#ATTENDANCE QUIZ FOR LECTURE 16 of Dr. Z.'s Math454(02) Rutgers University

# Please Edit this .txt page with Answers

#Email  ShaloshBEkhad@gmail.com 
#Subject: p16
#with an attachment called
#p16FirstLast.txt
#(e.g. p16DoronZeilberger.txt)

#Right after finishing watching the lecture but no later than Oct. 30, 2020, 8:00pm


THE NUMBER OF ATTENDANCE QUESTIONS WERE: 8

PLEASE LIST ALL THE QUESTIONS FOLLOWED, AFTER EACH, BY THE ANSWER

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Attendance Question 1:

Explicit format = f(n+k) = b1*f(n+k-1) + ... + bk*f(n)

Convert the following recurrence to explicit form:
6*f(n)+12*f(n+1)+18*f(n+3)=0 


f(n+3) = (-2/3)*f(n+1) - (1/3)*f(n).


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Attendance Question 2:

Find f(5) and f(6) of the above sequence:
f(n) = 3*f(n-1)-4*f(n-2), f(0)=1, f(1)=-2

f(2) = -10
f(3) = -22
f(4) = -26
f(5) = 3*(-26) - 4*(-22) = 10
f(6) = 3*(10) - 4*(-26) = 134


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Attendance Question 3:
What is F(10^6)? How long did it take?
What is f(10^6)? How long did it take?

I couldn't find the answers to the above problems. F(10^6) was taking a very long time to evaluate whereas f(10^6) had too many levels of recursion for Maple to handle.

The biggest power of 10 this worked for was 10^4 with F(10^4) = f(10^4) = -54635525252036814086233701871706016920677387869578154278314186625790929546573001236511580399095815348073980671027651330320316638179877740264160623881386920701556845879929718109118483596249681052820294244712588895979537683225668477370829577126752936013817228059609324496942780635503050133282487015858670599230521101877164742574537638932435579953287019613340124637844396156118333126613280019096408515504519200309270092213514284229004009003410236020962526731797479180587911817822812297627947389364867191448555233306943653488237651122372016521152919862749321389744459508853223665681444103474239601489615695682907613866153758494422621184438080206120314926750126471369077595814266855286329381556129869882076145777070655403644321780961229515395524539542279735017717637959180803857223061032456506277953478061688015117409827467442867342908860781863725412630487219594970232766621215352327493183766922380952253049089393904007324708249774078543018649446408427298157877383551057530139930196329045358735107142198861611435207919712103291237599953174372793930108554109964491045497728924857641738663184489303687494109014425328728333246071639352990446490682660184412285764784727929252626557316764742111465484398270605294091591990359088712987596349859757101675953072404787991958245064364549807877378220877445430870988127834382762209717695591445233050531872684784350085832892671535173113152935265038165085361068355892807530648796307139168154535206917093400342086705053825253499861248502315427472139151983340660623539394491157896554790138507157806626966624707980183418823607940304327526605513578898033300533898051193880849936552048245914410700406410748094676152665674133435561789640361443356544929287631701491275951694185000827435489559585328988333708765183315334415555334698075509310103892801300406691224906578108781587694669828396113966908222281337306010539347715318936192119673138830665587257739236031073640415321510445908800193711474642222608105560972270413732338061408408456822346635759390106363332291392355496067562367204195010903938859809096276283046463833852055174925958068506463632353537975449995175532517883230348947448363255685806129415278275568126773222761092371058929333455626093200830819739431585782022479458743564680120612936223832945049037789721855309589556296396073071023849828481875982953323349694253175662487044676603642700505140990274677812466514508109839760945973085690512920512821227929373021020555582981345129831726150428588505502974283155143864892062717066618675330531013028429655624689303931696863376805109614333617325103775630432030241361759204052376349572221750985258229700660402538477863831850260984360906489245821990809843281143933650120500675151689663750279446230318625535278643264208856661988229029692374087944000509728270640227436583181044168711260414587969016772222129984929022018067021758447445775403603375094633026573757424650998491999014827326959495816957080727065376783938271089271572403256100206963965664255030063476157894849043678081781980902530291971054696640280228881346283444282246861622106

time(f(10^4)) = 0.109
time(F(10^4)) = 0.15e-1

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Attendance Question 4:
Consider the sequence that is defined by the recurrence 
f(n+1000) = f(n+999) + 5*f(n)
What is the operator ope(N) such that ope(N) f(n) = 0?

f(n+1000) - f(n+999) - 5*f(n) = 0
ope(N) = N^1000 - N^999 - 5


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Attendance Question 5:
Characterize the sequence(S) that satisfy a (Homogeneous) recurrence of order zero.
Order zero means f(n) doesn't depend on any of the previous values f(n-1),f(n-2),...

f(n) = c = 0.

The only sequence that satisfies a homogenouse recurrence of order zero is the sequence of all zeros i.e. f(n) = 0 for all n >= 1.

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Attendance Question 6:
Can you prove that d(n)/n! -> 1/e?

d(n)=n*d(n-1)+(-1)^n, d(0) = 1, d(1) = 0
d(n)=n*((n-1) * d(n-2) + (-1)^(n-1)) + (-1)^n
...
d(n)=n! * Sum((-1)^i/i!, i = 0..n)

d(n) / n! = Sum((-1)^i/i!, i = 0..n).
We know that the taylor expansion of e^x = Sum(x^i/i!, i = 0..infty).

So, by plugging in -1 for x, we get that
lim_{n->infty} d(n) / n! = Sum((-1)^i/i!, i = 0..infty) = e^-1 = 1/e as desired.


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Attendance Question 6:
What is the OEIS A number of this sequence?

A85.

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Attendance Question 7:
Find the Operators in N and N^(-1) annihilated by w(n).

w(n+2)-w(n+1)-(n+1)*w(n)=0

ope(n,N) := N^2 - N - (n+1)I
ope(n,N) := I - N^(-1) - (n+1)N^(-2)

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Attendance Question 8:
?SumTools[Hypergeometric][ZeilbergerRecurrence]
Look up the syntax of ZeilbergerRecurrence

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Calling Sequences: 

Zeilberger(T, n, k, En)

Zeilberger(T, n, k, En, 'gosper')

ZeilbergerRecurrence(T, n, k, f, l..u)

Verify(T, 'Zpair', n, k, En)
_____

Parameters:

T-hypergeometric term of n and k

n-name

k-name

En-name; denote the shift operator with respect to n

f-name of the recurrence function

l..u-range for k

'Zpair'-list of two elements specifying a Z-pair f