#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


1. Write down manually without a computer the set of all integer partitions of 7.
7, 6+1, 5+2, 5+1+1, 4+3, 4+2+1, 4+1+1+1, 3+3+1, 3+2+2, 3+2+1+1, 3+1+1+1+1, 2+2+2+1, 2+2+1+1+1, 2+1+1+1+1+1, 1+1+1+1+1+1+1

2. What is the A-number of this sequence?
1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,231,297,385,490,627
A000041	

3. Who wrote the classic book on the theory of partitions? What is his birthdate?
George Andrews
December 4, 1938

4. What is the A-number of this sequence?
1,1,2,2,3,4,5,6,8,10,12,15,18,22,27,32,38,46,54,64
A000009

5. What is the A-number of this?
1,1,1,2,2,3,3,4,5,6,7,9,10,12,14,17,19,23,26,31
A003114

6. Whose theorem is it that these above two sequences are the same?
Rogers-Ramanujan

7. Is it in the OEIS? What is the A-number of this sequence?
6, 27, 98, 315, 913, 2462, 6237, 15035, 34705, 77231, 166364, 348326, 710869, 1417900, 2769730, 5308732, 9999185, 18533944, 33845975, 60960273, 108389248, 190410133, 330733733, 568388100, 967054374, 1629808139, 2722189979, 4508130889, 7405471040, 12071334656, 19532545711, 31383895059, 50087785023, 79425014950, 125169350624, 196092576086, 305454719925, 473204548369, 729214486425, 1118017663499, 1705716260475, 2590019185179, 3914771232229, 5890909988350, 8826586976851, 13170317194055, 19572586740717, 28973538499289, 42727383964125, 62778398261333, 91909150089735, 134089624612034, 194966873988925, 282549913034690, 408165170515015, 587785958785911, 843877705717419, 1207952776419103, 1724099255093005, 2453843603845893
A071734

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? What is the A-number (if it exists)
k=5
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 a k between 0 and 10 such that seq(pn(11*n+k)/11,n=1..60); is an integer sequence?
k=6
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

9. Why should Dr.Z from now on make you watch the lectures before the recitation?
Doing so will give students more topics to discuss and ask questions about during recitation.