0132_0213_1230_1302
Counting sequence:
1, 1, 2, 6, 20, 63, 187, 534, 1491, 4111, 11255, 30686, 83442, 226471, 613760, 1661268, 4491542, 12131289, 32734501, 88250676, 237720585, 639841621, 1720892657, 4625170736, 12422534100, 33343840573, 89445299822, 239799586194, 642539728136, 1720770878691, 4606040479615, 12323201994306, 32954852233647, 88089363463051, 235365977693915, 628618779476978, 1678268250772902, 4478909433090499, 11948852539208396, 31866102196414656, 84954423594964970, 226413623211387741, 603230840362990537, 1606695626226052416, 4278161829036779205, 11388290504700653233, 30306905825792671157, 80632514751043463204, 214470705621708535656, 570319715918828491321, 1516228716354645162650, 4030047141999965320926, 10709214561989957547452, 28451821392149169237831, 75573622306650629166595, 200696938756202693345838, 532873500954964297120587, 1414564591859310761679559, 3754387050881822832657791, 9962655861812101706847686, 26432191661373459354805530, 70115693201739263587774111, 185961599055318316032210392, 493127463219206651149730796, 1307449157255800552715907710, 3465946749253700786253893793, 9186542945968320228514550317, 24345410914334809387440183564, 64508724418623737975788500801, 170906137380615745177722391501, 452726778218873889428787388985, 1199100093723877758085068846296, 3175534101800724497884297644732, 8408558111774319239765030499061, 22262347334962338620944534588550, 58934049297336989317472589061098, 155994289668281402015151539239472, 412856721636981242084612932794219, 1092545091919851627624901664906311, 2890878302224667525592104375074266, 7648419842383243300171233993999015, 20233271559710245543179052894817731, 53519735813473950483118468020453347, 141552068476105794199176525868641626, 374346526424299539839659090361766990, 989891548629766960202544512843443003, 2617330176154467637690957770272614052, 6919701112069060708756534996659768024, 18292577500069522459314282493658740946, 48352842275954505825507012107487234981, 127799577651222184515433217424363249745, 337751964760181617059151331329112587416, 892540910553303185178870386457944442237, 2358418474589199924689649966566120451241, 6231273042358732029009800318909646113405, 16462528532230830496486723268649105778508, 43489137664420827022771565516827305684432, 114876231911549018924644589092715426767729, 303420775311691267247290853164177186630706, 801358398297402526952797308797508153662646, 2116290115160684508521680437018565123955300
Generating function in Maple syntax:
(x^7-5*x^6+15*x^5-30*x^4+38*x^3-25*x^2+8*x-1)/(x^2-3*x+1)^2/(x-1)^3
Generating function in latex syntax:
\frac{x^{7}-5 x^{6}+15 x^{5}-30 x^{4}+38 x^{3}-25 x^{2}+8 x -1}{\left(x^{2}-3 x +1\right)^{2} \left(x -1\right)^{3}}
Generating function in sympy syntax:
(x**7 - 5*x**6 + 15*x**5 - 30*x**4 + 38*x**3 - 25*x**2 + 8*x - 1)/((x - 1)**3*(x**2 - 3*x + 1)**2)
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)^2*(x-1)^3*F(x)-x^7+5*x^6-15*x^5+30*x^4-38*x^3+25*x^2-8*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)-x^{7}+5 x^{6}-15 x^{5}+30 x^{4}-38 x^{3}+25 x^{2}-8 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/50*((-15*n-1)*5^(1/2)+35*n+25)*(3/2-1/2*5^(1/2))^(-n)+1/50*((15*n+1)*5^(1/2)+35*n+25)*(3/2+1/2*5^(1/2))^(-n)-1/2*n^2+1/2*n-1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(-15 n -1\right) \sqrt{5}+35 n +25\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{50}+\\\frac{\left(\left(15 n +1\right) \sqrt{5}+35 n +25\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{50}-\frac{n^{2}}{2}+\frac{n}{2}-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) = 20
a(5) = 63
a(6) = 187
a(7) = 534
a(n+4) = -a(n)+6*a(n+1)-11*a(n+2)+6*a(n+3)-1/2*n*(n+3), 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) = 20
a \! \left(5\right) = 63
a \! \left(6\right) = 187
a \! \left(7\right) = 534
a \! \left(n +4\right) = -a \! \left(n \right)+6 a \! \left(n +1\right)-11 a \! \left(n +2\right)+6 a \! \left(n +3\right)-\frac{n \left(n +3\right)}{2}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15276/
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[23,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[14,x]+F[21,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[1,x]+F[12,x]
F[18,x] = F[14,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[12,x]*F[14,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[13,x]
F[23,x] = F[24,x]+F[38,x]
F[24,x] = F[25,x]
F[25,x] = F[12,x]*F[26,x]
F[26,x] = F[27,x]+F[30,x]
F[27,x] = F[1,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[12,x]*F[27,x]
F[30,x] = F[24,x]+F[31,x]
F[31,x] = F[32,x]+F[33,x]+F[37,x]
F[32,x] = 0
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[35,x]+F[36,x]
F[35,x] = F[28,x]
F[36,x] = F[31,x]
F[37,x] = F[12,x]*F[30,x]
F[38,x] = F[32,x]+F[39,x]+F[66,x]
F[39,x] = F[12,x]*F[40,x]
F[40,x] = F[41,x]+F[45,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[10,x]
F[45,x] = F[38,x]+F[46,x]
F[46,x] = 2*F[32,x]+F[47,x]+F[51,x]
F[47,x] = F[12,x]*F[48,x]
F[48,x] = F[49,x]+F[50,x]
F[49,x] = F[42,x]
F[50,x] = F[46,x]
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[32,x]+F[54,x]+F[64,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] = 2*F[32,x]+F[51,x]+F[60,x]
F[60,x] = F[12,x]*F[61,x]
F[61,x] = F[62,x]+F[63,x]
F[62,x] = F[57,x]
F[63,x] = F[59,x]
F[64,x] = F[12,x]*F[65,x]
F[65,x] = F[24,x]+F[53,x]
F[66,x] = F[12,x]*F[67,x]
F[67,x] = F[65,x]+F[68,x]
F[68,x] = F[69,x]+F[82,x]
F[69,x] = F[32,x]+F[33,x]+F[70,x]
F[70,x] = F[12,x]*F[71,x]
F[71,x] = F[72,x]+F[79,x]
F[72,x] = F[24,x]+F[73,x]
F[73,x] = F[32,x]+F[74,x]+F[78,x]
F[74,x] = F[12,x]*F[75,x]
F[75,x] = F[76,x]+F[77,x]
F[76,x] = F[12,x]
F[77,x] = F[73,x]
F[78,x] = F[12,x]*F[24,x]
F[79,x] = F[69,x]+F[80,x]
F[80,x] = F[81,x]
F[81,x] = F[12,x]*F[69,x]
F[82,x] = F[83,x]
F[83,x] = F[12,x]*F[68,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_{23}\! \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_{14}\! \left(x \right)+F_{21}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{38}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{30}\! \left(x \right)
F_{27}\! \left(x \right) = F_{1}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{30}\! \left(x \right) = F_{24}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)+F_{37}\! \left(x \right)
F_{32}\! \left(x \right) = 0
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_{36}\! \left(x \right)
F_{35}\! \left(x \right) = F_{28}\! \left(x \right)
F_{36}\! \left(x \right) = F_{31}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{30}\! \left(x \right)
F_{38}\! \left(x \right) = F_{32}\! \left(x \right)+F_{39}\! \left(x \right)+F_{66}\! \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_{45}\! \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_{10}\! \left(x \right)
F_{45}\! \left(x \right) = F_{38}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = 2 F_{32}\! \left(x \right)+F_{47}\! \left(x \right)+F_{51}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{50}\! \left(x \right)
F_{49}\! \left(x \right) = F_{42}\! \left(x \right)
F_{50}\! \left(x \right) = F_{46}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{32}\! \left(x \right)+F_{54}\! \left(x \right)+F_{64}\! \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) = 2 F_{32}\! \left(x \right)+F_{51}\! \left(x \right)+F_{60}\! \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_{63}\! \left(x \right)
F_{62}\! \left(x \right) = F_{57}\! \left(x \right)
F_{63}\! \left(x \right) = F_{59}\! \left(x \right)
F_{64}\! \left(x \right) = F_{12}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{24}\! \left(x \right)+F_{53}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{65}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)+F_{82}\! \left(x \right)
F_{69}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \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_{79}\! \left(x \right)
F_{72}\! \left(x \right) = F_{24}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{32}\! \left(x \right)+F_{74}\! \left(x \right)+F_{78}\! \left(x \right)
F_{74}\! \left(x \right) = F_{12}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)+F_{77}\! \left(x \right)
F_{76}\! \left(x \right) = F_{12}\! \left(x \right)
F_{77}\! \left(x \right) = F_{73}\! \left(x \right)
F_{78}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{79}\! \left(x \right) = F_{69}\! \left(x \right)+F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{12}\! \left(x \right) F_{69}\! \left(x \right)
F_{82}\! \left(x \right) = F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{12}\! \left(x \right) F_{68}\! \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_23(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_14(x) + F_21(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_1(x) + F_12(x))
Eq(F_18(x), F_14(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_12(x)*F_14(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_12(x)*F_13(x))
Eq(F_23(x), F_24(x) + F_38(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_12(x)*F_26(x))
Eq(F_26(x), F_27(x) + F_30(x))
Eq(F_27(x), F_1(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_12(x)*F_27(x))
Eq(F_30(x), F_24(x) + F_31(x))
Eq(F_31(x), F_32(x) + F_33(x) + F_37(x))
Eq(F_32(x), 0)
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_36(x))
Eq(F_35(x), F_28(x))
Eq(F_36(x), F_31(x))
Eq(F_37(x), F_12(x)*F_30(x))
Eq(F_38(x), F_32(x) + F_39(x) + F_66(x))
Eq(F_39(x), F_12(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_45(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_10(x))
Eq(F_45(x), F_38(x) + F_46(x))
Eq(F_46(x), 2*F_32(x) + F_47(x) + F_51(x))
Eq(F_47(x), F_12(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_50(x))
Eq(F_49(x), F_42(x))
Eq(F_50(x), F_46(x))
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_32(x) + F_54(x) + F_64(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), 2*F_32(x) + F_51(x) + F_60(x))
Eq(F_60(x), F_12(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_63(x))
Eq(F_62(x), F_57(x))
Eq(F_63(x), F_59(x))
Eq(F_64(x), F_12(x)*F_65(x))
Eq(F_65(x), F_24(x) + F_53(x))
Eq(F_66(x), F_12(x)*F_67(x))
Eq(F_67(x), F_65(x) + F_68(x))
Eq(F_68(x), F_69(x) + F_82(x))
Eq(F_69(x), F_32(x) + F_33(x) + F_70(x))
Eq(F_70(x), F_12(x)*F_71(x))
Eq(F_71(x), F_72(x) + F_79(x))
Eq(F_72(x), F_24(x) + F_73(x))
Eq(F_73(x), F_32(x) + F_74(x) + F_78(x))
Eq(F_74(x), F_12(x)*F_75(x))
Eq(F_75(x), F_76(x) + F_77(x))
Eq(F_76(x), F_12(x))
Eq(F_77(x), F_73(x))
Eq(F_78(x), F_12(x)*F_24(x))
Eq(F_79(x), F_69(x) + F_80(x))
Eq(F_80(x), F_81(x))
Eq(F_81(x), F_12(x)*F_69(x))
Eq(F_82(x), F_83(x))
Eq(F_83(x), F_12(x)*F_68(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": 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:
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