0123_0213_1230_1320_3012
Counting sequence:
1, 1, 2, 6, 19, 53, 127, 277, 573, 1152, 2284, 4503, 8868, 17483, 34534, 68358, 135573, 269325, 535761, 1066941, 2126607, 4241632, 8464774, 16899951, 33752414, 67428483, 134734032, 269269862, 538219863, 1075923157, 2151011619, 4300673973, 8599166273, 17194804208, 34383901344, 68758570583, 137502205688, 274980247915, 549921401466, 1099779550166, 2199456758777, 4398747929357, 8797227935669, 17594022367421, 35187343315859, 70373551717488, 140745267111162, 281487562994447, 562970318448178, 1125932858139587, 2251853129994404, 4503685894960918, 9007338838623803, 18014624360937333, 36029162454283079, 72058185324682133, 144116144797907653, 288231924160494880, 576463257034234196, 1152925557346415191, 2305849566684045772, 4611696628637279307, 9223389204534988814, 18446771851599624166, 36893533092989355229, 73787049018298495693, 147574070258706917017, 295148095571843580157, 590296118420226869143, 1180592119171423232192, 2361184047950355708014, 4722367787839190196463, 9444735077224368461958, 18889469347933203820483, 37778937390896862655864, 75557872670308647274758, 151115741924162671581343, 302231478320385642213845, 604462947696376960569451, 1208925880920419896393141, 2417851738424111444246217, 4835703438959160515273232, 9671406816612530308856584, 19342813534030207522876311, 38685626907559771229301024, 77371253555424045547391723, 154742506690651950367202722, 309485012701412263095698742, 618970024302736747825197633, 1237940046825494079645579597, 2475880090770920964920237949, 4951760176881795319464836733, 9903520346223476834183214363, 19807040680246793253244435408, 39614081340753312286620527074, 79228162649566189938250829583, 158456325247451671021643207258, 316912650411282198553437858435, 633825300687262544762168833356, 1267650601155602093689782355766, 2535301201956979339200302649987
Generating function in Maple syntax:
-(4*x^9-2*x^8-10*x^7+7*x^6-2*x^5-2*x^4+13*x^3-14*x^2+6*x-1)/(2*x-1)/(x^2+x-1)/(x-1)^4
Generating function in latex syntax:
-\frac{4 x^{9}-2 x^{8}-10 x^{7}+7 x^{6}-2 x^{5}-2 x^{4}+13 x^{3}-14 x^{2}+6 x -1}{\left(2 x -1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(-4*x**9 + 2*x**8 + 10*x**7 - 7*x**6 + 2*x**5 + 2*x**4 - 13*x**3 + 14*x**2 - 6*x + 1)/((x - 1)**4*(2*x - 1)*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^2+x-1)*(x-1)^4*F(x)+4*x^9-2*x^8-10*x^7+7*x^6-2*x^5-2*x^4+13*x^3-14*x^2+6*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+4 x^{9}-2 x^{8}-10 x^{7}+7 x^{6}-2 x^{5}-2 x^{4}+13 x^{3}-14 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/30*(-12*5^(1/2)+30)*(-1/2*5^(1/2)-1/2)^(-n)+1/30*(12*5^(1/2)+30)*(1/2*5^(1/2)-1/2)^(-n)+1/6*n^3-n^2-43/6*n+2^(n+1)+8)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(-12 \sqrt{5}+30\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{30}+\frac{\left(12 \sqrt{5}+30\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{30}+\frac{n^{3}}{6}-n^{2}-\\\frac{43 n}{6}+2^{n +1}+8 & \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) = 53
a(6) = 127
a(7) = 277
a(8) = 573
a(9) = 1152
a(n+3) = 1/6*n^3-2*n^2-2*a(n)-a(n+1)+3*a(n+2)-25/6*n+25, n >= 10
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) = 53
a \! \left(6\right) = 127
a \! \left(7\right) = 277
a \! \left(8\right) = 573
a \! \left(9\right) = 1152
a \! \left(n +3\right) = \frac{n^{3}}{6}-2 n^{2}-2 a \! \left(n \right)-a \! \left(n +1\right)+3 a \! \left(n +2\right)-\frac{25 n}{6}+25, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16921/
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[17,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[14,x]+F[7,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[7,x]
F[17,x] = F[18,x]+F[45,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[10,x]+F[21,x]
F[21,x] = F[18,x]+F[22,x]
F[22,x] = F[23,x]+F[24,x]+F[43,x]
F[23,x] = 0
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]+F[30,x]
F[26,x] = F[11,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[11,x]
F[30,x] = F[22,x]+F[31,x]
F[31,x] = 2*F[23,x]+F[32,x]+F[36,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]+F[35,x]
F[34,x] = F[27,x]
F[35,x] = F[31,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[42,x]
F[41,x] = F[11,x]
F[42,x] = F[38,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[18,x]+F[38,x]
F[45,x] = F[23,x]+F[46,x]+F[83,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]+F[55,x]
F[48,x] = F[49,x]+F[7,x]
F[49,x] = F[23,x]+F[50,x]+F[51,x]
F[50,x] = F[26,x]*F[4,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[29,x]+F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[4,x]*F[41,x]
F[55,x] = F[45,x]+F[56,x]
F[56,x] = F[23,x]+F[57,x]+F[61,x]+F[81,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]+F[60,x]
F[59,x] = F[49,x]
F[60,x] = F[56,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[63,x]+F[76,x]
F[63,x] = F[64,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[66,x]+F[68,x]
F[66,x] = F[11,x]+F[67,x]
F[67,x] = F[54,x]
F[68,x] = F[63,x]+F[69,x]
F[69,x] = 2*F[23,x]+F[70,x]+F[74,x]
F[70,x] = F[4,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[4,x]*F[75,x]
F[75,x] = F[63,x]
F[76,x] = F[77,x]
F[77,x] = F[4,x]*F[78,x]
F[78,x] = F[79,x]+F[80,x]
F[79,x] = F[27,x]
F[80,x] = F[76,x]
F[81,x] = F[4,x]*F[82,x]
F[82,x] = F[38,x]
F[83,x] = F[4,x]*F[84,x]
F[84,x] = F[85,x]+F[86,x]
F[85,x] = F[18,x]+F[63,x]
F[86,x] = F[87,x]+F[88,x]
F[87,x] = F[39,x]
F[88,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_{17}\! \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_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{7}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{45}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{18}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{43}\! \left(x \right)
F_{23}\! \left(x \right) = 0
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{30}\! \left(x \right)
F_{26}\! \left(x \right) = F_{11}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{11}\! \left(x \right)
F_{30}\! \left(x \right) = F_{22}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = 2 F_{23}\! \left(x \right)+F_{32}\! \left(x \right)+F_{36}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{35}\! \left(x \right)
F_{34}\! \left(x \right) = F_{27}\! \left(x \right)
F_{35}\! \left(x \right) = F_{31}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{42}\! \left(x \right)
F_{41}\! \left(x \right) = F_{11}\! \left(x \right)
F_{42}\! \left(x \right) = F_{38}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{18}\! \left(x \right)+F_{38}\! \left(x \right)
F_{45}\! \left(x \right) = F_{23}\! \left(x \right)+F_{46}\! \left(x \right)+F_{83}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{55}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{7}\! \left(x \right)
F_{49}\! \left(x \right) = F_{23}\! \left(x \right)+F_{50}\! \left(x \right)+F_{51}\! \left(x \right)
F_{50}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{29}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{55}\! \left(x \right) = F_{45}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{23}\! \left(x \right)+F_{57}\! \left(x \right)+F_{61}\! \left(x \right)+F_{81}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{60}\! \left(x \right)
F_{59}\! \left(x \right) = F_{49}\! \left(x \right)
F_{60}\! \left(x \right) = F_{56}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{76}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \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_{11}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{54}\! \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_{23}\! \left(x \right)+F_{70}\! \left(x \right)+F_{74}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \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_{4}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{63}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{4}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)+F_{80}\! \left(x \right)
F_{79}\! \left(x \right) = F_{27}\! \left(x \right)
F_{80}\! \left(x \right) = F_{76}\! \left(x \right)
F_{81}\! \left(x \right) = F_{4}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{38}\! \left(x \right)
F_{83}\! \left(x \right) = F_{4}\! \left(x \right) F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{85}\! \left(x \right)+F_{86}\! \left(x \right)
F_{85}\! \left(x \right) = F_{18}\! \left(x \right)+F_{63}\! \left(x \right)
F_{86}\! \left(x \right) = F_{87}\! \left(x \right)+F_{88}\! \left(x \right)
F_{87}\! \left(x \right) = F_{39}\! \left(x \right)
F_{88}\! \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_17(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_14(x) + F_7(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_7(x))
Eq(F_17(x), F_18(x) + F_45(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_10(x) + F_21(x))
Eq(F_21(x), F_18(x) + F_22(x))
Eq(F_22(x), F_23(x) + F_24(x) + F_43(x))
Eq(F_23(x), 0)
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x) + F_30(x))
Eq(F_26(x), F_11(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_11(x))
Eq(F_30(x), F_22(x) + F_31(x))
Eq(F_31(x), 2*F_23(x) + F_32(x) + F_36(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_35(x))
Eq(F_34(x), F_27(x))
Eq(F_35(x), F_31(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_42(x))
Eq(F_41(x), F_11(x))
Eq(F_42(x), F_38(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_18(x) + F_38(x))
Eq(F_45(x), F_23(x) + F_46(x) + F_83(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_55(x))
Eq(F_48(x), F_49(x) + F_7(x))
Eq(F_49(x), F_23(x) + F_50(x) + F_51(x))
Eq(F_50(x), F_26(x)*F_4(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_29(x) + F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_4(x)*F_41(x))
Eq(F_55(x), F_45(x) + F_56(x))
Eq(F_56(x), F_23(x) + F_57(x) + F_61(x) + F_81(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_60(x))
Eq(F_59(x), F_49(x))
Eq(F_60(x), F_56(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_63(x) + F_76(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_66(x) + F_68(x))
Eq(F_66(x), F_11(x) + F_67(x))
Eq(F_67(x), F_54(x))
Eq(F_68(x), F_63(x) + F_69(x))
Eq(F_69(x), 2*F_23(x) + F_70(x) + F_74(x))
Eq(F_70(x), F_4(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_4(x)*F_75(x))
Eq(F_75(x), F_63(x))
Eq(F_76(x), F_77(x))
Eq(F_77(x), F_4(x)*F_78(x))
Eq(F_78(x), F_79(x) + F_80(x))
Eq(F_79(x), F_27(x))
Eq(F_80(x), F_76(x))
Eq(F_81(x), F_4(x)*F_82(x))
Eq(F_82(x), F_38(x))
Eq(F_83(x), F_4(x)*F_84(x))
Eq(F_84(x), F_85(x) + F_86(x))
Eq(F_85(x), F_18(x) + F_63(x))
Eq(F_86(x), F_87(x) + F_88(x))
Eq(F_87(x), F_39(x))
Eq(F_88(x), F_77(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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