This the guessing part of the Maple files that accompany the paper "Walks avoiding a quadrant and the reflection principl" by Mireille Bousquet-M\303\251lou and Michael Wallner. ############################################################# Abstract: We continue the enumeration of plane lattice walks with small steps avoiding the negative quadrant, initiated by the first author in 2016. We solve in detail a new case, namely the king model where all 8 nearest neighbour steps are allowed. The associated generating function is proved to be the sum of a simple, explicit D-finite series (related to the number of walks confined to the first quadrant), and an algebraic one. This was already the case for the two models solved by the first author in 2016. The principle of the approach is also the same, but challenging theoretical and computational difficulties arise as we now handle algebraic series of larger degree.
We expect a similar algebraicity phenomenon to hold for the 7 Weyl step sets, which are those for which walks confined to the first quadrant can be counted using the reflection principle. With this paper, this is now proved for 3 of them. For the remaining 4, we predict the D-finite part of the solution, and in 3 of the 4 cases, give evidence for the algebraicity of the remaining part. ############################################################# restart;
with(gfun): with(plots): gfun:-version;
with(LinearAlgebra):
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShsaWJuYW1lR0YoNidRMEQ6XGxpYlxnZnVuLm1sYUYoUSdEOlxsaWJGKFFCL2hvbWUvbWljaGFlbHdhbGxuZXIvbGliL2dmdW4ubWxhRihROS9ob21lL21pY2hhZWx3YWxsbmVyL2xpYkYoUUBDOlxQcm9ncmFtfkZpbGVzXE1hcGxlfjIwMjBcbGliRig3J0YvRjBGMUYyRjM= JCIkJVEhIiM=
parseData := proc(data::list)
local vec,i,n;
n := nops(data):
vec := Vector(n):
for i from 1 to n do
vec[i] := op(i,data);
end:
return vec;
end:
loadData := proc(filename)
local fd,data;
#read file i
fd := fopen(filename, READ, TEXT);
data := readdata(fd,integer,1);
fclose(fd);
return parseData(data);
end:
computeParams := proc(algeq)
local degF, degC, numUndet, nn;
degF := degree(op(1,algeq),F(t));
degC := degree(subs(F(t)=F,op(1,algeq)),t);
numUndet := (degF+1)*(degC+1);
nn := nops(expand(op(1,algeq)));
return [degF,degC,numUndet,nn];
end:
infolevel[gfun] := 1;
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkqaW5mb2xldmVsR0YnNiNJJWdmdW5HRiciIiI3I0Yx
fprefix := "data_King_A/":
<Text-field style="Heading 1" layout="Heading 1">R0</Text-field>
Recall that R(x) = t^2*M(x,0).
Hence, R0 = R(0) = t^2*M(0,0) = t^2*A_{-1,0}, and we will guess the GF for walks ending at (-1,0).
filename := "king-(-1,0)-2000.txt";
CC := loadData(filename):
seq(CC[i],i=1..10);
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSlmaWxlbmFtZUdGKFE1a2luZy0oLTEsMCktMjAwMC50eHRGKDcjRi4= NiwiIiEiIiQiIiciI14iJFMjIiUzOyIlLSgqIiZtWCciJythVSIocjMhSA==
Divide by 3 CC := CC/3:
seq(CC[i],i=1..10);
NiwiIiEiIiIiIiMiIzwiIyEpIiRPJiIlTUsiJkE6IyInKz05IidkcCcq
Guess the algebraic relation.
As we are interested in R0 = t^2*A_{-1,0} we append to initial 0s. algeq := listtoalgeq([0,0,seq(CC[n],n=1..Dimension(CC))],F(t));
listtoalgeq: looking for equation of order 1
listtoalgeq: looking for equation of order 2
listtoalgeq: looking for equation of order 3
listtoalgeq: looking for equation of order 4
listtoalgeq: looking for equation of order 5
listtoalgeq: looking for equation of order 6
listtoalgeq: looking for equation of order 7
listtoalgeq: looking for equation of order 8
listtoalgeq: looking for equation of order 9
listtoalgeq: looking for equation of order 10
listtoalgeq: looking for equation of order 11
listtoalgeq: looking for equation of order 12
listtoalgeq: looking for equation of order 13
listtoalgeq: looking for equation of order 14
listtoalgeq: looking for equation of order 15
listtoalgeq: looking for equation of order 16
listtoalgeq: looking for equation of order 17
listtoalgeq: looking for equation of order 18
listtoalgeq: looking for equation of order 19
listtoalgeq: looking for equation of order 20
listtoalgeq: looking for equation of order 21
listtoalgeq: looking for equation of order 22
listtoalgeq: looking for equation of order 23
listtoalgeq: looking for equation of order 24
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computeParams(algeq);
NyYiI0MiI08iJEQqIiRCJA==
algR0 := primpart(subs(F(t)=R0,op(1,algeq)),R0):
algR0 := collect(algR0,R0,factor):
<Text-field style="Heading 1" layout="Heading 1">R1</Text-field>
Recall that R(x) = t^2*M(x,0).
Hence, R1 = R'(0) = t^2*M_x(0,0) = t^2*A_{-2,0}, and we will guess the GF for walks ending at (-2,0).
filename := "king-(-2,0)-2000.txt";
CC := loadData(filename):
seq(CC[i],i=1..10);
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSlmaWxlbmFtZUdGKFE1a2luZy0oLTIsMCktMjAwMC50eHRGKDcjRi4= NiwiIiIiIiEiIioiI0MiJDcjIiUlMyIiJTV3IiZNdyUiJzpuSyIoazs/Iw==
Divide by 3 CC := CC/3:
seq(CC[i],i=1..10);
NiwjIiIiIiIkIiIhRiUiIikjIiQ3I0YlIyIlJTMiRiUjIiU1d0YlIiZ5ZSIiJzAqMyIiJykpUXQ=
Guess the algebraic relation.
As we are interested in R1 = t^2*A_{-2,0} we append to initial 0s. algeq := listtoalgeq([0,0,seq(CC[n],n=1..Dimension(CC))],F(t));
listtoalgeq: looking for equation of order 1
listtoalgeq: looking for equation of order 2
listtoalgeq: looking for equation of order 3
listtoalgeq: looking for equation of order 4
listtoalgeq: looking for equation of order 5
listtoalgeq: looking for equation of order 6
listtoalgeq: looking for equation of order 7
listtoalgeq: looking for equation of order 8
listtoalgeq: looking for equation of order 9
listtoalgeq: looking for equation of order 10
listtoalgeq: looking for equation of order 11
listtoalgeq: looking for equation of order 12
listtoalgeq: looking for equation of order 13
listtoalgeq: looking for equation of order 14
listtoalgeq: looking for equation of order 15
listtoalgeq: looking for equation of order 16
listtoalgeq: looking for equation of order 17
listtoalgeq: looking for equation of order 18
listtoalgeq: looking for equation of order 19
listtoalgeq: looking for equation of order 20
listtoalgeq: looking for equation of order 21
listtoalgeq: looking for equation of order 22
listtoalgeq: looking for equation of order 23
listtoalgeq: looking for equation of order 24
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=F2FQF2F6*&"FgBxrix&Hi^rRi,oEmL)*F2FaxF2F6*&"C!o?&=PFY-q`[,Fv]\DF2F_oF2F2*&"B+9e$f![n'*[1sV`'HoTF2FEF2F6*&"?#***H_[;wZ?i(fEN8$F2F4F2F6"<e6W(**))Q?_h^U:`F6F2)FjxF5F2F2*&,L*&"Kc#G(\!)oO&Rfz2Va(4ky)H*)["F2FUF2F2*&"Lo:y;hqPK4qCfg<**)p3z.K8F2FYF2F2*&"L[+,()Q42YfFp_pDt0`l\">`F2FgnF2F2*&"M?&>A?D)f()y[dFAU!\[m3GXE"F2F[oF2F2*&"MOf:,ec$)oKpg#pD+q^f\9m,#F2FcoF2F2*&"M'px%e?)>lLe&)[b<zq3!o5"=J#F2F3F2F2*&"M'\Ul#R;31Q;+1+GKBNT/r-?F2F[sF2F2*&"M'HZu8n`kd"fY!Hwx+3a]$)pO"F2FgrF2F2*&"L>pe;Oz=41oq['3W6P.HE6wF2FAF2F2*&"LUSM(4a8etkZ1q'e1)eMZBONF2F]wF2F2*&"L^,K,0P)RmaOG'za^7!Hi?"Q"F2FawF2F2*&"Koo_K&=lWId)y\$4PDWm=o_%F2FgoF2F2*&"K:!>7`@#Q\p%RkC)f'=IZ)GT7F2F_sF2F2*&"J5$GBNah'H=GlxJc_+6!H8GF2F9F2F2*&"I`Wim^Kp(=y-s/=)HjHb"*\F2F[xF2F2*&"H)*po\D7q<!\&\9(3Y-rbOgF2FQF2F2*&"G#[>"H!3*H9A%>[R<cW>7i$F2FaxF2F2*&"DAp&)Qc?q;v[ud5FvN*pF2F_oF2F2*&"C+ctvJ"*=IyQC#>.l+E"F2FEF2F2*&"@3[bZ4NWV_GO/!)3C"F2F4F2F2"=`o_3^^W_0"z#e[8F2F2)FjxFdoF2F2*&,J*&"K?6mAWXclR])GoC/"4$f6VB$F2FYF2F6*&"L![?/Nj$)QH1yT1wE-s%QT(oDF2FgnF2F6*&"LO^bQo+*e4$[g5<#oG#oyg/6*F2F[oF2F6*&"M!oJIL*>%f1Q)[80_5v_="3&>>F2FcoF2F6*&"MKS+^*\#*z*>c-Z4u0kgpEf/FF2F3F2F6*&"M;'>PTqFWx)>n&QKgxQ^*pQMFF2F[sF2F6*&"M/23E,H*y*p>Zq6TI.y0[U$4#F2FgrF2F6*&"M+o"*zWyaFto1'*)ew"Q:6Ctr7F2FAF2F6*&"L)oWgdzn0m?$4-WQ5uvR1\N'F2F]wF2F6*&"LkQ"f:`TP7ZB%zY;d(Q!4;al#F2FawF2F6*&"KE&*4D'phsB&o`A2K1AjBej#*F2FgoF2F6*&"K&R3q49<9l)elM5is[L4B&o#F2F_sF2F6*&"J_+j(RMi8uF/du#e--h4wV'F2F9F2F6*&"J;HsQbNO%z*=.Xu5.C9rrA"F2F[xF2F6*&"IQsqJ-0#y,MS!3v:XA(\!H;F2FQF2F6*&"H3(\X"*=,\#\+,JyyOU%G$4"F2FaxF2F6*&"E%>Mvw)4,:!*QED'Q)4F+TF2F_oF2F6*&"CZs"p)[cgZ-_.P&G"GLLF2FEF2F6*&"@#48uwE+/L!4\W:-t$F2F4F2F6"=qCK%z+)e!>C)G$)HHF6F2)FjxF\oF2F2*&,H*&"KK%eUj#>NL>gp$*yNlO^w:s^F2FgnF2F2*&"Lk%=>W7"41S![\/%z:"GYPK(p$F2F[oF2F2*&"M[_%eC.Uh@Nr3^EU%)['H\Ft6F2FcoF2F2*&"M3%4m%GWbvlH*4<(e]0B&z#z&>#F2F3F2F2*&"Mk/2-5rI=Nn\>nVY9qqAkHFF2F[sF2F2*&"M315(Gn6!)R*zB"R>sD@>l%GFCF2FgrF2F2*&"Msq_-3O4bu(3Lb5T&\5v1JT;F2FAF2F2*&"Ls'oGz$fSN$GaUc$o&*=ie.I!*)F2F]wF2F2*&"LWVM8&p.-w4@)=)[$>"=+%p[+%F2FawF2F2*&"L'\!4!\l7I(oH0F8o4")f,v*)\"F2FgoF2F2*&"KH5(\Kz&o[>hL.7uq,DQzAYF2F_sF2F2*&"KOi#pi@QTRn='RMP)\bHEO<"F2F9F2F2*&"Ja_Fxt))Hl$)fj8eohYE0kS#F2F[xF2F2*&"I;6=>/'Q<h!>0<x%GWI`W`$F2FQF2F2*&"Hm5bg7[F>Uc(Hf6q:P#)zEF2FaxF2F2*&"FY5r0f-(p7_IB5,;Ai69F2F_oF2F2*&"Cx%)>G\)4z=L:lGytHd(F2FEF2F2*&"@i(fyp7Ps8'f2\js$))F2F4F2F2"=.5"fe3H9V6ANxR&F2F2)FjxFhnF2F2*&,F*&"K%Q)3f]U1->p"HHz&y9!ze"*='F2F[oF2F6*&"LO"z:BFK'\l0Q(p$[p(G#oY-+%F2FcoF2F6*&"M%=ljw6"[%R)y$zp0a8VKlRg8"F2F3F2F6*&"MCu#zvWdTCbC-K=/6:;!=^")=F2F[sF2F6*&"MG`>m!*zI>j,.k")HtD`/,\]?F2FgrF2F6*&"Mo()3CUGx,TB?%[Di)oL9)R\f"F2FAF2F6*&"LS=@Xzk;&*e5j:cI$3*Q#=sF&*F2F]wF2F6*&"LC552>LIk*RC(\C>>d^)H8RYF2FawF2F6*&"Lk_/D$**y'Hn#)z9Qi&y"fC6w(=F2FgoF2F6*&"K_,=k2i`)>9$yTHfOkLS]<iF2F_sF2F6*&"K))GcFrNZk"3inJ@k$y&\#Ru;F2F9F2F6*&"Jgs!y)RPv(pgQ!y_@4PFTdo$F2F[xF2F6*&"Is$)es**p7t@EX"*HXu<giRgF2FQF2F6*&"H;x8P*4")QC@`)o>%Ra$e9D&F2FaxF2F6*&"FkKgzPOw'>`9<<$Qu$eFMF2F_oF2F6*&"D!R)y')*\*e$4"**)*QzDKYV"F2FEF2F6*&"AKg!pMXyEZUt`N#[g;F2F4F2F6"=t#RrYk._XUKE(4$)F6F2)FjxFZF2F2*&,D*&"K/$=7QA>#4zc">!QO%\)**\q.bF2FcoF2F2*&"L3A%*e*yT'=%R:xk#=P^x$=0;KF2F3F2F2*&"LoV$yw(z\`jnF^t+<$>7AgR")F2F[sF2F2*&"MSu"zj%H#oCc5$y())*\hXTf1$="F2FgrF2F2*&"MsY-l07quO"ohg2rA/PBw&=6F2FAF2F2*&"Lci(><=ZB8856'=ofUr?hVc(F2F]wF2F2*&"L#>)\pzo*y:ksSL<Zzr#[f7,%F2FawF2F2*&"LOz&3C$=h)e_Il#prZp&=2mk<F2FgoF2F2*&"Ko]n^hYUkKP6g@]K54URajF2F_sF2F2*&"K_>]5&Q[9YrbS!)[lHe'>LC=F2F9F2F2*&"J!f>z^"e@R@AD,hM;nr4;G%F2F[xF2F2*&"Ia&)["zM2<=JL$*)QF&o"e=")yF2FQF2F2*&"H!RaZQqQGHq4!G$=D_gyN!)F2FaxF2F2*&"F#Gik\)o'y![)3C*=1PF/2'F2F_oF2F2*&"Diu"RI2=my!Qv'*)fY$4=#F2FEF2F2*&"A!GV"R$yy%))>=r%Q;cW#F2F4F2F2">Lj]-'oL*fj*Q8[X5F2F2)FjxFVF2F2*&,B*&"K))Qhk)4H(*3>s%[p4H%z2!HSNF2F3F2F6*&"LkK[4'41:z/@'R=&oJVt[xk=F2F[sF2F6*&"L%yQSRY8VK#R&RcIep!fK?H<%F2FgrF2F6*&"L[kh!)>dhZwv(3n*p!HiLb@`_F2FAF2F6*&"L3AY'Q0SiY$QPHN&)*G")[l2[UF2F]wF2F6*&"Lc1j]EZ9>))[L%e_#fulY02\#F2FawF2F6*&"L_n%eL@b'4ZP-Odb*e#G$z$))="F2FgoF2F6*&"K#R<[,5;`m+0a>`NV&yw(3s%F2F_sF2F6*&"KW8)4Wc"fctoz]`p&Hj$=[l9F2F9F2F6*&"JK#\$\#)[g)>u^=>M9!e_1&>OF2F[xF2F6*&"ISekL=!f(\eE4&p`Xp4ZY[(F2FQF2F6*&"H?F1vQ7f\\2-1;I\K>kD*F2FaxF2F6*&"FC^>n82VB3G%Rqo$oW(RxF2F_oF2F6*&"Dimh+"f*oW&o><V8]fLDF2FEF2F6*&"A+DhJTU&Hb=v@pu3u#F2F4F2F6">&y^;c'pZE'>R%R%Q5F6F2)FjxFNF2F2*&,@*&"K+Ksc`UK?H))GE*GFZ(\Nlf:F2F[sF2F2*&"K![kHCM'\23*GC$*3^N[ndgO(F2FgrF2F2*&"L[c"**G.n`l$of(>+`([cNr+W"F2FAF2F2*&"L)GkNYyb1pRa-k7,$[@K<,a"F2F]wF2F2*&"L;/uVW]D$Q?A+TM)>/S_7n/"F2FawF2F2*&"K%e^[+l0u:#)[(*y<A#*R+mLP&F2FgoF2F2*&"K'*oS'3Na'Qr`*fDr%yr3c8iBF2F_sF2F2*&"JOZ_kEL!RF8Mp2&o1kT?q=)F2F9F2F2*&"Jg\$))QLR\3O7j6.V+&*p#*)4#F2F[xF2F2*&"I;/*R`$\)fhqf()31%)3rF3x%F2FQF2F2*&"Hs)R.9z_96**e?R@!\]<Sb(F2FaxF2F2*&"F_$=b.M7!*\T9Mi!>9`/w'F2F_oF2F2*&"DmMjMtSmH5UAZ/xg()4#F2FEF2F2*&"AuCE[_4HI$)*peBV&3AF2F4F2F2"=Vg0jzmlm@aP`'o(F2F2)FjxFJF2F2*&,>*&"J/6^Qe&*y<tqJ1hx(=gX!R@%F2FgrF2F6*&"K[)y+W4ee"[FK*>Ps)4q:4r<F2FAF2F6*&"K_6gF"Rz1okA]c-o*p[2:tHF2F]wF2F6*&"K;[>'>r'*y#zVYGA!\2u6a]i#F2FawF2F6*&"KSuN(GAe$ykTM/D]s))=E+u9F2FgoF2F6*&"JSI90b8jC"[[qU*HD?y&Q7pF2F_sF2F6*&"J?f#eFZ&eMG0aSYC([!z__"GF2F9F2F6*&"IGhDs_/'\dz(eGsfBjr"zi(F2F[xF2F6*&"Io^$\(3*R-G&Q,H$f*yL@YX<F2FQF2F6*&"HCyj_FCwKWM/:Ra))eKk*QF2Fa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computeParams(algeq);
NyYiI0MiI08iJEQqIiRCJw==
algR1 := primpart(subs(F(t)=R1,op(1,algeq)),R1):
algR1 := collect(algR1,R1,factor):
<Text-field style="Heading 1" layout="Heading 1">B1 and B2</Text-field>
Recall that (1+t)^2*zeta*S'(zeta) = B1 + I*sqrt(3)*B2, where zeta = exp(2*Pi*I/3)
exp(2*Pi*I/3);
LCYjIiIiIiIjISIiKiYqJkYjRiReI0YkRiRGJCkiIiRGI0YkRiQ=
In order to ocmpute it we first generated the counting sequence for the real and imaginary parts of S'(zeta), i.e. S'(zeta) = A1 + I*A2. For this purpose recall that S(x) = t*x*M(0,x). Hence S'(x) = t*M(0,x) + t*x*M_2(0,x), and we are therefore interested in the walks A_{-1,i} for i>=0 and A_{-2,j} for j>=0 evaluated at x=zeta. In order to count them using modular arithmetic, we again multiply by 3 to have integers as initial values (recall the starting polynom at t=0: 2/3 + x^{-2}/3 + y^{-2}/3. Moreover, we use the relation 1 + zeta + zeta^2 = 0 to reduce the powers of zeta to 1 and zeta, i.e. zeta^{3*k} = 1 zeta^{3*k+1} = zeta zeta^{3*k+2} = -1 - zeta. Hence, the coefficients of S'(zeta) are elements of Q + zeta*Q. Hence, after multiplying by 2 we get that 6*S'(zeta) is part of Z + I*sqrt(3)*Z and we write 6*S'(zeta) = A1 + A2*I*sqrt(3). The integer sequences A1 and A2 have been computied in a seperate C++ programm using modular arithmetic and Chinese remaindering for the first n=2000 steps.
Let's load these numbers filename := "king-A1-2000.txt";
CCA1 := loadData(filename):
seq(CCA1[i],i=1..10);
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSlmaWxlbmFtZUdGKFExa2luZy1BMS0yMDAwLnR4dEYoNyNGLg== NiwiIiFGIyEjPUYjISRNIyEkdichJW9lISZjOCQhJyNcPiMhKGUrUyI=
Divide by 6 CCA1 := CCA1/6:
seq(CCA1[i],i=1..10);
NiwiIiFGIyEiJEYjISNSIyEkRCMiIiMhJHkqISVFXyEmI2VPISdWTEI=
filename := "king-A2-2000.txt";
CCA2 := loadData(filename):
seq(CCA2[i],i=1..10);
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSlmaWxlbmFtZUdGKFExa2luZy1BMi0yMDAwLnR4dEYoNyNGLg== NiwiIiEiIichIiciI0kiI2EiJCRbIiVvRCImbW8iIichRzIiIicnNDwo
Divide by 6 CCA2 := CCA2/6:
seq(CCA2[i],i=1..10);
NiwiIiEiIiIhIiIiIiYiIiojIiRoIiIiIyIkRyUiJTZHIiYhKXkiIic7Jj4i
We are actually interested in B1 and B2.
First we compute the coefficients of zeta*S'(zeta) CCzA1 := -CCA1/2 - 3/2*CCA2:
CCzA2 := CCA1/2 - CCA2/2:
And then we compute (1+t)^2*zeta*S'(zeta) by a simple shift CCB1 := Vector(Dimension(CCzA1)):
CCB1[1] := CCzA1[1]:
CCB1[2] := 2*CCzA1[1]+CCzA1[2]:
for i from 3 to Dimension(CCB1) do CCB1[i] := CCzA1[i-2]+2*CCzA1[i-1]+CCzA1[i]: end:
seq(CCB1[i],i=1..10);
NiwiIiEjISIkIiIjRiNGJSEiJyEjZyEkdyMhJXU+ISYqKT0iISZrNyk=
This should be equal to (see B1ex(n) in king-weg-part2.mw) -3/2*t^2 - 3*t^4 - 6*t^5 - 60*t^6 - 276*t^7 - 1974*t^8 - 11889*t^9 - 81264*t^10;
LDIqJiMiIiQiIiMiIiIpSSJ0RzYiRiZGJyEiIiomRiVGJylGKSIiJUYnRisqJiIiJ0YnKUYpIiImRidGKyomIiNnRicpRilGMEYnRisqJiIkdyNGJylGKSIiKEYnRisqJiIldT5GJylGKSIiKUYnRisqJiImKik9IkYnKUYpIiIqRidGKyomIiZrNylGJylGKSIjNUYnRis=
CCB2 := Vector(Dimension(CCzA2)):
CCB2[1] := CCzA2[1]:
CCB2[2] := 2*CCzA2[1]+CCzA2[2]:
for i from 3 to Dimension(CCB2) do CCB2[i] := CCzA2[i-2]+2*CCzA2[i-1]+CCzA2[i]: end:
seq(CCB2[i],i=1..10);
NiwiIiEjISIiIiIjISIjISImISNJISRaIiEkPyohJUBiISZyZiQhJzVcQg==
This should be equal to (see B2ex(n) in king-weg-part2.mw) -1/2*t^2 - 2*t^3 - 5*t^4 - 30*t^5 - 147*t^6 - 920*t^7 - 5521*t^8 - 35971*t^9 - 234910*t^10;
LDQqJiMiIiIiIiNGJSlJInRHNiJGJkYlISIiKiZGJkYlKUYoIiIkRiVGKiomIiImRiUpRigiIiVGJUYqKiYiI0lGJSlGKEYvRiVGKiomIiRaIkYlKUYoIiInRiVGKiomIiQ/KkYlKUYoIiIoRiVGKiomIiVAYkYlKUYoIiIpRiVGKiomIiZyZiRGJSlGKCIiKkYlRioqJiInNVxCRiUpRigiIzVGJUYq
Guess the algebraic relation for B1
(wee add another 0 at the beginning, as there is another shift by t in S(x)) algeq := listtoalgeq([0,seq(CCB1[n],n=1..Dimension(CCB1))],F(t));
listtoalgeq: looking for equation of order 1
listtoalgeq: looking for equation of order 2
listtoalgeq: looking for equation of order 3
listtoalgeq: looking for equation of order 4
listtoalgeq: looking for equation of order 5
listtoalgeq: looking for equation of order 6
listtoalgeq: looking for equation of order 7
listtoalgeq: looking for equation of order 8
listtoalgeq: looking for equation of order 9
listtoalgeq: looking for equation of order 10
listtoalgeq: looking for equation of order 11
listtoalgeq: looking for equation of order 12
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
computeParams(algeq);
NyYiIzciI0MiJEQkIiRIIw==
algB1 := primpart(subs(F(t)=B1,op(1,algeq)),B1):
algB1 := collect(algB1,B1,factor):
Guess the algebraic relation for B2
(wee add another 0 at the beginning, as there is another shift by t in S(x)) algeq := listtoalgeq([0,seq(CCB2[n],n=1..Dimension(CCB2))],F(t));
listtoalgeq: looking for equation of order 1
listtoalgeq: looking for equation of order 2
listtoalgeq: looking for equation of order 3
listtoalgeq: looking for equation of order 4
listtoalgeq: looking for equation of order 5
listtoalgeq: looking for equation of order 6
listtoalgeq: looking for equation of order 7
listtoalgeq: looking for equation of order 8
listtoalgeq: looking for equation of order 9
listtoalgeq: looking for equation of order 10
listtoalgeq: looking for equation of order 11
listtoalgeq: looking for equation of order 12
listtoalgeq: looking for equation of order 13
listtoalgeq: looking for equation of order 14
listtoalgeq: looking for equation of order 15
listtoalgeq: looking for equation of order 16
listtoalgeq: looking for equation of order 17
listtoalgeq: looking for equation of order 18
listtoalgeq: looking for equation of order 19
listtoalgeq: looking for equation of order 20
listtoalgeq: looking for equation of order 21
listtoalgeq: looking for equation of order 22
listtoalgeq: looking for equation of order 23
listtoalgeq: looking for equation of order 24
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computeParams(algeq);
NyYiI0MiI2ciJUQ6IiR4JQ==
algB2 := primpart(subs(F(t)=B2,op(1,algeq)),B2):
algB2 := collect(algB2,B2,factor):
Save all computed equations save(algR0,algR1,algB1,algB2,guessedeqs):
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=