#Please do not post homework
#Ravali Bommanaboina, 10/04/2020, Assignment 7

#Question 1
#GFseq(1/(1-x-x^2-x^3),x,30);
#[1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, 5768, 10609, 19513, 35890, 66012, 121415, 223317, 410744, 755476, 1389537, 2555757, 4700770, 8646064, 15902591, 29249425, 53798080]

#GFseq(1/(1-x+x^2-x^3),x,30);
#[1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0]

#GFseq(exp(-x)/(1-x),x,30);
#[1, 0, 1/2, 1/3, 3/8, 11/30, 53/144, 103/280, 2119/5760, 16687/45360, 16481/44800, 1468457/3991680, 16019531/43545600, 63633137/172972800, 2467007773/6706022400, 34361893981/93405312000, 15549624751/42268262400, 8178130767479/22230464256000, 138547156531409/376610217984000, 92079694567171/250298560512000, 4282366656425369/11640679464960000, 72289643288657479/196503623737344000, 6563440628747948887/17841281393295360000, 39299278806015611311/106826515449937920000, 9923922230666898717143/26976017466662584320000, 79253545592131482810517/215433472824041472000000, 5934505493938805432851513/16131658445064225423360000, 14006262966463963871240459/38072970106357874688000000, 461572649528573755888451011/1254684545727217532928000000, 116167945043852116348068366947/315777214062132212662272000000, 3364864615063302680426807870189/9146650338351415815045120000000]

#Question 2
#GFseq(x/(-x^2 - x + 1), x, 30);
#[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610,987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025,121393, 196418, 317811, 514229, 832040]

SeqRec([[1, 1], [0, 1]], 30);
#[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610,987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040]

#Both sequences are the same because SeqRec represents the fibonacci sequence and GFseq represents the first n+1 coefficients of the taylor series

#Question 3
#taylor series coefficients can be represented by the sequence
#[1,x,x^2/2!,...,x^n/n!]
#Sum(a(n)*x^n, n=0..infinity) = 1/(1-c1*x-... -ck*x^k)
#a(n)=c1*a(n-1)+c2*a(n-2)+ ... + ck*a(n-k)
#since n! can be represented by the a(n-1), a(n-2), a(n-3)...
#c1, c2, c3,...,ck comes from the formula sum(a(n)*x^n)