#ATTENDANCE QUIZ FOR LECTURE 14 of Dr. Z.'s Math454(02) Rutgers University

# Please Edit this .txt page with Answers

#Email  ShaloshBEkhad@gmail.com 
#Subject: p24
#with an attachment called
#p24FirstLast.txt
#(e.g. p24DoronZeilberger.txt)

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


THE NUMBER OF ATTENDANCE QUESTIONS WERE: 9

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

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1. Write down manually the set of all integer partitions of 7. 
A1. Partition(7) := {[7],[6,1],[5,2],[4,3],[5,1,1],[4,2,1],[3,2,2],[3,3,1],[4,1,1,1],[3,2,1,1],
	[2,2,2,1],[2,2,1,1,1],[3,1,1,1,1],[2,1,1,1,1,1],[1,1,1,1,1,1,1]}

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2. Write down manually the set of partitions of 7 into distinct part. Write explicitly the number 
	of integer partition of 7 into odd parts. Find if they are the same.
A2. pnDistinctSeq(7);
                     [1, 1, 2, 2, 3, 4, 5]

pnOddSeq(7);
                     [1, 1, 2, 2, 3, 4, 5]


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3. What is the A-number of seq(nops(Pn(i)),i=1..20) in OEIS?
A3. seq(nops(Pn(i)), i = 1 .. 20);
1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 

  297, 385, 490, 627


A-number := A000041

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4. Who wrote the classic book on the theory of partitions? When was his birthday?
A4. George Eyre Andres wrote the famous book The Theory of Partitions. 
	His birthday is on Decemeber 4th 1938

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5. What is the A-number of this sequence seq(nops(PnD(i,1)),i=1..20).
A5. seq(nops(PnD(i, 1)), i = 1 .. 20);
1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, 32, 38, 46, 

  54, 64

A-number := A000009

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6. Whose theorem is it that this above two sequences are the same?
A6. Euler found that these two sequence of PnD and PnC always remains the same.

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7. What is the A-number of this sequence?
A7. seq(pn(5*n + 4)/5, n = 1 .. 20);
6, 27, 98, 315, 913, 2462, 6237, 15035, 34705, 77231, 166364, 

  348326, 710869, 1417900, 2769730, 5308732, 9999185, 18533944, 

  33845975, 60960273

A-number := A071734
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8. (i) Does there exist a k between 0 and 6 such that seq(pn(7*n + k)/7, n = 1 .. 60) is an integer
	sequence?
   (ii) Does there exist a k between 0 and 10 such that seq(pn(11*n + k)/11, n = 1 .. 60) is an integer
	sequence?
A8. (i) seq(pn(7*n + 5)/7, n = 1 .. 60);
11, 70, 348, 1449, 5334, 17822, 55165, 160215, 441105, 1159752, 2929465, 7142275, 16873472, 38749850, 
86737678, 189672868, 405991500, 852077072, 1756048833, 3558408287, 7098041203, 13951818365, 
27047831797, 51760979985, 97851055848, 182858720324, 338003248835, 618337665521, 1120093746591, 
2010077957041, 3575124822873, 6304704983465, 11027984666216, 19139751334984, 32970507766383, 
56389096665051, 95778303294310, 161605500562658, 270937729084512, 451448266735571, 747767295679096, 
1231499467923575, 2016951809325715, 3285715236498191, 5324915110964011, 8586478548523792, 
13778587170314680, 22006228197082290, 34986362493626058, 55376076131289889, 87271200535983443, 
136961242558905435, 214068391941511583, 333260171220556048, 516816109123943489, 798461886085057784, 
1229078822764093580, 1885179852417919386, 2881454716972276237, 4389283592442150157

A-number := A071746

(ii) seq(pn(11*n + 6)/11, n = 1 .. 60);
27, 338, 2835, 18566, 101955, 490253, 2121679, 8424520, 31120519, 108082568, 355805845, 1117485621, 
3366123200, 9767105406, 27398618368, 74534264393, 197147918679, 508189847045, 1279140518117, 
3149375120229, 7596463993261, 17975137880152, 41776886404425, 95472499084647, 214747195030701, 
475851915432152, 1039594734347464, 2240918740984590, 4769276748075331, 10027987311390252, 
20842935170110650, 42846733115308980, 87157154355667095, 175514188395496892, 
350048948606129926, 691709349394534479, 1354741480937268088, 2630730659734981650, 
5066678649512389696, 9681186536839919871, 18357579935010426239, 34554133523920054407, 
64579072566966203455, 119865491055660930293, 221006373119026120716, 404870518142812813398, 
737082082866134011286, 1333788661167343930225, 2399432817727762088014, 4291959601287308912758, 
7634832928240080000211, 13508538678919990017965, 23776449251021876569610, 41636798909831300777602, 
72553524269818107818797, 125819244166493359868989, 217168633063630216612362, 
373130181055092154587010, 638243703049886314995807, 1086984023480585956148135

A-number :=  A076394

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9. Why should Dr.Z from now on make you watch lectures before recitations and not after?
A9. Its always helpful to watch lectures first and understand the maple code and carry any questions 
you might have with the code for the recitations. This wa we can better understand the concepts being
covered.

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