[10,x]+F[57,x]
F[57,x] = F[43,x]
F[58,x] = F[53,x]+F[59,x]
F[59,x] = F[50,x]
F[60,x] = F[12,x]*F[61,x]
F[61,x] = F[21,x]+F[53,x]
F[62,x] = F[63,x]
F[63,x] = F[12,x]*F[52,x]
F[64,x] = F[12,x]*F[65,x]
F[65,x] = F[61,x]+F[66,x]
F[66,x] = F[67,x]+F[74,x]
F[67,x] = F[68,x]
F[68,x] = F[12,x]*F[69,x]
F[69,x] = F[70,x]
F[70,x] = F[10,x]+F[71,x]
F[71,x] = F[72,x]
F[72,x] = F[12,x]*F[73,x]
F[73,x] = F[17,x]
F[74,x] = 2*F[15,x]+F[75,x]+F[84,x]
F[75,x] = F[12,x]*F[76,x]
F[76,x] = F[77,x]
F[77,x] = F[18,x]+F[78,x]
F[78,x] = F[79,x]
F[79,x] = F[12,x]*F[80,x]
F[80,x] = F[81,x]
F[81,x] = F[18,x]+F[82,x]
F[82,x] = F[83,x]
F[83,x] = F[12,x]*F[81,x]
F[84,x] = F[12,x]*F[66,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_{20}\! \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_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{19}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{10}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{38}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{31}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{9}\! \left(x \right)
F_{28}\! \left(x \right) = F_{10}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{21}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{21}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{38}\! \left(x \right) = F_{15}\! \left(x \right)+F_{39}\! \left(x \right)+F_{64}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{48}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{6}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{17}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{18}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{48}\! \left(x \right) = F_{38}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{62}\! \left(x \right)
F_{53}\! \left(x \right) = F_{15}\! \left(x \right)+F_{54}\! \left(x \right)+F_{60}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{58}\! \left(x \right)
F_{56}\! \left(x \right) = F_{10}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{43}\! \left(x \right)
F_{58}\! \left(x \right) = F_{53}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{50}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{21}\! \left(x \right)+F_{53}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{64}\! \left(x \right) = F_{12}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{61}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)+F_{74}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{12}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{10}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{12}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{17}\! \left(x \right)
F_{74}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{75}\! \left(x \right)+F_{84}\! \left(x \right)
F_{75}\! \left(x \right) = F_{12}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{18}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{12}\! \left(x \right) F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{18}\! \left(x \right)+F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{12}\! \left(x \right) F_{81}\! \left(x \right)
F_{84}\! \left(x \right) = F_{12}\! \left(x \right) F_{66}\! \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_20(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_10(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_19(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_12(x)*F_17(x))
Eq(F_17(x), F_10(x) + F_18(x))
Eq(F_18(x), F_16(x))
Eq(F_19(x), F_12(x)*F_13(x))
Eq(F_20(x), F_21(x) + F_38(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_12(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_31(x))
Eq(F_24(x), F_1(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_12(x)*F_27(x))
Eq(F_27(x), F_28(x) + F_9(x))
Eq(F_28(x), F_10(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_12(x)*F_28(x))
Eq(F_31(x), F_21(x) + F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_21(x) + F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_12(x)*F_35(x))
Eq(F_38(x), F_15(x) + F_39(x) + F_64(x))
Eq(F_39(x), F_12(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_48(x))
Eq(F_41(x), F_42(x) + F_6(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_17(x) + F_45(x))
Eq(F_45(x), F_18(x) + F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_12(x)*F_45(x))
Eq(F_48(x), F_38(x) + F_49(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_53(x) + F_62(x))
Eq(F_53(x), F_15(x) + F_54(x) + F_60(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_58(x))
Eq(F_56(x), F_10(x) + F_57(x))
Eq(F_57(x), F_43(x))
Eq(F_58(x), F_53(x) + F_59(x))
Eq(F_59(x), F_50(x))
Eq(F_60(x), F_12(x)*F_61(x))
Eq(F_61(x), F_21(x) + F_53(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_12(x)*F_52(x))
Eq(F_64(x), F_12(x)*F_65(x))
Eq(F_65(x), F_61(x) + F_66(x))
Eq(F_66(x), F_67(x) + F_74(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_12(x)*F_69(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_10(x) + F_71(x))
Eq(F_71(x), F_72(x))
Eq(F_72(x), F_12(x)*F_73(x))
Eq(F_73(x), F_17(x))
Eq(F_74(x), 2*F_15(x) + F_75(x) + F_84(x))
Eq(F_75(x), F_12(x)*F_76(x))
Eq(F_76(x), F_77(x))
Eq(F_77(x), F_18(x) + F_78(x))
Eq(F_78(x), F_79(x))
Eq(F_79(x), F_12(x)*F_80(x))
Eq(F_80(x), F_81(x))
Eq(F_81(x), F_18(x) + F_82(x))
Eq(F_82(x), F_83(x))
Eq(F_83(x), F_12(x)*F_81(x))
Eq(F_84(x), F_12(x)*F_66(x))
Pack JSON:
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Specification JSON:
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