#OK to post homework
#Ariana Yousafzai, November 8, Assignment 16
1.  
ope:=-N^3+N^2+(n-1)*N+(n(n-1)/2) is order 3
p_3(200) = 1128229342224250409977571477237602211005224534912356350930443615542857213566496891997776513837813244158753769620566503215847518434710389045756669563685354991503389333552591617962022597374604028294248769634453145378584484248158483438208939923996110880239517696
3. 
S6 := proc(n) local i, ans, k:
add(binomial(n, k)^6, k = 0 .. n-1): 
end:
S6seq := proc(N) local i, ans:
seq(S6(i), i = 0 .. N-1):
end proc:
S6seqclever := proc(k) local i, n, ope, ans: 
ope := -8*n^3*(6*n - 1)*(2*n + 1)*(510578*n + 701841)*(6*n + 1)/((n + 2)*(68821*n + 89555)*(n + 3)^5) + (1220155462*n^7 + 7630525053*n^6 + 24102407415*n^5 + 48083389345*n^4 + 61153339323*n^3 + 47245035102*n^2 + 20060238220*n + 3581016180)*N/((3*n + 6)*(68821*n + 89555)*(n + 3)^5) - (329726969*n^7 + 4510478951*n^6 + 25961601810*n^5 + 81977886100*n^4 + 153836556389*n^3 + 171830219751*n^2 + 105853624100*n + 27749509650)*N^2/((3*n + 6)*(68821*n + 89555)*(n + 3)^5) - (8356172*n^6 + 112044504*n^5 + 630368112*n^4 + 1895699256*n^3 + 3199049376*n^2 + 2858515470*n + 1050878290)*N^3/((206463*n + 268665)*(n + 3)^5) + N^4:
SeqFromRec(ope, n, N, [1, 2, 66, 1460], k):
end:


#time(S6seq(1000)) = 7.343


time(S6seqclever(1000)) = 0.296