#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:

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

#1

#write integer partition of 7

#attached to the email.

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#2. Write integer partition of 7 into distinct parts and odd parts.

#attached to the email

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#3.seq(nops(Pn(i)),i=1..20) <- is this in OEIS?
{1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627)

#A000041

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#4. Who wrote the classic book on the theory of partitions? what is his birthdate?

#answer: The book "Theory of Partition" is by George Andrews, who was born in December 4, 1938


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#5. what is the A-number of 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]

#A000009

#seq(nops(PnC(i,{1},2)),i=1..20) is the same seq.

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#6. what is the A-number of seq(nops(PnD(i,2)),i=1..20)
[1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 12, 14, 17, 19, 23, 26, 31]

#A003114

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#7. Whose theroem is it that these above two sequences are the same?
seq(nops(PnD(i,2)),i=1..20)
seq(nops(PnC(i,{1,4},5)),i=1..20)

#this is known as Rogers?Ramanujan identities

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#8. what is the A-number of this seq? is it in the OEIS?

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]

#A071734

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#9.
#i) Does there exist k between 0 and 6 such that seq(pn(7*n+k)/7,n=1..60) is an integer sequence?
#for k=5,
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

#A071746

#ii) Does there exist k between 0 and 10 such that seq(pn(11*n+k)/11,n=1..60) is an integer sequence?
#for k=6,
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

#A076394

#iii) Can you find other A and B such that pn(A*n+B)/A is ALways an integer?

#According to the Rogers?Ramanujan identities, the given numbers output mods 1,4,7 which is all the option we have.

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#10.
#I hope we can summarize the whole class section by section for us to easily see the whole picture of the semester.