0123_0132_0213_1023
Counting sequence:
1, 1, 2, 6, 20, 71, 260, 971, 3674, 14032, 53968, 208692, 810492, 3158760, 12346628, 48377494, 189952216, 747180999, 2943648824, 11612917815, 45869337526, 181372345723, 717856746216, 2843678131629, 11273602645942, 44725291921541, 177551518494116, 705264937798343, 2802952203816934, 11145403903727517, 44337984726749664, 176457932173728907, 702551462011587162, 2798174400690013280, 11148597264029208464, 44432898092084996348, 177140993462500307372, 706407638541072236868, 2817773838010493369156, 11242541563542808522362, 44866711977811623757200, 179093411648158956320114, 715031708782122868238612, 2855330778644113423324760, 11404295432416808584720140, 45557218883694861157487474, 182019855735646495657819828, 727360182148761769243513572, 2907007159488183219329132628, 11619979307802093096734242136, 46454131312475934249953891172, 185737808616631351853162056398, 742729547579656026336766374720, 2970387308395219943613266767376, 11880786364948798289328793136396, 47525302277393982506632265104922, 190129794029270295086966967175712, 760710071501355892043342882729524, 3043897826491281901751718096328812, 12180951326913448171743167632702910, 48749592278752546100720099078885632, 195118256317948943849221853896368116, 781017093683146660980457575697216076, 3126493551176595419371982200572905974, 12516635952360322765419276508350764648, 50112904611755791916436482504413283623, 200651424931969908080845796322197770056, 803460520515594902775136724124746646823, 3217476720114001965133748268470286929190, 12885280278303535589849210682465561894883, 51605853482518678634089791864586781196088, 206694917375060216374661790555075353639525, 827914534182915192956451215084007653584790, 3316387517365967421339698462676665592803793, 13285206470598966470740626821437814150408260, 53222327959701783816210683901343478338620315, 213226525625559082981116305055106158695244110, 854298637119184636513859877140423370706050233, 3422935057829802347670056691439201699117975392, 13715367203036944303286740555240265038564431783, 54958573399777820764606637265394949028866111042, 220232858525105327993991533427702346515234180931, 882565364174341715794336299681330291439618965116, 3536952364613345549554851148383016471932941017397, 14175180507159619264912417974807528247300567022930, 56812584985949580318783841858289809244118224943251, 227707112389652090706650550024964145751569480811200, 912692078877087864109418841611583117239503433090697, 3658366151085070608178098372813592968263018669981918, 14664418360641531262929105759837513504777244708442823, 58783696584627126849723923966139175303652702847217292, 235647552423075403210943098845456387435695036789016325, 944675359665392564162602690985058771371044928456782218, 3787176051428723072867539264185457324103371457303619695, 15183129785409059164759987547348126887601214076483862072, 60872295789645403862999756217156814252974779719881331613, 244056456061835758290082658328863401419247929591551167470, 978527078933222833599256272592356340309353327681389488325, 3923440061372421505755347079857860885548624275760705083372, 15731586751716412514468390850848729890769502683658809617351, 63079622802941219353121860734881046795980102516002854821374
Generating function in Maple syntax:
(-x*(x+1)^2*(-4*x+1)^(1/2)-x^3+4*x^2+7*x-2)/(2*x^4+8*x-2)
Generating function in latex syntax:
\frac{-x \left(x +1\right)^{2} \sqrt{-4 x +1}-x^{3}+4 x^{2}+7 x -2}{2 x^{4}+8 x -2}
Generating function in sympy syntax:
(-x**3 + 4*x**2 - x*sqrt(1 - 4*x)*(x + 1)**2 + 7*x - 2)/(2*x**4 + 8*x - 2)
Implicit equation for the generating function in Maple syntax:
(x^4+4*x-1)*F(x)^2+(x^3-4*x^2-7*x+2)*F(x)+x^3+4*x^2+3*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{4}+4 x -1\right) F \! \left(x \right)^{2}+\left(x^{3}-4 x^{2}-7 x +2\right) F \! \left(x \right)+x^{3}+4 x^{2}+3 x -1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 71
a(n+6) = -2*(1+2*n)/(5+n)*a(n)+(32+7*n)/(5+n)*a(5+n)-3*(n+4)/(5+n)*a(n+1)+a(n+2)-8*(1+2*n)/(5+n)*a(n+3)-2*(23+4*n)/(5+n)*a(n+4), n >= 6
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) = 71
a \! \left(n +6\right) = -\frac{2 \left(1+2 n \right) a \! \left(n \right)}{5+n}+\frac{\left(32+7 n \right) a \! \left(5+n \right)}{5+n}-\frac{3 \left(n +4\right) a \! \left(n +1\right)}{5+n}+a \! \left(n +2\right)-\frac{8 \left(1+2 n \right) a \! \left(n +3\right)}{5+n}-\frac{2 \left(23+4 n \right) a \! \left(n +4\right)}{5+n}, \quad n \geq 6
Specification 1
Strategy pack name: insertion_point_placements_tracked_fusion
Tree: http://permpal.com/tree/21123/
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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]
F[13,x] = F[4,x]*F[7,x]
F[14,x] = F[15,x]+F[2,x]
F[15,x] = F[16,x]
F[16,x] = F[10,x]*F[17,x]*F[4,x]
F[17,x] = F[14,x]+F[18,x]
F[18,x] = F[19,x,1]
F[19,x,k[0]] = F[20,x,k[0]]+F[31,x,k[0]]
F[20,x,k[0]] = F[21,x,k[0]]+F[29,x]
F[21,x,k[0]] = F[22,x]*F[25,x,k[0]]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[1,x]+F[23,x]
F[25,x,k[0]] = F[1,x]+F[26,x,k[0]]
F[26,x,k[0]] = F[27,x,k[0]]
F[27,x,k[0]] = F[25,x,k[0]]*F[28,x,k[0]]
F[28,x,k[0]] = k[0]*x
F[29,x] = F[30,x]
F[30,x] = F[4,x]^2*F[24,x]*F[6,x]
F[31,x,k[0]] = F[32,x]+F[39,x,k[0]]
F[32,x] = F[33,x]
F[33,x] = F[10,x]*F[34,x]*F[4,x]
F[34,x] = F[35,x]+F[36,x]
F[35,x] = F[14,x]+F[7,x]
F[36,x] = F[37,x]+F[38,x]
F[37,x] = F[22,x]^2
F[38,x] = F[31,x,1]
F[39,x,k[0]] = F[40,x,k[0]]
F[40,x,k[0]] = F[4,x]*F[41,x,k[0]]
F[41,x,k[0]] = F[42,x,k[0]]+F[67,x,k[0]]
F[42,x,k[0]] = F[43,x,k[0]]+F[52,x,k[0]]
F[43,x,k[0]] = F[44,x]+F[45,x,k[0]]+F[50,x,k[0]]
F[44,x] = 0
F[45,x,k[0]] = F[4,x]*F[46,x,k[0]]
F[46,x,k[0]] = F[47,x,k[0]]+F[48,x,k[0]]
F[47,x,k[0]] = F[28,x,k[0]]
F[48,x,k[0]] = F[49,x,k[0]]
F[49,x,k[0]] = F[28,x,k[0]]*F[7,x]
F[50,x,k[0]] = F[28,x,k[0]]*F[51,x,k[0]]
F[51,x,k[0]] = F[43,x,k[0]]+F[7,x]
F[52,x,k[0]] = F[53,x,k[0]]+F[57,x,k[0]]
F[53,x,k[0]] = F[54,x,k[0]]
F[54,x,k[0]] = F[28,x,k[0]]*F[55,x,k[0]]
F[55,x,k[0]] = F[19,x,k[0]]+F[56,x,k[0]]
F[56,x,k[0]] = F[2,x]+F[53,x,k[0]]
F[57,x,k[0]] = F[58,x,k[0]]
F[58,x,k[0]] = F[28,x,k[0]]*F[59,x,k[0]]
F[59,x,k[0]] = F[60,x,k[0]]+F[66,x]
F[60,x,k[0]] = F[61,x,k[0]]+F[62,x,k[0]]
F[61,x,k[0]] = F[15,x]+F[57,x,k[0]]
F[62,x,k[0]] = F[63,x,k[0]]+F[65,x,k[0]]
F[63,x,k[0]] = F[64,x,k[0]]
F[64,x,k[0]] = F[22,x]^2*F[25,x,k[0]]
F[65,x,k[0]] = (F[39,x,k[0]]-F[39,x,1])/(-1+k[0])
F[66,x] = F[17,x]*F[4,x]
F[67,x,k[0]] = F[68,x,k[0]]+F[70,x,k[0]]
F[68,x,k[0]] = F[69,x,k[0]]
F[69,x,k[0]] = F[22,x]^2*F[26,x,k[0]]
F[70,x,k[0]] = -k[0]*(F[39,x,1]-F[39,x,k[0]])/(-1+k[0])
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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{2}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{10}\! \left(x \right) F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x , 1\right)
F_{19}\! \left(x , y\right) = F_{20}\! \left(x , y\right)+F_{31}\! \left(x , y\right)
F_{20}\! \left(x , y\right) = F_{21}\! \left(x , y\right)+F_{29}\! \left(x \right)
F_{21}\! \left(x , y\right) = F_{22}\! \left(x \right) F_{25}\! \left(x , y\right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{23}\! \left(x \right)
F_{25}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{26}\! \left(x , y\right)
F_{26}\! \left(x , y\right) = F_{27}\! \left(x , y\right)
F_{27}\! \left(x , y\right) = F_{25}\! \left(x , y\right) F_{28}\! \left(x , y\right)
F_{28}\! \left(x , y\right) = y x
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{4} \left(x \right)^{2} F_{24}\! \left(x \right) F_{6}\! \left(x \right)
F_{31}\! \left(x , y\right) = F_{32}\! \left(x \right)+F_{39}\! \left(x , y\right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{10}\! \left(x \right) F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)
F_{35}\! \left(x \right) = F_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{38}\! \left(x \right)
F_{37}\! \left(x \right) = F_{22} \left(x \right)^{2}
F_{38}\! \left(x \right) = F_{31}\! \left(x , 1\right)
F_{39}\! \left(x , y\right) = F_{40}\! \left(x , y\right)
F_{40}\! \left(x , y\right) = F_{4}\! \left(x \right) F_{41}\! \left(x , y\right)
F_{41}\! \left(x , y\right) = F_{42}\! \left(x , y\right)+F_{67}\! \left(x , y\right)
F_{42}\! \left(x , y\right) = F_{43}\! \left(x , y\right)+F_{52}\! \left(x , y\right)
F_{43}\! \left(x , y\right) = F_{44}\! \left(x \right)+F_{45}\! \left(x , y\right)+F_{50}\! \left(x , y\right)
F_{44}\! \left(x \right) = 0
F_{45}\! \left(x , y\right) = F_{4}\! \left(x \right) F_{46}\! \left(x , y\right)
F_{46}\! \left(x , y\right) = F_{47}\! \left(x , y\right)+F_{48}\! \left(x , y\right)
F_{47}\! \left(x , y\right) = F_{28}\! \left(x , y\right)
F_{48}\! \left(x , y\right) = F_{49}\! \left(x , y\right)
F_{49}\! \left(x , y\right) = F_{28}\! \left(x , y\right) F_{7}\! \left(x \right)
F_{50}\! \left(x , y\right) = F_{28}\! \left(x , y\right) F_{51}\! \left(x , y\right)
F_{51}\! \left(x , y\right) = F_{43}\! \left(x , y\right)+F_{7}\! \left(x \right)
F_{52}\! \left(x , y\right) = F_{53}\! \left(x , y\right)+F_{57}\! \left(x , y\right)
F_{53}\! \left(x , y\right) = F_{54}\! \left(x , y\right)
F_{54}\! \left(x , y\right) = F_{28}\! \left(x , y\right) F_{55}\! \left(x , y\right)
F_{55}\! \left(x , y\right) = F_{19}\! \left(x , y\right)+F_{56}\! \left(x , y\right)
F_{56}\! \left(x , y\right) = F_{2}\! \left(x \right)+F_{53}\! \left(x , y\right)
F_{57}\! \left(x , y\right) = F_{58}\! \left(x , y\right)
F_{58}\! \left(x , y\right) = F_{28}\! \left(x , y\right) F_{59}\! \left(x , y\right)
F_{59}\! \left(x , y\right) = F_{60}\! \left(x , y\right)+F_{66}\! \left(x \right)
F_{60}\! \left(x , y\right) = F_{61}\! \left(x , y\right)+F_{62}\! \left(x , y\right)
F_{61}\! \left(x , y\right) = F_{15}\! \left(x \right)+F_{57}\! \left(x , y\right)
F_{62}\! \left(x , y\right) = F_{63}\! \left(x , y\right)+F_{65}\! \left(x , y\right)
F_{63}\! \left(x , y\right) = F_{64}\! \left(x , y\right)
F_{64}\! \left(x , y\right) = F_{22} \left(x \right)^{2} F_{25}\! \left(x , y\right)
F_{65}\! \left(x , y\right) = \frac{F_{39}\! \left(x , y\right)-F_{39}\! \left(x , 1\right)}{-1+y}
F_{66}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{67}\! \left(x , y\right) = F_{68}\! \left(x , y\right)+F_{70}\! \left(x , y\right)
F_{68}\! \left(x , y\right) = F_{69}\! \left(x , y\right)
F_{69}\! \left(x , y\right) = F_{22} \left(x \right)^{2} F_{26}\! \left(x , y\right)
F_{70}\! \left(x , y\right) = -\frac{y \left(F_{39}\! \left(x , 1\right)-F_{39}\! \left(x , y\right)\right)}{-1+y}
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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_4(x)*F_7(x))
Eq(F_14(x), F_15(x) + F_2(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_10(x)*F_17(x)*F_4(x))
Eq(F_17(x), F_14(x) + F_18(x))
Eq(F_18(x), F_19(x, 1))
Eq(F_19(x, k_0), F_20(x, k_0) + F_31(x, k_0))
Eq(F_20(x, k_0), F_21(x, k_0) + F_29(x))
Eq(F_21(x, k_0), F_22(x)*F_25(x, k_0))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_1(x) + F_23(x))
Eq(F_25(x, k_0), F_1(x) + F_26(x, k_0))
Eq(F_26(x, k_0), F_27(x, k_0))
Eq(F_27(x, k_0), F_25(x, k_0)*F_28(x, k_0))
Eq(F_28(x, k_0), k_0*x)
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_24(x)*F_4(x)**2*F_6(x))
Eq(F_31(x, k_0), F_32(x) + F_39(x, k_0))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_10(x)*F_34(x)*F_4(x))
Eq(F_34(x), F_35(x) + F_36(x))
Eq(F_35(x), F_14(x) + F_7(x))
Eq(F_36(x), F_37(x) + F_38(x))
Eq(F_37(x), F_22(x)**2)
Eq(F_38(x), F_31(x, 1))
Eq(F_39(x, k_0), F_40(x, k_0))
Eq(F_40(x, k_0), F_4(x)*F_41(x, k_0))
Eq(F_41(x, k_0), F_42(x, k_0) + F_67(x, k_0))
Eq(F_42(x, k_0), F_43(x, k_0) + F_52(x, k_0))
Eq(F_43(x, k_0), F_44(x) + F_45(x, k_0) + F_50(x, k_0))
Eq(F_44(x), 0)
Eq(F_45(x, k_0), F_4(x)*F_46(x, k_0))
Eq(F_46(x, k_0), F_47(x, k_0) + F_48(x, k_0))
Eq(F_47(x, k_0), F_28(x, k_0))
Eq(F_48(x, k_0), F_49(x, k_0))
Eq(F_49(x, k_0), F_28(x, k_0)*F_7(x))
Eq(F_50(x, k_0), F_28(x, k_0)*F_51(x, k_0))
Eq(F_51(x, k_0), F_43(x, k_0) + F_7(x))
Eq(F_52(x, k_0), F_53(x, k_0) + F_57(x, k_0))
Eq(F_53(x, k_0), F_54(x, k_0))
Eq(F_54(x, k_0), F_28(x, k_0)*F_55(x, k_0))
Eq(F_55(x, k_0), F_19(x, k_0) + F_56(x, k_0))
Eq(F_56(x, k_0), F_2(x) + F_53(x, k_0))
Eq(F_57(x, k_0), F_58(x, k_0))
Eq(F_58(x, k_0), F_28(x, k_0)*F_59(x, k_0))
Eq(F_59(x, k_0), F_60(x, k_0) + F_66(x))
Eq(F_60(x, k_0), F_61(x, k_0) + F_62(x, k_0))
Eq(F_61(x, k_0), F_15(x) + F_57(x, k_0))
Eq(F_62(x, k_0), F_63(x, k_0) + F_65(x, k_0))
Eq(F_63(x, k_0), F_64(x, k_0))
Eq(F_64(x, k_0), F_22(x)**2*F_25(x, k_0))
Eq(F_65(x, k_0), -F_39(x, k_0) + (k_0*F_39(x, k_0) - F_39(x, 1))/(k_0 - 1))
Eq(F_66(x), F_17(x)*F_4(x))
Eq(F_67(x, k_0), F_68(x, k_0) + F_70(x, k_0))
Eq(F_68(x, k_0), F_69(x, k_0))
Eq(F_69(x, k_0), F_22(x)**2*F_26(x, k_0))
Eq(F_70(x, k_0), -F_39(x, 1) + (k_0*F_39(x, k_0) - F_39(x, 1))/(k_0 - 1))
Pack JSON:
{"expansion_strats": [[{"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.rearrange_assumption", "strategy_class": "RearrangeAssumptionFactory"}, {"class_module": "tilings.strategies.assumption_insertion", "strategy_class": "AddAssumptionFactory"}, {"class_module": "tilings.strategies.fusion.fusion", "strategy_class": "FusionFactory", "tracked": true}, {"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}, {"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": true, "maxreqlen": 1, "one_cell_only": true, "strategy_class": "CellInsertionFactory"}], "iterative": false, "name": "insertion_point_placements_tracked_fusion", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
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