02314_02341_03124_03142_03214_03241_03412_03421_20314_20341_23014_23041_23401_30124_30142_30214_30241_30412_30421_32014_32041_32401_34012_34021_34201
Counting sequence:
1, 1, 2, 6, 24, 95, 358, 1312, 4756, 17189, 62110, 224518, 811912, 2936791, 10624154, 38436500, 139061328, 503123637, 1820310926, 6585935694, 23828122940, 86210935163, 311914059862, 1128515622036, 4083007809184, 14772460933805, 53447265620898, 193374023432962, 699633789057496, 2531298828119499, 9158325195306470, 33135131835930908, 119884033203117812, 433744506835929681, 1569302368164054014, 5677789306640615810, 20542434692382802568, 74323226928710926787, 268903961181640613466, 972903671264648425104, 3519998550415039049200, 12735474395751953110753, 46077379226684570297262, 166709524154663085921226, 603160724639892578107644, 2182256002426147460919015, 7895476388931274414042838, 28566101932525634765604112, 103353127717971801757790336, 373935128927230834960914009, 1352910006046295166015600130, 4894874385595321655273411198, 17709821897745132446289034712, 64074737560749053955078095671, 231824578315019607543945281574, 838749203771352767944335904920, 3034623127281665802001953090708, 10979369617551565170288085901437, 39723732451349496841430664024606, 143721814168989658355712890585214, 519990408588200807571411132770760, 1881342972096055746078491210893743, 6806762817539274692535400390579162, 24627099227216094732284545898389516, 89101682048384100198745727539012304, 322372914105840027332305908203072525, 1166356160287279635667800903320257678, 4219916230907198041677474975585880262, 15267800353099592030048370361328065276, 55239420610961969941854476928710875219, 199858101289311889559030532836913997590, 723093403391749598085880279541015557388, 2616176510512188542634248733520507742112, 9465415535602194722741842269897460864261, 34246195125450030900537967681884765548834, 123903897949239180888980627059936523358330, 448288514118945749942213296890258788980248, 1621923080848532845266163349151611328039587, 5868172833647935476619750261306762695223846, 21231248763997013156767934560775756835845524, 76815379651745388400740921497344970703029620, 277920654438741876219864934682846069335838633, 1005526373934982439095620065927505493163960062, 3638028597481202814378775656223297119140518906, 13162511117731101876415777951478958129882702664, 47622412601249495310185011476278305053710823835, 172299507417591967168846167623996734619140507418, 623385474081712359293305780738592147827148315912, 2255429833320601960622298065572977066040038936816, 8160221796194448006644961424171924591064452995129, 29523959814369230230113316792994737625122070178350, 106818690090873911117341776844114065170288085798978, 386473651382523404436142300255596637725830077982012, 1398274806458247466594002617057412862777709960789951, 5059005775378620310789301584009081125259399413910294, 18303654844601864220976494834758341312408447265468040, 66223245346116219550935966253746300935745239257650688, 239597952507571776649797357094939798116683959960770737, 866873535807277785494306954205967485904693603515453186, 3136377916498530044222547985555138438940048217773260534, 11347521903456261293641205156745854765176773071288880280
Generating function in Maple syntax:
(5*x^6-17*x^5+29*x^4-35*x^3+24*x^2-8*x+1)/(5*x^2-5*x+1)/(x-1)^4
Generating function in latex syntax:
\frac{5 x^{6}-17 x^{5}+29 x^{4}-35 x^{3}+24 x^{2}-8 x +1}{\left(5 x^{2}-5 x +1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(5*x**6 - 17*x**5 + 29*x**4 - 35*x**3 + 24*x**2 - 8*x + 1)/((x - 1)**4*(5*x**2 - 5*x + 1))
Implicit equation for the generating function in Maple syntax:
(5*x^2-5*x+1)*(x-1)^4*F(x)-5*x^6+17*x^5-29*x^4+35*x^3-24*x^2+8*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(5 x^{2}-5 x +1\right) \left(x -1\right)^{4} F \! \left(x \right)-5 x^{6}+17 x^{5}-29 x^{4}+35 x^{3}-24 x^{2}+8 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/30*(-132*5^(1/2)+300)*(1/2-1/10*5^(1/2))^(-n)+1/30*(132*5^(1/2)+300)*(1/2+1/10*5^(1/2))^(-n)-1/6*n^3-3/2*n^2-16/3*n-20)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(-132 \sqrt{5}+300\right) \left(\frac{1}{2}-\frac{\sqrt{5}}{10}\right)^{-n}}{30}+\frac{\left(132 \sqrt{5}+300\right) \left(\frac{1}{2}+\frac{\sqrt{5}}{10}\right)^{-n}}{30}-\frac{n^{3}}{6}-\frac{3 n^{2}}{2}\\-\frac{16 n}{3}-20 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 24
a(5) = 95
a(6) = 358
a(n+2) = -1/6*n^3-5*a(n)+5*a(n+1)+25/6*n-3, n >= 7
Recurrence in latex format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 24
a(5) = 95
a(6) = 358
a{\left(n + 2 \right)} = - \frac{n^{3} - 25 n + 18}{6} - 5 a{\left(n \right)} + 5 a{\left(n + 1 \right)}, \quad n \geq 7
Specification 1
Strategy pack name: regular_insertion_encoding_left
Tree: http://permpal.com/tree/25255/
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[16,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[21,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[16,x]*F[8,x]
F[8,x] = F[10,x]+F[9,x]
F[9,x] = F[1,x]+F[6,x]
F[10,x] = F[11,x]+F[6,x]
F[11,x] = F[12,x]+F[13,x]+F[19,x]
F[12,x] = 0
F[13,x] = F[14,x]*F[16,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]+F[17,x]
F[16,x] = x
F[17,x] = F[12,x]+F[13,x]+F[18,x]
F[18,x] = F[16,x]*F[6,x]
F[19,x] = F[16,x]*F[20,x]
F[20,x] = F[6,x]
F[21,x] = F[12,x]+F[22,x]+F[66,x]
F[22,x] = F[16,x]*F[23,x]
F[23,x] = F[24,x]+F[57,x]
F[24,x] = F[25,x]+F[35,x]
F[25,x] = F[26,x]
F[26,x] = F[16,x]*F[27,x]
F[27,x] = F[28,x]+F[31,x]
F[28,x] = F[1,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[16,x]*F[28,x]
F[31,x] = F[16,x]+F[32,x]
F[32,x] = F[12,x]+F[33,x]+F[34,x]
F[33,x] = F[16,x]*F[29,x]
F[34,x] = F[16,x]*F[31,x]
F[35,x] = F[12,x]+F[22,x]+F[36,x]
F[36,x] = F[16,x]*F[37,x]
F[37,x] = F[38,x]+F[51,x]
F[38,x] = F[39,x]+F[6,x]
F[39,x] = F[12,x]+F[40,x]+F[56,x]
F[40,x] = F[16,x]*F[41,x]
F[41,x] = F[42,x]+F[43,x]
F[42,x] = F[29,x]+F[39,x]
F[43,x] = F[39,x]+F[44,x]
F[44,x] = F[12,x]+F[45,x]+F[52,x]+F[54,x]
F[45,x] = F[16,x]*F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[32,x]+F[48,x]
F[48,x] = F[12,x]+F[45,x]+F[49,x]+F[50,x]
F[49,x] = F[16,x]*F[39,x]
F[50,x] = F[16,x]*F[51,x]
F[51,x] = F[17,x]+F[48,x]
F[52,x] = F[16,x]*F[53,x]
F[53,x] = F[39,x]
F[54,x] = F[16,x]*F[55,x]
F[55,x] = F[11,x]+F[44,x]
F[56,x] = F[16,x]*F[38,x]
F[57,x] = F[58,x]+F[61,x]
F[58,x] = F[12,x]+F[40,x]+F[59,x]
F[59,x] = F[16,x]*F[60,x]
F[60,x] = F[38,x]
F[61,x] = F[12,x]+F[45,x]+F[62,x]+F[64,x]
F[62,x] = F[16,x]*F[63,x]
F[63,x] = F[58,x]
F[64,x] = F[16,x]*F[65,x]
F[65,x] = F[55,x]
F[66,x] = F[16,x]*F[67,x]
F[67,x] = F[68,x]+F[72,x]
F[68,x] = F[6,x]+F[69,x]
F[69,x] = F[12,x]+F[40,x]+F[70,x]
F[70,x] = F[16,x]*F[71,x]
F[71,x] = F[68,x]
F[72,x] = F[11,x]+F[73,x]
F[73,x] = F[12,x]+F[45,x]+F[62,x]+F[74,x]
F[74,x] = F[16,x]*F[75,x]
F[75,x] = F[76,x]
F[76,x] = F[11,x]+F[77,x]
F[77,x] = F[12,x]+F[45,x]+F[52,x]+F[78,x]
F[78,x] = F[16,x]*F[79,x]
F[79,x] = F[76,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_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{21}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{16}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{10}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{6}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{19}\! \left(x \right)
F_{12}\! \left(x \right) = 0
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{16}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = x
F_{17}\! \left(x \right) = F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{16}\! \left(x \right) F_{6}\! \left(x \right)
F_{19}\! \left(x \right) = F_{16}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{6}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right)+F_{22}\! \left(x \right)+F_{66}\! \left(x \right)
F_{22}\! \left(x \right) = F_{16}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{57}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{35}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{16}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{31}\! \left(x \right)
F_{28}\! \left(x \right) = F_{1}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{16}\! \left(x \right) F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{16}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right)+F_{33}\! \left(x \right)+F_{34}\! \left(x \right)
F_{33}\! \left(x \right) = F_{16}\! \left(x \right) F_{29}\! \left(x \right)
F_{34}\! \left(x \right) = F_{16}\! \left(x \right) F_{31}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right)+F_{22}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{16}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{51}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{6}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right)+F_{40}\! \left(x \right)+F_{56}\! \left(x \right)
F_{40}\! \left(x \right) = F_{16}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{43}\! \left(x \right)
F_{42}\! \left(x \right) = F_{29}\! \left(x \right)+F_{39}\! \left(x \right)
F_{43}\! \left(x \right) = F_{39}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right)+F_{45}\! \left(x \right)+F_{52}\! \left(x \right)+F_{54}\! \left(x \right)
F_{45}\! \left(x \right) = F_{16}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{32}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right)+F_{45}\! \left(x \right)+F_{49}\! \left(x \right)+F_{50}\! \left(x \right)
F_{49}\! \left(x \right) = F_{16}\! \left(x \right) F_{39}\! \left(x \right)
F_{50}\! \left(x \right) = F_{16}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{17}\! \left(x \right)+F_{48}\! \left(x \right)
F_{52}\! \left(x \right) = F_{16}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{39}\! \left(x \right)
F_{54}\! \left(x \right) = F_{16}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{11}\! \left(x \right)+F_{44}\! \left(x \right)
F_{56}\! \left(x \right) = F_{16}\! \left(x \right) F_{38}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{61}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right)+F_{40}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{16}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{38}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right)+F_{45}\! \left(x \right)+F_{62}\! \left(x \right)+F_{64}\! \left(x \right)
F_{62}\! \left(x \right) = F_{16}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{58}\! \left(x \right)
F_{64}\! \left(x \right) = F_{16}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{55}\! \left(x \right)
F_{66}\! \left(x \right) = F_{16}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)+F_{72}\! \left(x \right)
F_{68}\! \left(x \right) = F_{6}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right)+F_{40}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{16}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{68}\! \left(x \right)
F_{72}\! \left(x \right) = F_{11}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{12}\! \left(x \right)+F_{45}\! \left(x \right)+F_{62}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{16}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{11}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{12}\! \left(x \right)+F_{45}\! \left(x \right)+F_{52}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{16}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{76}\! \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_16(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_21(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_16(x)*F_8(x))
Eq(F_8(x), F_10(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_6(x))
Eq(F_10(x), F_11(x) + F_6(x))
Eq(F_11(x), F_12(x) + F_13(x) + F_19(x))
Eq(F_12(x), 0)
Eq(F_13(x), F_14(x)*F_16(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x) + F_17(x))
Eq(F_16(x), x)
Eq(F_17(x), F_12(x) + F_13(x) + F_18(x))
Eq(F_18(x), F_16(x)*F_6(x))
Eq(F_19(x), F_16(x)*F_20(x))
Eq(F_20(x), F_6(x))
Eq(F_21(x), F_12(x) + F_22(x) + F_66(x))
Eq(F_22(x), F_16(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_57(x))
Eq(F_24(x), F_25(x) + F_35(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_16(x)*F_27(x))
Eq(F_27(x), F_28(x) + F_31(x))
Eq(F_28(x), F_1(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_16(x)*F_28(x))
Eq(F_31(x), F_16(x) + F_32(x))
Eq(F_32(x), F_12(x) + F_33(x) + F_34(x))
Eq(F_33(x), F_16(x)*F_29(x))
Eq(F_34(x), F_16(x)*F_31(x))
Eq(F_35(x), F_12(x) + F_22(x) + F_36(x))
Eq(F_36(x), F_16(x)*F_37(x))
Eq(F_37(x), F_38(x) + F_51(x))
Eq(F_38(x), F_39(x) + F_6(x))
Eq(F_39(x), F_12(x) + F_40(x) + F_56(x))
Eq(F_40(x), F_16(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_43(x))
Eq(F_42(x), F_29(x) + F_39(x))
Eq(F_43(x), F_39(x) + F_44(x))
Eq(F_44(x), F_12(x) + F_45(x) + F_52(x) + F_54(x))
Eq(F_45(x), F_16(x)*F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_32(x) + F_48(x))
Eq(F_48(x), F_12(x) + F_45(x) + F_49(x) + F_50(x))
Eq(F_49(x), F_16(x)*F_39(x))
Eq(F_50(x), F_16(x)*F_51(x))
Eq(F_51(x), F_17(x) + F_48(x))
Eq(F_52(x), F_16(x)*F_53(x))
Eq(F_53(x), F_39(x))
Eq(F_54(x), F_16(x)*F_55(x))
Eq(F_55(x), F_11(x) + F_44(x))
Eq(F_56(x), F_16(x)*F_38(x))
Eq(F_57(x), F_58(x) + F_61(x))
Eq(F_58(x), F_12(x) + F_40(x) + F_59(x))
Eq(F_59(x), F_16(x)*F_60(x))
Eq(F_60(x), F_38(x))
Eq(F_61(x), F_12(x) + F_45(x) + F_62(x) + F_64(x))
Eq(F_62(x), F_16(x)*F_63(x))
Eq(F_63(x), F_58(x))
Eq(F_64(x), F_16(x)*F_65(x))
Eq(F_65(x), F_55(x))
Eq(F_66(x), F_16(x)*F_67(x))
Eq(F_67(x), F_68(x) + F_72(x))
Eq(F_68(x), F_6(x) + F_69(x))
Eq(F_69(x), F_12(x) + F_40(x) + F_70(x))
Eq(F_70(x), F_16(x)*F_71(x))
Eq(F_71(x), F_68(x))
Eq(F_72(x), F_11(x) + F_73(x))
Eq(F_73(x), F_12(x) + F_45(x) + F_62(x) + F_74(x))
Eq(F_74(x), F_16(x)*F_75(x))
Eq(F_75(x), F_76(x))
Eq(F_76(x), F_11(x) + F_77(x))
Eq(F_77(x), F_12(x) + F_45(x) + F_52(x) + F_78(x))
Eq(F_78(x), F_16(x)*F_79(x))
Eq(F_79(x), F_76(x))
Pack JSON:
{"name": "regular_insertion_encoding_left", "initial_strats": [{"class_module": "tilings.strategies.factor", "strategy_class": "FactorFactory", "interleaving": null, "unions": false, "ignore_parent": true, "workable": true, "tracked": false}, {"class_module": "tilings.strategies.requirement_insertion", "strategy_class": "CellInsertionFactory", "ignore_parent": true, "maxreqlen": 1, "extra_basis": [], "one_cell_only": true}], "inferral_strats": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}], "expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "strategy_class": "RowAndColumnPlacementFactory", "partial": false, "ignore_parent": false, "dirs": [0], "place_row": false, "place_col": true}]], "symmetries": [], "iterative": false}
Specification JSON:
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