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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 29 "TP n\2602 : La suite de S
yracuse" }}{PARA 19 "" 0 "" {TEXT -1 20 "Serge Dupont  (2004)" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 179 "L'objectif de ce TP est d'\351tud
ier une suite ultra-classique : la suite de Syracuse. (Je ne connais p
as l'origine de ce nom.) Il faudra pour cela manipuler listes et proc
\351dures... " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }
}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 31 "Probl\350me: La suite de Syracu
se." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT 
-1 11 "Question 1:" }}{PARA 0 "" 0 "" {TEXT -1 21 "D\351finir une fonc
tion " }{TEXT 256 4 "syr " }{TEXT -1 49 "d\351finie sur les entiers st
rictement positifs par " }{TEXT 260 9 "syr(1)=1," }{TEXT -1 1 " " }
{TEXT 257 11 "syr(n)=n/2 " }{TEXT -1 13 "si n pair et " }{TEXT 258 5 "
3n+1 " }{TEXT -1 50 "si n est impair diff\351rent de 1. Afficher la su
ite " }{TEXT 259 7 "syr(n) " }{TEXT -1 36 "pour n variant de 0 \340 30
. (Utiliser " }{TEXT 261 4 "seq." }{TEXT -1 1 ")" }}{SECT 0 {PARA 5 "
" 0 "" {TEXT -1 9 "R\351ponse: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 85 "syr:=proc(n::nonnegint)\n  if n=1 then 1 elif irem(n,2)=0 then
 n/2 else 3*n+1 fi \nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
23 "s:=seq(syr(k),k=1..30);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG6
@\"\"\"F&\"#5\"\"#\"#;\"\"$\"#A\"\"%\"#G\"\"&\"#M\"\"'\"#S\"\"(\"#Y\"
\")\"#_\"\"*\"#eF'\"#k\"#6\"#q\"#7\"#w\"#8\"##)\"#9\"#))\"#:" }}}}}
{SECT 0 {PARA 4 "" 0 "" {TEXT -1 29 "Question 2: Version it\351rative
" }}{PARA 0 "" 0 "" {TEXT -1 20 "Ecrire une fonction " }{TEXT 262 15 "
syracuse1(n,u0)" }{TEXT -1 53 " qui calcule le n-i\350me terme de la s
uite d\351finie par " }{TEXT 264 4 "u(0)" }{TEXT -1 8 " et par " }
{TEXT 265 16 "u(n+1)=syr(u(n))" }{TEXT -1 56 ". Il s'agit donc d'une f
onction de deux variables. Pour " }{TEXT 266 4 "u(0)" }{TEXT -1 103 " \+
compris entre 1 et 100, regarder  le 100e terme puis le 1000e terme de
 la suite. Que constatez-vous ? " }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 
8 "R\351ponse:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "syracuse1
:=proc(p::nonnegint,u0::posint)\n  local i,s: s:=u0:  \n  for i from 1
 to p do s:=syr(s) od: end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
31 "seq(syracuse1(100,i),i=1..100);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6
`q\"\"\"F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#\"#`F#F#F#\"
#5F#F#F#F#F#F#F#F#F#\"#!)F#F#F#F#F#\"#;F#F#F#F#F#F#\"$1\"F(F#F#F#F#F#F
#\"#?F)F#F#F#F#F#F#F#\"\"%F#\"#BF#F#F#F#F#F#F#F#\"$g\"F,F#F#F#F#F#F#F#
F#F#F#\"\"&F-F#\"$%=F#F#F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
32 "seq(syracuse1(1000,i),i=1..100);" }}{PARA 12 "" 1 "" {XPPMATH 20 "
6`q\"\"\"F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F
#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F
#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#" }
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 
29 "Question 3: Version r\351cursive" }}{PARA 0 "" 0 "" {TEXT -1 28 "E
crire un version r\351cursive " }{TEXT 267 15 "syracuse2(n,u0)" }
{TEXT -1 151 " du programme pr\351c\351dent. Comparer les performances
 sur un exemple sur le couple (500,10^100+1). Laquelle choisissez-vous
 pour la suite ? Explication ? " }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 8 
"R\351ponse:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "syracuse2:=
proc(p::nonnegint,u0::posint)\noption remember:  \nif p=1 then u0 else
 syracuse2(p-1,syr(u0)) fi end: " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "seq(syracuse2(1000,i),i=1..100);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6`q\"\"\"F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#
F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#
F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#
F#F#F#F#F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "t:=time() : s
yracuse1(500,10^100+1); time() -t; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"#I!\"$" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 47 "t:=time() : syracuse2(500,10^100+1); time
() -t;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"#L!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 
0 {PARA 4 "" 0 "" {TEXT -1 25 "Question 4: Stationnarit\351" }}{PARA 
0 "" 0 "" {TEXT -1 236 "Montrer avec Maple que pour toute valeur initi
ale comprise entre 0 et 1000, la suite vaut constamment 1 au bout d'un
 certain temps. (On ne demande pas ce temps pr\351cisement, mais le fa
it qu'il en existe un.) On pourra \351cire un programme " }{TEXT 263 
13 "syracuseegal1" }{TEXT -1 148 " qui renvoie 'true' si syracuse(n,u0
)= 1 et 'false' sinon. Quel est la plus petite valeur u0 pour laquelle
 le 200e terme de la suite ne vaut pas 1 ?" }}{SECT 0 {PARA 5 "" 0 "" 
{TEXT -1 9 "R\351ponse: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "
syracuseegal1:=proc(n::nonnegint,u0::posint)\n  if syracuse2(n,u0)=1 t
hen true else false fi end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
106 "u:=1 : bool:=true : for j from 1 to 200 do \n  if not syracuseega
l1(200,u) then bool:=false fi : od : bool;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "u
:=1: while syracuseegal1(200,u) do u:=u+1 od: u;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"%jC" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "syr
acuse2(200,2463);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#8" }}}}}{SECT 
0 {PARA 4 "" 0 "" {TEXT -1 30 "Question 5: Calcul des orbites" }}
{PARA 0 "" 0 "" {TEXT -1 6 " Soit " }{TEXT 280 2 "u0" }{TEXT -1 33 " u
n entier. On appelle orbite de " }{TEXT 278 2 "u0" }{TEXT -1 21 " la s
uite des termes " }{TEXT 277 14 "syracuse(k,u0)" }{TEXT -1 51 " jusqu'
\340 ce qu'on tombe sur 1. Ecrire un programme " }{TEXT 281 7 "orb(u0)
" }{TEXT -1 24 " qui renvoie l'orbite de" }{TEXT 279 3 " u0" }{TEXT 
-1 50 " sous forme d'une s\351quence. Donner l'orbite de 27." }}{SECT 
0 {PARA 5 "" 0 "" {TEXT -1 8 "R\351ponse:" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 109 "orb:=proc(n::posint)\n  local s,k,Orb ; \n  Orb:=n :
 s:=n:\n  while s<>1 do s:=syr(s): Orb:= Orb,s od: Orb: end:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "orb(27);  " }}{PARA 12 "" 1 
"" {XPPMATH 20 "6\\r\"#F\"##)\"#T\"$C\"\"#i\"#J\"#%*\"#Z\"$U\"\"#r\"$9
#\"$2\"\"$A$\"$h\"\"$%[\"$U#\"$@\"\"$k$\"$#=\"#\"*\"$u#\"$P\"\"$7%\"$1
#\"$.\"\"$5$\"$b\"\"$m%\"$L#\"$+(\"$]$\"$v\"\"$E&\"$j#\"$!z\"$&R\"%'=
\"\"$$f\"%!y\"\"$!*)\"$X%\"%O8\"$o'\"$M$\"$n\"\"$-&\"$^#\"$a(\"$x$\"%K
6\"$m&\"$$G\"$])\"$D%\"%w7\"$Q'\"$>$\"$e*\"$z%\"%Q9\"$>(\"%e@\"%z5\"%Q
K\"%>;\"%e[\"%HC\"%)G(\"%WO\"%A=\"$6*\"%MF\"%n8\"%-T\"%^?\"%ah\"%xI\"%
K#*\"%;Y\"%3B\"%a6\"$x&\"%K<\"$m)\"$L%\"%+8\"$]'\"$D$\"$w*\"$)[\"$W#\"
$A\"\"#h\"$%=\"##*\"#Y\"#B\"#q\"#N\"$1\"\"#`\"$g\"\"#!)\"#S\"#?\"#5\"
\"&\"#;\"\")\"\"%\"\"#\"\"\"" }}}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 
27 "Question 6: Dessin d'orbite" }}{PARA 0 "" 0 "" {TEXT -1 20 "Ecrire
 un programme " }{TEXT 270 12 "dessinorbite" }{TEXT -1 9 " gr\342ce \+
\340 " }{TEXT 268 15 "plots[listplot]" }{TEXT -1 26 " le dessin de l'o
rbite de " }{TEXT 269 3 "u0." }{TEXT -1 26 " Afficher l'orbite de 27. \+
" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 8 "R\351ponse:" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 54 "plots[listplot]([seq(sin(n),n=1..500)],conn
ect=false);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6#-
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Question 7: A la recherche de la plus grande orbite" }}{PARA 0 "" 0 "
" {TEXT -1 20 "Ecrire un programme " }{TEXT 274 16 "rechlongorb(n,m)" 
}{TEXT -1 69 " qui renvoie une liste \340 deux termes: d'abord l'entie
r compris entre " }{TEXT 275 1 "n" }{TEXT -1 4 " et " }{TEXT 276 1 "m
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 1" }}}}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 45 "Question 8: A la recher
che du maximum calcul\351" }}{PARA 0 "" 0 "" {TEXT -1 20 "Ecrire un pr
ogramme " }{TEXT 273 9 "maxorb(n)" }{TEXT -1 62 " qui renvoie le maxim
um de l'orbite de n. Ecrire un programme " }{TEXT 272 15 "rechmaxorb(n
,m)" }{TEXT -1 312 " qui d\351termine le maximum des orbites des entie
rs compris entre n et m et affiche une liste \340 deux \351l\351ments \+
: d'abord l'entier en lequel il est atteint puis le maximum de son orb
ite. L'appliquer \340 l'intervalle d'entiers [1,30]. Quel est le maxim
um que vous arrivez \340 atteindre ? Dessiner l'orbite correspondante.
" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 8 "R\351ponse:" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 128 "maxorb:=proc(n::integer)\n  local s,max; \+
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fi: od: \n  max: end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "ma
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\nfor k from n+1 to m do\n  p:=maxorb(k) : \n  if p>max then L:=[k,p]:
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{SECT 0 {PARA 4 "" 0 "" {TEXT -1 25 "Question 9: Ouverture... " }}
{PARA 0 "" 0 "" {TEXT -1 174 "Rechercher au choix : la plus longue orb
ite possible ou le maximum d'une orbite le plus grand possible ; dessi
ner les orbites ainsi trouv\351e ; donner vos records personnels..." }
}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 8 "R\351ponse:" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 8 "a:=1000:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 36 "MOrb:=rechmaxorb(a*10^3,(a+1)*10^3);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%MOrbG7$\"(z/+\"\"*;Wca)" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 45 "nops([orb(MOrb[1])]) ; dessinorbite(MOrb[1]);" }}
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