[seqfan] Re: G.f. for trees with degree at most 3

[Georgi Guninski]

On Sun, May 23, 2010 at 04:35:44PM -0600, Richard Guy wrote:
> It looks as though it is a divisibility sequence.  Could
> someone check if it satisfies a linear recurrence of order
> 6 or less?  (or more, but that would be very tentative)  R.
>

based on the Douglas McNeil sage code the minimal relation i found is
degree 3, width 7, 53 monomials (attached).

verified it to 2000 terms (the rel. coefficients are strangely large)

there may be 14 potential relations: degree 4, width 6.

the first 2000 terms failed the divisibility test for me.

strongly doubt there is linear recurrence with constant coefficients
with width < 16.


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A003692
Message-ID: <AANLkTimdAODME_KElsFx2QtLJzKQdcfzp4m21xqDVMmy at mail.gmail.com>
'w0*fq[i-2]**2*fq[i]+   -(w1*fq[i-3]*fq[i-1]*fq[i]+w2*fq[i-2]*fq[i-1]*fq[i]+w3*fq[i-1]**2*fq[i]+w4*fq[i-3]*fq[i]**2+w5*fq[i-2]*fq[i]**2+w6*fq[i-1]*fq[i]**2+w7*fq[i]**3+w8*fq[i-3]*fq[i-2]*fq[i+1]+w9*fq[i-2]**2*fq[i+1]+w10*fq[i-3]*fq[i-1]*fq[i+1]+w11*fq[i-2]*fq[i-1]*fq[i+1]+w12*fq[i-1]**2*fq[i+1]+w13*fq[i-3]*fq[i]*fq[i+1]+w14*fq[i-2]*fq[i]*fq[i+1]+w15*fq[i-1]*fq[i]*fq[i+1]+w16*fq[i]**2*fq[i+1]+w17*fq[i-3]*fq[i+1]**2+w18*fq[i-2]*fq[i+1]**2+w19*fq[i-1]*fq[i+1]**2+w20*fq[i]*fq[i+1]**2+w21*fq[i-3]*fq[i-2]*fq[i+2]+w22*fq[i-2]**2*fq[i+2]+w23*fq[i-3]*fq[i-1]*fq[i+2]+w24*fq[i-2]*fq[i-1]*fq[i+2]+w25*fq[i-1]**2*fq[i+2]+w26*fq[i-3]*fq[i]*fq[i+2]+w27*fq[i-2]*fq[i]*fq[i+2]+w28*fq[i-1]*fq[i]*fq[i+2]+w29*fq[i]**2*fq[i+2]+w30*fq[i-3]*fq[i+1]*fq[i+2]+w31*fq[i-2]*fq[i+1]*fq[i+2]+w32*fq[i-1]*fq[i+1]*fq[i+2]+w33*fq[i]*fq[i+1]*fq[i+2]+w34*fq[i-3]*fq[i+2]**2+w35*fq[i-2]*fq[i+2]**2+w36*fq[i-1]*fq[i+2]**2+w37*fq[i-3]*fq[i-2]*fq[i+3]+w38*fq[i-2]**2*fq[i+3]+w39*fq[i-3]*fq[i-1]*fq[i+3]+w40*fq[i-2]*fq[i-1]*fq[i+3]+w41*fq[i-1]**2*fq[i+3]+w42*fq[i-3]*fq[i]*fq[i+3]+w43*fq[i-2]*fq[i]*fq[i+3]+w44*fq[i-1]*fq[i]*fq[i+3]+w45*fq[i]**2*fq[i+3]+w46*fq[i-3]*fq[i+1]*fq[i+3]+w47*fq[i-2]*fq[i+1]*fq[i+3]+w48*fq[i-1]*fq[i+1]*fq[i+3]+w49*fq[i-3]*fq[i+2]*fq[i+3]+w50*fq[i-2]*fq[i+2]*fq[i+3])', [[w0 == -13058976677880832766200000/130554002609461132917*r21, w1 == 4988810855977117591500500/43518000869820377639*r21, w2 == -70649512572012775359447380/391662007828383398751*r21, w3 == 5123966405183185715371360/130554002609461132917*r21, w4 == -14966432567931352774501500/43518000869820377639*r21, w5 == 20090398973114538489872860/130554002609461132917*r21, w6 == -9052010015687277537417265/391662007828383398751*r21, w7 == 436449290056010202449545/130554002609461132917*r21, w8 == 6902601958308440176420000/43518000869820377639*r21, w9 == -15647452412246497832357500/43518000869820377639*r21, w10 == 56176993883067833598171650/130554002609461132917*r21, w11 == 21044571534185269960315765/174072003479281510556*r21, w12 == -29937642596989712046978485/522216010437844531668*r21, w13 == -4116912328571144001539595/87036001739640755278*r21, w14 == 121218191384573886128547715/1566648031313533595004*r21, w15 == -4628705144015596233988765/261108005218922265834*r21, w16 == -612885875388273560934115/391662007828383398751*r21, w17 == -7344863284115794235806955/87036001739640755278*r21, w18 == 3978639348800647590697295/261108005218922265834*r21, w19 == 676342887724232320811925/174072003479281510556*r21, w20 == 19604065036918150942730/130554002609461132917*r21, w21 == -29834126014568979438416000/391662007828383398751*r21, w22 == 9302642318193852201800500/43518000869820377639*r21, w23 == -37458432877359117868583375/174072003479281510556*r21, w24 == -54826799572715649044868755/261108005218922265834*r21, w25 == 43422531043349343714946355/522216010437844531668*r21, w26 == 433487472986584224538019615/1566648031313533595004*r21, w27 == -42135175364829372059766760/391662007828383398751*r21, w28 == 3480205792339199796771835/783324015656766797502*r21, w29 == -1215831350176539255860/130554002609461132917*r21, w30 == 887541624317930608523135/43518000869820377639*r21, w31 == -2775134029563192744801352/391662007828383398751*r21, w32 == -263024196295058932963/43518000869820377639*r21, w33 == 6*r21, w34 == 13733465444141933097345/43518000869820377639*r21, w35 == -284/3*r21, w36 == -5*r21, w37 == 978488029911504705766000/130554002609461132917*r21, w38 == -8936982035982350679556000/391662007828383398751*r21, w39 == 20472385290180865302954005/1566648031313533595004*r21, w40 == 52788905474284612624628755/1566648031313533595004*r21, w41 == -725644644298161886195905/87036001739640755278*r21, w42 == -4307873003046859557434360/130554002609461132917*r21, w43 == 6561008422833745282776125/783324015656766797502*r21, w44 == -36489717542953065296170/130554002609461132917*r21, w45 == -10*r21, w46 == -100368695682344538479635/1566648031313533595004*r21, w47 == 9191977929446466566470/43518000869820377639*r21, w48 == 8*r21, w49 == 3673734903786217958255/130554002609461132917*r21, w50 == r21, w51 == r20, w52 == r19