0132_2301_3120
Counting sequence:
1, 1, 2, 6, 21, 71, 218, 610, 1585, 3895, 9186, 21022, 47061, 103663, 225618, 486626, 1042305, 2220303, 4708714, 9949294, 20956661, 44022231, 92253162, 192912386, 402622961, 838825351, 1744789138, 3623830750, 7516137525, 15569193055, 32212182306, 66571910722, 137438860161, 283467736223, 584115433818, 1202590710126, 2473901014165, 5085241113223, 10445360280314, 21440476538274, 43980464886321, 90159953229911, 184717953193922, 378231999655966, 774056185626901, 1583296743639631, 3236962231782258, 6614661952275490, 13510798881649665, 27584547717143215, 56294995341588426, 114841790497360622, 234187180622631861, 477381560500588983, 972777519511291018, 1981583836042226626, 4035225266123114225, 8214565720322872743, 16717361816798304114, 34011184385900940254, 69175290276409700981, 140656423562034137791, 285924533142496777282, 581072438321849519746, 1180591620717409859841, 2398076729582240174911, 4869940435459319995578, 9887454823508317934830, 20070057552195990322005, 40730410914750688022375, 82641413450218789179546, 167644010141872402906722, 340010386766614453082161, 689465506498968198765943, 1397820478929414980685218, 2833419889721787125508126, 5742397643169488576998485, 11635911013790805803538863, 23574053482485268903604114, 47752569874777852397563234, 96714065569170333972992641, 195845982777569926298721871, 396527668833598369299763242, 802726744224113772000847726, 1624796301562061610800849781, 3288278229351791355196343191, 6653927711158918977578125098, 13462597927228510489523088770, 27234680864278366047775618545, 55088331748199422233005678535, 111414603535684224740915587666, 225305087149939210031634765022, 455561934457019941162871611061, 921027389228322924524942051103, 1861861819085211933448276184418, 3763337719427556035693330706626, 7605903601369376408980212002945, 15370263527767281493147518831583, 31057439705591620336669220684314, 62748704711297355374086800495214, 126765060022822940149670312033301
Generating function in Maple syntax:
-(6*x^6-30*x^5+50*x^4-48*x^3+27*x^2-8*x+1)/(2*x-1)^2/(x-1)^5
Generating function in latex syntax:
-\frac{6 x^{6}-30 x^{5}+50 x^{4}-48 x^{3}+27 x^{2}-8 x +1}{\left(2 x -1\right)^{2} \left(x -1\right)^{5}}
Generating function in sympy syntax:
(-6*x**6 + 30*x**5 - 50*x**4 + 48*x**3 - 27*x**2 + 8*x - 1)/((x - 1)**5*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^5*F(x)+6*x^6-30*x^5+50*x^4-48*x^3+27*x^2-8*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{5} F \! \left(x \right)+6 x^{6}-30 x^{5}+50 x^{4}-48 x^{3}+27 x^{2}-8 x +1 = 0
Explicit closed form in Maple syntax:
1+2^n*n-1/12*n^4-1/6*n^3-5/12*n^2-4/3*n
Explicit closed form in latex syntax:
1+2^{n} n -\frac{n^{4}}{12}-\frac{n^{3}}{6}-\frac{5 n^{2}}{12}-\frac{4 n}{3}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 71
a(6) = 218
a(n+2) = -1/12*n^4+1/2*n^3+7/12*n^2-4*a(n)+4*a(n+1)-n+2, 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) = 21
a \! \left(5\right) = 71
a \! \left(6\right) = 218
a \! \left(n +2\right) = -\frac{n^{4}}{12}+\frac{n^{3}}{2}+\frac{7 n^{2}}{12}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-n +2, \quad n \geq 7
Specification 1
Strategy pack name: insertion_row_placements
Tree: http://permpal.com/tree/303/
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[8,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[8,x]*F[9,x]
F[8,x] = x
F[9,x] = F[10,x]+F[16,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[13,x]*F[8,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[8,x]
F[16,x] = F[17,x]
F[17,x] = F[18,x]*F[8,x]
F[18,x] = F[19,x]+F[9,x]
F[19,x] = F[13,x]*F[20,x]*F[8,x]
F[20,x] = F[13,x]+F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[13,x]*F[20,x]*F[8,x]
F[23,x] = F[24,x]+F[53,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]*F[8,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[20,x]*F[8,x]
F[30,x] = F[14,x]+F[31,x]
F[31,x] = F[32,x]+F[33,x]+F[35,x]
F[32,x] = 0
F[33,x] = F[34,x]*F[8,x]
F[34,x] = F[28,x]+F[31,x]
F[35,x] = F[36,x]*F[8,x]
F[36,x] = F[37,x]+F[45,x]
F[37,x] = F[14,x]+F[38,x]
F[38,x] = F[32,x]+F[39,x]+F[44,x]
F[39,x] = F[40,x]*F[8,x]
F[40,x] = F[38,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[43,x]*F[8,x]
F[43,x] = F[1,x]+F[41,x]
F[44,x] = F[37,x]*F[8,x]
F[45,x] = F[46,x]+F[49,x]
F[46,x] = F[47,x]
F[47,x] = F[48,x]*F[8,x]
F[48,x] = F[14,x]+F[46,x]
F[49,x] = 2*F[32,x]+F[50,x]+F[52,x]
F[50,x] = F[51,x]*F[8,x]
F[51,x] = F[38,x]+F[49,x]
F[52,x] = F[45,x]*F[8,x]
F[53,x] = F[111,x]+F[32,x]+F[54,x]
F[54,x] = F[55,x]*F[8,x]
F[55,x] = F[56,x]+F[94,x]
F[56,x] = F[57,x]+F[6,x]
F[57,x] = F[32,x]+F[58,x]+F[80,x]
F[58,x] = F[59,x]
F[59,x] = F[60,x]*F[8,x]
F[60,x] = F[61,x]+F[69,x]
F[61,x] = F[62,x]
F[62,x] = F[63,x]*F[8,x]
F[63,x] = F[43,x]+F[64,x]
F[64,x] = F[14,x]+F[65,x]
F[65,x] = F[32,x]+F[66,x]+F[68,x]
F[66,x] = F[67,x]*F[8,x]
F[67,x] = F[41,x]+F[65,x]
F[68,x] = F[64,x]*F[8,x]
F[69,x] = F[32,x]+F[66,x]+F[70,x]
F[70,x] = F[71,x]*F[8,x]
F[71,x] = F[64,x]+F[72,x]
F[72,x] = F[73,x]+F[76,x]
F[73,x] = F[74,x]
F[74,x] = F[75,x]*F[8,x]
F[75,x] = F[14,x]+F[73,x]
F[76,x] = 2*F[32,x]+F[77,x]+F[79,x]
F[77,x] = F[78,x]*F[8,x]
F[78,x] = F[65,x]+F[76,x]
F[79,x] = F[72,x]*F[8,x]
F[80,x] = F[81,x]
F[81,x] = F[8,x]*F[82,x]
F[82,x] = F[83,x]+F[86,x]
F[83,x] = F[84,x]+F[85,x]
F[84,x] = F[13,x]*F[14,x]
F[85,x] = F[39,x]
F[86,x] = F[57,x]+F[87,x]
F[87,x] = F[14,x]*F[88,x]
F[88,x] = F[32,x]+F[89,x]+F[91,x]
F[89,x] = F[90,x]
F[90,x] = F[13,x]*F[41,x]*F[8,x]
F[91,x] = F[92,x]
F[92,x] = F[43,x]*F[8,x]*F[93,x]
F[93,x] = F[14,x]+F[88,x]
F[94,x] = F[53,x]+F[95,x]
F[95,x] = 2*F[32,x]+F[96,x]+F[98,x]
F[96,x] = F[97,x]
F[97,x] = F[14,x]*F[60,x]*F[8,x]
F[98,x] = F[8,x]*F[99,x]
F[99,x] = F[100,x]+F[105,x]
F[100,x] = F[101,x]
F[101,x] = F[102,x]+F[103,x]
F[102,x] = F[14,x]*F[83,x]
F[103,x] = F[104,x]*F[41,x]
F[104,x] = F[33,x]
F[105,x] = F[106,x]+F[95,x]
F[106,x] = F[107,x]
F[107,x] = F[108,x]*F[8,x]
F[108,x] = F[109,x]+F[110,x]
F[109,x] = F[87,x]
F[110,x] = F[106,x]
F[111,x] = F[112,x]*F[8,x]
F[112,x] = F[113,x]+F[64,x]
F[113,x] = F[114,x]+F[120,x]
F[114,x] = F[115,x]+F[117,x]+F[32,x]
F[115,x] = F[116,x]
F[116,x] = F[13,x]*F[28,x]*F[8,x]
F[117,x] = F[118,x]
F[118,x] = F[119,x]*F[43,x]*F[8,x]
F[119,x] = F[114,x]+F[14,x]
F[120,x] = F[121,x]+F[123,x]+F[132,x]+F[32,x]
F[121,x] = F[122,x]*F[8,x]
F[122,x] = F[65,x]+F[76,x]
F[123,x] = F[124,x]
F[124,x] = F[125,x]*F[8,x]
F[125,x] = F[126,x]+F[127,x]
F[126,x] = F[89,x]
F[127,x] = 2*F[32,x]+F[128,x]+F[131,x]
F[128,x] = F[129,x]*F[8,x]
F[129,x] = F[130,x]+F[65,x]
F[130,x] = F[128,x]
F[131,x] = F[125,x]*F[8,x]
F[132,x] = F[113,x]*F[8,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_{8}\! \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_{8}\! \left(x \right) F_{9}\! \left(x \right)
F_{8}\! \left(x \right) = x
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{16}\! \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_{13}\! \left(x \right) F_{8}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{8}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{8}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{9}\! \left(x \right)
F_{19}\! \left(x \right) = F_{13}\! \left(x \right) F_{20}\! \left(x \right) F_{8}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{13}\! \left(x \right) F_{20}\! \left(x \right) F_{8}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{53}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{8}\! \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_{20}\! \left(x \right) F_{8}\! \left(x \right)
F_{30}\! \left(x \right) = F_{14}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)+F_{35}\! \left(x \right)
F_{32}\! \left(x \right) = 0
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{8}\! \left(x \right)
F_{34}\! \left(x \right) = F_{28}\! \left(x \right)+F_{31}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{8}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{45}\! \left(x \right)
F_{37}\! \left(x \right) = F_{14}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{32}\! \left(x \right)+F_{39}\! \left(x \right)+F_{44}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right) F_{8}\! \left(x \right)
F_{40}\! \left(x \right) = F_{38}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right) F_{8}\! \left(x \right)
F_{43}\! \left(x \right) = F_{1}\! \left(x \right)+F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{37}\! \left(x \right) F_{8}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{49}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right) F_{8}\! \left(x \right)
F_{48}\! \left(x \right) = F_{14}\! \left(x \right)+F_{46}\! \left(x \right)
F_{49}\! \left(x \right) = 2 F_{32}\! \left(x \right)+F_{50}\! \left(x \right)+F_{52}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right) F_{8}\! \left(x \right)
F_{51}\! \left(x \right) = F_{38}\! \left(x \right)+F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{45}\! \left(x \right) F_{8}\! \left(x \right)
F_{53}\! \left(x \right) = F_{111}\! \left(x \right)+F_{32}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right) F_{8}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{94}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{6}\! \left(x \right)
F_{57}\! \left(x \right) = F_{32}\! \left(x \right)+F_{58}\! \left(x \right)+F_{80}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right) F_{8}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{69}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right) F_{8}\! \left(x \right)
F_{63}\! \left(x \right) = F_{43}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{14}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{32}\! \left(x \right)+F_{66}\! \left(x \right)+F_{68}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right) F_{8}\! \left(x \right)
F_{67}\! \left(x \right) = F_{41}\! \left(x \right)+F_{65}\! \left(x \right)
F_{68}\! \left(x \right) = F_{64}\! \left(x \right) F_{8}\! \left(x \right)
F_{69}\! \left(x \right) = F_{32}\! \left(x \right)+F_{66}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right) F_{8}\! \left(x \right)
F_{71}\! \left(x \right) = F_{64}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)+F_{76}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right) F_{8}\! \left(x \right)
F_{75}\! \left(x \right) = F_{14}\! \left(x \right)+F_{73}\! \left(x \right)
F_{76}\! \left(x \right) = 2 F_{32}\! \left(x \right)+F_{77}\! \left(x \right)+F_{79}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right) F_{8}\! \left(x \right)
F_{78}\! \left(x \right) = F_{65}\! \left(x \right)+F_{76}\! \left(x \right)
F_{79}\! \left(x \right) = F_{72}\! \left(x \right) F_{8}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{8}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{83}\! \left(x \right)+F_{86}\! \left(x \right)
F_{83}\! \left(x \right) = F_{84}\! \left(x \right)+F_{85}\! \left(x \right)
F_{84}\! \left(x \right) = F_{13}\! \left(x \right) F_{14}\! \left(x \right)
F_{85}\! \left(x \right) = F_{39}\! \left(x \right)
F_{86}\! \left(x \right) = F_{57}\! \left(x \right)+F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{14}\! \left(x \right) F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{32}\! \left(x \right)+F_{89}\! \left(x \right)+F_{91}\! \left(x \right)
F_{89}\! \left(x \right) = F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{13}\! \left(x \right) F_{41}\! \left(x \right) F_{8}\! \left(x \right)
F_{91}\! \left(x \right) = F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{43}\! \left(x \right) F_{8}\! \left(x \right) F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{14}\! \left(x \right)+F_{88}\! \left(x \right)
F_{94}\! \left(x \right) = F_{53}\! \left(x \right)+F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = 2 F_{32}\! \left(x \right)+F_{96}\! \left(x \right)+F_{98}\! \left(x \right)
F_{96}\! \left(x \right) = F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{14}\! \left(x \right) F_{60}\! \left(x \right) F_{8}\! \left(x \right)
F_{98}\! \left(x \right) = F_{8}\! \left(x \right) F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right)+F_{105}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)+F_{103}\! \left(x \right)
F_{102}\! \left(x \right) = F_{14}\! \left(x \right) F_{83}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right) F_{41}\! \left(x \right)
F_{104}\! \left(x \right) = F_{33}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right)+F_{95}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right) F_{8}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right)+F_{110}\! \left(x \right)
F_{109}\! \left(x \right) = F_{87}\! \left(x \right)
F_{110}\! \left(x \right) = F_{106}\! \left(x \right)
F_{111}\! \left(x \right) = F_{112}\! \left(x \right) F_{8}\! \left(x \right)
F_{112}\! \left(x \right) = F_{113}\! \left(x \right)+F_{64}\! \left(x \right)
F_{113}\! \left(x \right) = F_{114}\! \left(x \right)+F_{120}\! \left(x \right)
F_{114}\! \left(x \right) = F_{115}\! \left(x \right)+F_{117}\! \left(x \right)+F_{32}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right)
F_{116}\! \left(x \right) = F_{13}\! \left(x \right) F_{28}\! \left(x \right) F_{8}\! \left(x \right)
F_{117}\! \left(x \right) = F_{118}\! \left(x \right)
F_{118}\! \left(x \right) = F_{119}\! \left(x \right) F_{43}\! \left(x \right) F_{8}\! \left(x \right)
F_{119}\! \left(x \right) = F_{114}\! \left(x \right)+F_{14}\! \left(x \right)
F_{120}\! \left(x \right) = F_{121}\! \left(x \right)+F_{123}\! \left(x \right)+F_{132}\! \left(x \right)+F_{32}\! \left(x \right)
F_{121}\! \left(x \right) = F_{122}\! \left(x \right) F_{8}\! \left(x \right)
F_{122}\! \left(x \right) = F_{65}\! \left(x \right)+F_{76}\! \left(x \right)
F_{123}\! \left(x \right) = F_{124}\! \left(x \right)
F_{124}\! \left(x \right) = F_{125}\! \left(x \right) F_{8}\! \left(x \right)
F_{125}\! \left(x \right) = F_{126}\! \left(x \right)+F_{127}\! \left(x \right)
F_{126}\! \left(x \right) = F_{89}\! \left(x \right)
F_{127}\! \left(x \right) = 2 F_{32}\! \left(x \right)+F_{128}\! \left(x \right)+F_{131}\! \left(x \right)
F_{128}\! \left(x \right) = F_{129}\! \left(x \right) F_{8}\! \left(x \right)
F_{129}\! \left(x \right) = F_{130}\! \left(x \right)+F_{65}\! \left(x \right)
F_{130}\! \left(x \right) = F_{128}\! \left(x \right)
F_{131}\! \left(x \right) = F_{125}\! \left(x \right) F_{8}\! \left(x \right)
F_{132}\! \left(x \right) = F_{113}\! \left(x \right) F_{8}\! \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_8(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_8(x)*F_9(x))
Eq(F_8(x), x)
Eq(F_9(x), F_10(x) + F_16(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_13(x)*F_8(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_8(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_18(x)*F_8(x))
Eq(F_18(x), F_19(x) + F_9(x))
Eq(F_19(x), F_13(x)*F_20(x)*F_8(x))
Eq(F_20(x), F_13(x) + F_21(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_13(x)*F_20(x)*F_8(x))
Eq(F_23(x), F_24(x) + F_53(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x)*F_8(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_20(x)*F_8(x))
Eq(F_30(x), F_14(x) + F_31(x))
Eq(F_31(x), F_32(x) + F_33(x) + F_35(x))
Eq(F_32(x), 0)
Eq(F_33(x), F_34(x)*F_8(x))
Eq(F_34(x), F_28(x) + F_31(x))
Eq(F_35(x), F_36(x)*F_8(x))
Eq(F_36(x), F_37(x) + F_45(x))
Eq(F_37(x), F_14(x) + F_38(x))
Eq(F_38(x), F_32(x) + F_39(x) + F_44(x))
Eq(F_39(x), F_40(x)*F_8(x))
Eq(F_40(x), F_38(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_43(x)*F_8(x))
Eq(F_43(x), F_1(x) + F_41(x))
Eq(F_44(x), F_37(x)*F_8(x))
Eq(F_45(x), F_46(x) + F_49(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_48(x)*F_8(x))
Eq(F_48(x), F_14(x) + F_46(x))
Eq(F_49(x), 2*F_32(x) + F_50(x) + F_52(x))
Eq(F_50(x), F_51(x)*F_8(x))
Eq(F_51(x), F_38(x) + F_49(x))
Eq(F_52(x), F_45(x)*F_8(x))
Eq(F_53(x), F_111(x) + F_32(x) + F_54(x))
Eq(F_54(x), F_55(x)*F_8(x))
Eq(F_55(x), F_56(x) + F_94(x))
Eq(F_56(x), F_57(x) + F_6(x))
Eq(F_57(x), F_32(x) + F_58(x) + F_80(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_60(x)*F_8(x))
Eq(F_60(x), F_61(x) + F_69(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_63(x)*F_8(x))
Eq(F_63(x), F_43(x) + F_64(x))
Eq(F_64(x), F_14(x) + F_65(x))
Eq(F_65(x), F_32(x) + F_66(x) + F_68(x))
Eq(F_66(x), F_67(x)*F_8(x))
Eq(F_67(x), F_41(x) + F_65(x))
Eq(F_68(x), F_64(x)*F_8(x))
Eq(F_69(x), F_32(x) + F_66(x) + F_70(x))
Eq(F_70(x), F_71(x)*F_8(x))
Eq(F_71(x), F_64(x) + F_72(x))
Eq(F_72(x), F_73(x) + F_76(x))
Eq(F_73(x), F_74(x))
Eq(F_74(x), F_75(x)*F_8(x))
Eq(F_75(x), F_14(x) + F_73(x))
Eq(F_76(x), 2*F_32(x) + F_77(x) + F_79(x))
Eq(F_77(x), F_78(x)*F_8(x))
Eq(F_78(x), F_65(x) + F_76(x))
Eq(F_79(x), F_72(x)*F_8(x))
Eq(F_80(x), F_81(x))
Eq(F_81(x), F_8(x)*F_82(x))
Eq(F_82(x), F_83(x) + F_86(x))
Eq(F_83(x), F_84(x) + F_85(x))
Eq(F_84(x), F_13(x)*F_14(x))
Eq(F_85(x), F_39(x))
Eq(F_86(x), F_57(x) + F_87(x))
Eq(F_87(x), F_14(x)*F_88(x))
Eq(F_88(x), F_32(x) + F_89(x) + F_91(x))
Eq(F_89(x), F_90(x))
Eq(F_90(x), F_13(x)*F_41(x)*F_8(x))
Eq(F_91(x), F_92(x))
Eq(F_92(x), F_43(x)*F_8(x)*F_93(x))
Eq(F_93(x), F_14(x) + F_88(x))
Eq(F_94(x), F_53(x) + F_95(x))
Eq(F_95(x), 2*F_32(x) + F_96(x) + F_98(x))
Eq(F_96(x), F_97(x))
Eq(F_97(x), F_14(x)*F_60(x)*F_8(x))
Eq(F_98(x), F_8(x)*F_99(x))
Eq(F_99(x), F_100(x) + F_105(x))
Eq(F_100(x), F_101(x))
Eq(F_101(x), F_102(x) + F_103(x))
Eq(F_102(x), F_14(x)*F_83(x))
Eq(F_103(x), F_104(x)*F_41(x))
Eq(F_104(x), F_33(x))
Eq(F_105(x), F_106(x) + F_95(x))
Eq(F_106(x), F_107(x))
Eq(F_107(x), F_108(x)*F_8(x))
Eq(F_108(x), F_109(x) + F_110(x))
Eq(F_109(x), F_87(x))
Eq(F_110(x), F_106(x))
Eq(F_111(x), F_112(x)*F_8(x))
Eq(F_112(x), F_113(x) + F_64(x))
Eq(F_113(x), F_114(x) + F_120(x))
Eq(F_114(x), F_115(x) + F_117(x) + F_32(x))
Eq(F_115(x), F_116(x))
Eq(F_116(x), F_13(x)*F_28(x)*F_8(x))
Eq(F_117(x), F_118(x))
Eq(F_118(x), F_119(x)*F_43(x)*F_8(x))
Eq(F_119(x), F_114(x) + F_14(x))
Eq(F_120(x), F_121(x) + F_123(x) + F_132(x) + F_32(x))
Eq(F_121(x), F_122(x)*F_8(x))
Eq(F_122(x), F_65(x) + F_76(x))
Eq(F_123(x), F_124(x))
Eq(F_124(x), F_125(x)*F_8(x))
Eq(F_125(x), F_126(x) + F_127(x))
Eq(F_126(x), F_89(x))
Eq(F_127(x), F_128(x) + F_131(x) + 2*F_32(x))
Eq(F_128(x), F_129(x)*F_8(x))
Eq(F_129(x), F_130(x) + F_65(x))
Eq(F_130(x), F_128(x))
Eq(F_131(x), F_125(x)*F_8(x))
Eq(F_132(x), F_113(x)*F_8(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": false, "place_row": true, "strategy_class": "RowAndColumnPlacementFactory"}]], "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}, {"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": true, "maxreqlen": 1, "one_cell_only": true, "strategy_class": "CellInsertionFactory"}], "iterative": false, "name": "insertion_row_placements", "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:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [2, 3, 0, 1], "pos": [[2, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [3, 1, 2, 0], "pos": [[2, 1], [2, 1], [2, 1], [2, 1]]}], "requirements": [[{"patt": [0], "pos": [[1, 0]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1], [2, 1]], [[1, 0]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], 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