Taro
֣Ĥʬס
Τǰּʬǿְ㤨Ȥפޤ()
ľ塡 519ڡ 0:09:29 25070
ȥȥ
ޤäޤвǤ
ޤޤǽ񤤤604240
ʼ˸лԡ 519ڡ 0:09:53 MAIL:h-sakai@zb3.so-net.ne.jp 25071
ޥ
ޤޤݤ󤫤Ǥġ

äݤˡꤷƤޤ
519ڡ 0:10:30 25072
ڼ
ȤʤäƤ⤤ɤäΤǡ
ޤ
͡ 519ڡ 0:10:57 MAIL:iida@ae.keio.ac.jp 25073
Ҥܤ
Ĥʬ򣴿
Ǥ̤롩
̣Ǥ⡦
ʤǤޤ
519ڡ 0:11:44 25075
ݤ
#25070
ۥȤǼ͡ĤˤʤäƤ޼
Ǽ򤫡ФƲ͡
ʧϤƤޤפޤ
Ǥϡ䤷Ƥ䤷Ƥä
ʧŷ񡡡 519ڡ 0:13:30 MAIL:hopes@mba.ocn.ne.jp HomePage:HOPES25076
DrK
ѡĤȤȤ߹碌äΤǤ͡
ñ˽ͤޤ
4ξϡ2սɤ뿧1֤Τ4̤
ʳϡ3ĤʬФɤȤ߹碌32=6̤
46=24
3ξϡ3սƱɤϡ2ս꤬٤礦Բġ
äơ2ս2ȤȤˤʤ롣ξϡߤ2٤礦Ȥˤʤ롣
43̤4̤
32֤Τ3̤
2¤2̤
ä432=24
24+24=48
ȤʤȤǤ뤬
5ʬΤȤΤߤǡ
ºݤ礭ۤʤ뤿ᡢ줾̤ΤΤȤƹͤ
485=240
ϳڱफ?㤦ʡ 519ڡ 0:13:33 MAIL:satoka@star.odn.ne.jp 25077
ߥƥ
ǧڤޤ̣ʬʤġġ
210̤ꤸʤΤʡ
519ڡ 0:14:17 HomePage:ߤ25078
ߤ
ꡢ郎ޤǤϤʤǤ礦

դܤȤϿѤʤФʤʤΤ
ɤʬǤ̤οˤΤQǤʸΤɤˤ
Ƥʤ
٤ƻȤɬפΤ
ޤƤ㤬ǤֻȤʤäƤ⤤
ȤȤʤΤǤ礦ʸ˽񤤤ߤ
ɤפȡ֤ޤäɤʤפƱ뤷ƤΤ
̡ɤפȤΤϹͤʤΤǤϤʤǤ礦
519ڡ 0:23:52 25079
ߥƥ
> դܤȤϿѤʤФʤʤΤ
ս񤭤Ρ֥γʬ϶̤ΤȤޤפ

> ٤ƻȤɬפΤ
㼨ФȤäƤʤΤǣϣ롩

ȻפޤץǤƤ210̤ꡣ
ɤ㤦

פˤʤäΤǡ00:30 ˾õ
519ڡ 0:30:59 HomePage:ߤ25080
ߥƥ
ɤ󤫤
ʤС240̤Ǵְ㤤ʤȻפޤ
519ڡ 0:26:46 HomePage:ߤ25081
ޥ
޵𤤤ޤʤȤꤢĤʬĤʬĺޤ

ʸꡢޤǤʲǤθǤ

դܤȤϿѤʤФʤʤΤ

ȡɤʬפȤդΰ̣˴ޤޤƤȻפäƤޤ٤ɤǡʥʤСޤϥʬˤǤƱȤäƤޤȡɤʬפȤȤˤʤʤʤ...ߤޤ󡣡ʼ¤ϣ٢ʬ˽񤤤ΤǤפʡפȻפäƾäƤޤޤ...

٤ƻȤɬפϡ

ͤŦ̤ǤɬפϤޤ󡣰Ǥ򼨤ĤǤϤʸϤǵ٤Ǥ
PowerBook G4 519ڡ 0:29:30 MAIL:masaru-y@sansu.org HomePage:25082
Ҥܤ
ɤʬ١٤礦ʬϰ㤦ɤ
ȤΤϡδ֤ǤϰۤλˤʤäƤߤǤ
Ǥ⡢ۤäǤ
餦ޤˡǤͤޤ
519ڡ 0:29:52 25083
ߤ
ɡ֣פϸȽǤ٤ʤΤǤ礦

ֶ̤룵Ʊ٤ʤ褦ˡɤ
Ȳ᤹Сξ礬̤ꡢξ⣱̤꤬ФƤ
Ȼפޤ

#25080ߥƥ
>> դܤȤϿѤʤФʤʤΤ
>ս񤭤Ρ֥γʬ϶̤ΤȤޤפ
ϡֱ߽ŬѤʤפȤȤɽƤǡƱ
ܤƤϤʤפȤϼʤǤ礦

>> ٤ƻȤɬפΤ
>㼨ФȤäƤʤΤǣϣ롩
㼨ϣˡȹͤƤ⤤ɡġ֤礦ɣǡפ
Ƥʤ飳Ǥ⤤ȹͤΤϻϰϤǤϤäȤĤ餤
519ڡ 0:33:30 25084
tomh
ʬǿޤ

[1] / (⥢) / (⥢, ⥤) / (⥤, ⥪) / (⥦, ⥪)
4x3x2x2x2 = 96.

[2] / (⥢) / (⥢, ⥤) / (ᥪ) / (⥪)
4x3x2x1x3 = 72.

[3] / (⥢) / (ᥤ) / (⥤) / (⥦, ⥪)
4x3x1x3x2 = 72.

96+72+72 = 240̤.
ԡ 519ڡ 0:37:15 MAIL:tomh@yahoo.co.jp HomePage:H to M25085
ҡΤ
ʬǤܤϰۤʤ뿧ˤΤQäΤǤ
֤ʤ1024ʤפȿǺǤʹ֤1͡ʶС
9ߤ濴 519ڡ 0:38:59 HomePage:25086
Ҥܤ
νɤȤơ
ߣߣߣߣᣳ
ڥȥƱۣߣߣߣᣱ
ڥȥȥƱۣߣߣᣳ
ڥȥȥƱۣߣᣱ
ڥȥȥȥƱۣߣᣱ
ݣܣݣᣲ̤
ƹͤˡǤʤȯŸƹͤޤ
519ڡ 0:39:39 25087
ޥ
#25084
> ɡ֣פϸȽǤ٤ʤΤǤ礦

ȡܿͤȤƤϴʡָפǤ֣...פΤޤޤȤȡ֣ĤΣǡ磴οȤäɤʬפȤˤʤ뵤ޤȤȡߣߣߣߣᣵ̤ꡢȤˤʤ뵤ΤǤ...
PowerBook G4 519ڡ 0:41:56 MAIL:masaru-y@sansu.org HomePage:25088
ߥƥ
>>25084
ϡ֥γʬ϶̤ΤȤޤפ
ֳʬ̤Ǥ褦㤦ɤʤפ
ᤷƤޤ(^^;;

ֱ߽ŬѤʤפǤʤۤɤͤ

ʤ256̤ꡢ1024̤ޤ
519ڡ 0:42:17 HomePage:ߤ25089
ҤȤ
ʬǤ­ǤޤϤҤܤͤʰ˼椫ޤ͡
43333324顢٤ä2꤬֤פޤ
84̤Ǥ褯ͤƤߤȤϡֱߤ4ʬ䤷ɤפ¾ʤޤ󡣡ʢ֤Ȥ1ȹͤ
519ڡ 0:47:54 MAIL:h-wada@cream.plala.or.jp 25090
ҡΤ
ǽŪˤϤʲˤʤޤʡ

5սˤ4ʤΤǡɬ12ĤФƤޤ
ޤ5սΤƱɤ2սȤʤꤢʤ褦֤
5C2-55̤ꡡ54332360̤
Ʊ22󤺤ĤĤŤͤƿƤΤ
Ʊ1Ĥ543215=120
2dzäƤޤ­ޤʡ(360-120)2+120240

ʤݽȤƤʤʶС
9ߤ濴 519ڡ 0:48:33 HomePage:25091
ޥ
ʸɵۤɤޤm(__)m

ʤߤ˻ꤷˡϡߤĤʬ䤷ȤɤʬAnȤơ

1ܣ2ᡡ
2ܣ3ᡡ
3ܣ4ᣱ
4ܣ5ᣳ

ȤǤ
PowerBook G4 519ڡ 0:49:58 MAIL:masaru-y@sansu.org HomePage:25092
ҤȤ
ʬǤ­ǤޤϤҤܤͤʰ˼椫ޤ͡
43333324顢٤ä2꤬֤פޤ
84̤Ǥ褯ͤƤߤȤϡֱߤ4ʬ䤷ɤפ¾ʤޤ󡣡ʢ֤Ȥ1ȹͤ
519ڡ 0:50:39 MAIL:h-wada@cream.plala.or.jp 25093
ߥƥ
>>25088
ĤΣӡ磴οȤäɤʬ롣
ȡϤοפʪȸʤ
540̤ꡣ

ĤΣӡ磴οȤäɤʬ롣
ȡϤοפƱ뤹롣
210̤ꡣ
ϤοפʾȤʤϤꤨޤ

ĤΣӡ磴οȤäɤʬ롣
ȡϤοפƱ뤷ģ˸¤롣
120̤ꡣ
ĤޤꡢȤ鷺ɤƱ̣ˤʤޤ(^^;;
519ڡ 1:40:58 HomePage:ߤ25094

ɤȤ
ȥƱ
ȥƱ
ȥƱ
ȥƱ
ȥƱ
äơ4*3*2*1*5=120̤

ɤȤ
ʳ򣲿ɤʥȥȥ줾Ʊ
ʳ򣲿ɤʥȥȥ줾Ʊ
ʳ򣲿ɤʥȥȥ줾Ʊ
ʳ򣲿ɤʥȥȥ줾Ʊ
ʳ򣲿ɤʥȥȥ줾Ʊ
äơ4*3*2*5=120̤

120+120240̤Ȳ򤭤ޤ
519ڡ 1:01:26 MAIL:ryoiti@sansu.org 25095
Ϥʤ
ȤΩƹǼꤴ̵Ƥޤ

̤­ޤ
43ߡ132+2(1322ˡˡ240
­ΤʤΤǡ

ϥƥǤ͡
ȡ硡 519ڡ 1:04:41 HomePage:http://hanau.daa.jp/blog2/25096
Ȥ
ϥ,,,,ʸäƤΤDzž
ƱͤʤƤ褤Ǥޤʶ路ʤäǤ
ʲʬʤβǤ
褺դοޤǥ,,Ʊ
,ƱǤȤƹͤ
ԥ󥯤P,忧B,ФG,WȤ,
ɤʬΤ٤οϰۤʤȤ
(,,)ХХǿ椬PΤΤ
(,,)=(B,P,G)(,)=(P,B),(P,W),(G,P),(G,B),(G,W),(W,P),(W,B)
(,,)=(G,P,B)(,)=(P,G),(P,W),(B,P),(B,G),(B,W),(W,P),(W,G)
(,,)=(B,P,W)(,)=(P,B),(P,G),(G,P),(G,B),(W,P),(W,B),(W,G)
(,,)=(W,P,B)(,)=(P,G),(P,W),(B,P),(B,G),(B,W),(G,P),(G,W)
(,,)=(G,P,W)(,)=(P,B),(P,G),(B,P),(B,G),(W,P),(W,B),(W,G)
(,,)=(W,P,G)(,)=(P,B),(P,W),(B,P),(B,W),(G,P),(G,B),(G,W)
42̤
ȿ椬B,G,WƱ̤42*4=168̤ꡦ
(,,)=ΤȤǥ==PΤΤ
(,,)=(P,B,P)(,)=(B,G),(G,B),(B,W),(W,B),(G,W),(W,G) B,G,W
(,,)=(P,G,P)(,)=(B,G),(G,B),(B,W),(W,B),(G,W),(W,G) B,G,W
(,,)=(P,W,P)(,)=(B,G),(G,B),(B,W),(W,B),(G,W),(W,G) B,G,W
18̤
ȥ=B,G,WƱ̤18*4=72̤
ȭ­72+168=240̤ꡦ()
οϻʤ󤫤̤Ǥʤݤ
٤ƽ񤭽ФǺ⤽ʲˡǤ
(ݤƱξϽ񤭽Фޤ)Ǥϡ
˭ԡ 519ڡ 1:08:40 MAIL:fttnm528@ybb.ne.jp 25097
ߤ
#25092
ɮȤʸʤ顢褯ɤʬªޤ
ɤפȤΤϰ´ꡣ̤ˡ֡ġФΣ
Ǥ⤤ʤȡʤ󤳤ˤϤοǤפȤפ
ߤäƤޤˡ
519ڡ 1:09:10 25098
Ȥ
ʤ󤫼ʬʸοޤ(,,)롼(,)롼
ʬƤߤʵ
ޤɤʬȤΤƱΤǤʤζ5Ѥ
㤦ХꥨΤΤǤ
Ȥ٤οƱǤɤ櫓ȲᤷƤޤǤɡ
äȿǤǤ礦׹ˡǤϡ
˭ԡ 519ڡ 1:51:04 MAIL:fttnm528@ybb.ne.jp 25099
Ȥ
οޤΤ褦ˡȤʸοޤǥ,,,,γ٤¬
=49,=98,=98,=65,=50
餤ǥ=ϤȤ⤫Ϥְ㤤ޤ
(Ϥ꿿̲ΤƱȸȻפޤ)
ȲˡȤƤ5ʤ3,2ȤʬΤ
(4,1Ȥ)
#25097
>դοޤǥ,,Ʊ
>,ƱǤȤƹͤ
ƱȤƹͤʤȤƱǤ
˭ԡ 519ڡ 2:13:18 MAIL:fttnm528@ybb.ne.jp 25100
̾õ
ĤǤϡ
n>=2ΤȤ
O(n)=3^n+(-1)^n+3
O(5)=243-3=240 ˤʤ褦Ǥ
519ڡ 8:38:01 25101
kasama
Ϥ褦ޤ(*^_^*)

public class Question451 {
private static int count = 0;
public static void main(String[] args) {
paint(0, new int[5]);
System.out.println(count);
}
private static final void paint(int n, int[] colors) {
if (n >= colors.length) {
for (int i = 0; i < colors.length; ++i) if (colors[i%colors.length] == colors[(i+1)%colors.length]) return;
++count;
return;
}
for (int i = 1; i <= 4; ++i) {
colors[n] = i;
paint(n+1, colors);
}
}
}
519ڡ 9:17:57 25102
̾õ

ĤǤϡ
n>=2ΤȤ
O(n)=3^n+(-1)^n*3
O(5)=243-3=240 ˤʤ褦Ǥ
519ڡ 11:09:36 25103
ޥ
ȡäˤʤޤ...

ǽν񤭤ߡ#25021
uchiyanβ#25061
Toru Fukatsu­#25065

Ǥ㤦ˡѤޤʤҥȤäʤǤ˴ְäƤʬФŦĺйǤ

nmȤm,nϸߤǡˡcos x=n/mȤcosǵɤͳuchiyamaƱ

m3m¸ߤȲꤹȡ

cos 2x=2n^2/m^2-1
=(2n^2-m^2)/m^2

m,nϸߤǤ顢mΤȤm^22n^2-m^2ϸߤǤǡդʬm^2Ǥꡢmm^2Ǥ롣

mΤȤդʬm^2/2Ǥꡢmm^2/2

äơm3ǤСcos 2^axФϤ٤ͭǡưۤʤ롣

Ȥp,qߤǤkȤȤ

cos k(q/p)=cos (2t/p)С ʢqk = 2tȤ
=cos (ps+r)/p2Сʢ2tpdzäs;rȤ
=cos (r/p)2

Ȥʤ뤬rϹ⡹p-1̤ǤΤǡ̷⡣

äơm=1,2

ʰʲά
PowerBook G4 519ڡ 13:10:34 MAIL:masaru-y@sansu.org HomePage:25104
uchinyan
Ϥˤϡ
ϡޤϡƻ˲򤤤ơǤǤäΤǡͤƤߤޤ

(ˡ1)
ƻˡǤ
ޤ餫ˡɤ뿧ϡ34ǤʤȥǤ

3֤Τ 4 ̤ꡢ֤Τ 3 ̤ꡢ֤Τ 2 ̤ꡢϡʬޤϥȥɤǤ
ƱξϡȤ⥤Ȥ㤦ɤ뤳Ȥˤʤ 1 * 1 ̤ǡΤȤ 2 ̤ꡣ
ۤʤ뿧ξϡ 1 ̤ǡƱϡ 2 ̤ꡢȰ㤦ϡ 1 ̤Ǥ
ǡ4 * 3 * 2 * (1 * 1 * 2 + (1 * 2 * 1 + 1 * 1 * 1)) = 120 ̤ˤʤޤ

֤Τ 4 ̤ꡢ֤Τ 3 ̤ꡢϡʬޤȥɤϤˤʤޤ
ƱξϡȤ⥤Ȥ㤦ɤ뤳Ȥˤʤ 2 * 1 ̤ǤΤȤϡ4ȤȤ狼 1 ̤Ǥ
ۤʤ뿧ξϡƱϡϤ4ȤȤ狼饦 2 ̤ˤʤꡢ 1 ̤ǡ1 * 2 * 1 ̤ꡢ
Ȱ㤦ϡƱȤˤϡϤ4Ȥ狼饨 1 ̤ 2 * 1 * 1 ̤ꡢ
Ȱ㤦Ȥˤϡ 2 ̤ 2 * 1 * 2 ̤ˤʤޤ
ǡ4 * 3 * (2 * 1 * 1 + (1 * 2 * 1 + 2 * 1 * 1 + 2 * 1 * 2)) = 120 ̤ˤʤޤ
ʾǤ٤ƤʤΤǡ120 + 120 = 240 ̤ꡣ

(ˡ2)
ͤޤ
̤ˡߤ򡢶̤ n Ĥζʬ 4 ɤʬο a(n) Ȥޤ
ȡa(n) ϡ
n-1 ξζʬΤȤڤ곫ƿʶɲä뤫
n-2 ξζΰĤĤڤ곫ƿʶɲä뤫
Τ줫ˤʤޤ
ΤȤɤʬڤ곫ϡоˤ̤ɬפϤޤ
(פˡ֥פˡ֥פɲä硢֥פȡ֥פȤϡ
ñʸդκʤĤζƱΤǤ)
ΤȤڤ곫ˤξ¦οϰۤʤΤ 2 ̤Ρ
ڤ곫ˤξ¦οƱʤΤ 3 ̤οζɲäǤޤ
ǡΩޤ
a(n) = 2 * a(n-1) + 3 * a(n-2)
郎̯ǡn = 1, 2, 3 ξϡɲäΩ
a(1) = 4, a(2) = 3 * a(1) = 12, a(3) = 2 * a(2) = 24
Ȥʤޤäơۤɤϡn >= 4 Ωޤ줫顢
a(4) = 2 * a(3) + 3 * a(2) = 2 * 24 + 3 * 12 = 48 + 36 = 84
a(5) = 2 * a(4) + 3 * a(3) = 2 * 84 + 3 * 24 = 168 + 72 = 240
Ȥʤꡢ240 ̤ˤʤޤ
ʤ⹻٥롩(¿ʬ̤ι⹻ˤϤä񤷤⡥)Ǥn >= 4 ǡ
a(n) + a(n-1) = 3 * (a(n-1) + a(n-2)) = ... = 3^(n-3) * (a(3) + a(2)) = 4 * 3^(n-1)
a(n) - (-1) * a(n-1) = 4 * 3^(n-1)
(-1)^n * a(n) - (-1)^(n-1) * a(n-1) = 4 * (-1)^n * 3^(n-1)
ꡢϲ򤱤ơ
a(n) = 3^n + (-1)^n * 3, n >= 4
Ȥʤޤȡa(1), a(2), a(3) ȤӤȡ
a(1) = 4, a(n) = a(n) = 3^n + (-1)^n * 3, n >= 2
̲ˤʤޤ
ͥνȡ 519ڡ 14:44:49 MAIL:uchi@sco.bekkoame.ne.jp 25105
uchinyan
Ǽɤߤޤ
#25095Τ餫ߤ#25087ΤҤܤβˡǤ͡ٶˤʤޤ
ʤǽͤƼԤˡľ顢ˤɤ夭ޤ
#25087ΤҤܤβˡˤäȻƤޤȯۤʬ㤤ޤ
ͤޤǤ˽񤤤Ƥޤ

(ˡ3)
ߤ˿ɤΤǤϤʤ5 Ĥζ˽֤ 1 5 ֹĤֹ񤤤ͤĤΰۤʤ뿧ɤäȢȹͤޤ
ȢäƤֹζˤȢοɤСդޤεοa(5) ̤Ȥޤ
⤷٤礦ƱɤäƤ⤤Ĥޤꡢ1, 2 Τ褦Ϣ³Ĥֹ桢5 ϡ4, 5 Ǥʤ 5, 1 ͤơ
ƱȢäƤƤ⤤Ȥȡ4^5 ̤Ǥ٤礦ɬפޤ
٤礦ϡ٤礦ֹ񤤤إǤäΤͤФǤΤȤ
Ĥ٤礦硧 4 ˤʤꡢκ 5 ̤ꡢȢؤ a(4) ̤ǡ5 * a(4) ̤ޤ
Ĥ٤礦硧 3 ˤʤꡢκ 5 + 5 = 10 ̤ꡢȢؤ a(3) ̤ǡ10 * a(3) ̤ޤ
Ĥ٤礦硧 2 ˤʤꡢκ 5 + 5 = 10 ̤ꡢȢؤ a(2) ̤ǡ10 * a(2) ̤ޤ
ͤĤ٤礦硧 1 ˤʤꡢκ 1 ̤ꡢȢؤ a(1) ̤ǡ1 * a(1) ̤ޤ
ɡ
a(5) = 4^5 - 5 * a(4) - 10 * a(3) - 10 * a(2) - a(1)
ƱͤΤȤa(1) a(4) ˹ͤȡ
a(1) = 4^1 = 4
a(2) = 4^2 - 1 * a(1) = 12
a(3) = 4^3 - 3 * a(2) - a(1) = 24
a(4) = 4^4 - 4 * a(3) - 6 * a(2) - a(1) = 84
a(5) = 4^5 - 5 * a(4) - 10 * a(3) - 10 * a(2) - a(1) = 240
ǡ240 ̤ꡣ
̲ǤǤ#25105(ˡ2)ǤäƤޤäΤǡάޤ

ͥνȡ 519ڡ 22:13:03 MAIL:uchi@sco.bekkoame.ne.jp 25106

ΤϴñǤ

ޤ¤ͤޤ
㤨С֤򥦤ɤȤȡ⤦Ĥ֤򥢤ɤ뤳Ȥˤʤޤȡ¤ĤλͳѤˡ֤ξüˤʤ褦ɤΤƱˤʤޤ
äơʣݣˡᣱ̤¤ޤ
˿ͤޤ
ĤοǡĤʬɤΤǡɬĤϣȤޤ
ȤͤФǤ顢ϣ̤ޤ
֤ϡޥĤ餻а㤦֤ˤʤФΤǡ̤ꤢޤ
äơᣲ̤Ȥʤޤ
519ڡ 16:34:41 25107
uchinyan
#25104ޥ뤵󡢲ˡ꤬Ȥޤ
٤äȵˤʤäΤǤå褦Ǥ
ˤʤä
m,nϸߤǤ顢mΤȤm^22n^2-m^2ϸߤǤǡդʬm^2Ǥꡢmm^2Ǥ롣
ξ⤽Ǥ֡פϰ̣ΤǤ礦
󡢤ʤΤǤ椬Ȥȡʲε̯ˤʤޤ
<פǤΤǤϡ
cos k(q/p)=cos (2t/p)С ʢqk = 2tȤ
䡡=cos (ps+r)/p2Сʢ2tpdzäs;rȤ
ѤǤ (^^;
2t = 2 * (ps+r) ǡt = ps+r Τ褦Ǥ
ξ硢kq Ǥʤȡt ˤʤ餺;꤬̣ʤʤޤ
¿ʬξ硢cos 2^axСפФƹͤΤǡk ϶ȤƤ褯ʤȤϻפޤ
Ȥʤ뤬rϹ⡹p-1̤ǤΤǡ̷⡣
⡹ p ̤ꡩcos(2) ϽƹͤƤΤǤ͡äȡ

¤ϡϡ4̤ˡϺƤޤ
ܤϡǼĤ˽񤤤ˡ
ܤϡ¿ѷγܱߤȾ¤դ椬ͭˤʤθƤ
ܤϡax = sin(x), a ͭξͭθƤ
ͤܤϡޥ뤵Τ褦ˡѴؿθ顢̵¸ĤͭиѴؿμ̷⤹Ȥ
Ǥ
ܤDz򤱤Τǡ괺­Ƥޤޤ
cos(/n) ͭ<=> cos(m/n) ͭ
ϡͤܤϺƤȤʪǤ
ͤܤϡޥ뤵βˡǥåꤷޤ
ʤܤϡ
¿ѷγܱߤȾ¤Ȥ¿ѷΰդȤ椬ͭȤʤΤϻѷ
Ȥ̣̿Ȥ꤬طȤ̣ޤäݤǤʵΤǤ
ޤܤ ax = sin(x) ͭϡ0 <= x <= 1/2 ˸ꤹȡa 2 3 ˸¤롢ȤϤ궽̣̤Ǥ
ˤ衢򤢤꤬Ȥޤ
󡢻⡢Ĥ򤤤Ǥ ^^/

ͥνȡ 519ڡ 17:46:33 MAIL:uchi@sco.bekkoame.ne.jp 25108
kobayashi
ɤ硡飳Ȥ߹碌ϣ̤ꡡǣսɤΤ֤ǣսꡡФǣսꡡǣսΤ褦ɤսοȤ߹碌ϡʣˤäƤΤΤҤȤĤƣܤФٹʤսϣ̤ǤγơФٹʤսˡϣ̤ꡡäƤΣɤʬˡϣߣᣱ̤ܤ̤ꡡΣϣ̤ꤢΤǣɤˡϣߣᣱ̤ꡡˣɤ硡ΤҤȤĤ򣲲Ȥ¾ϣ󤺤ĤʤΤǤο򣲲ȤƣܤФο򣲲Ȥˡϡս꤫ٹʤսƱɤˡ߻ĤΣսĤΰۤʤ룳ɤˡʤΤǣߣᣳ̤ꡡäƣɤˡϣߣᣱ̤ꡡɤʬϹפƣܣᣲ̤
519ڡ 21:14:18 25109
uchinyan
(ɲ)
(ˡ4)
#25106(ˡ3)֤򥤥᡼ʬ䤹Ǥ
4 ̤ꡢ 3 ̤ǡ
Ʊξϡ 1 ̤ꡢ 3 ̤ꡢ 2 ̤ꡣ
Ȱۤʤ뿧ξϡ 2 ̤ǡ
Ʊϡ 1 ̤ǡ 3 ̤ꡣ
Ȱۤʤϡ 2 ̤ǡ 2 ̤ꡣ
ɡ4 * 3 * (1 * 3 * 2 + 2 * (1 * 3 + 2 * 2)) = 4 * 3 * 20 = 240 ̤ꡣ
ϡ#25105(ˡ1)νǤǤ#25105(ˡ2)Τ褦ˡ̲ưפǤ
#25106(ˡ3)֤ǹͤޤʤn >= 4 Ȥޤ
դξο a(n)̤ˡn 1 ȰۤʤȢο b(n)n 1 ƱȢο c(n)Ȥޤ
ȡ餫ˡ
a(n) = b(n)
b(n) = 2 * b(n-1) + 3 * c(n-1)
c(n) = b(n-1)
줫顢
a(n) = 2 * a(n-1) + 3 * a(n-2)
Ȥʤꡢ#25105(ˡ2)Ƴޤ
ʤ#25087ΤҤܤβˡϡ
a(n) + a(n-1) = b(n) + c(n) = 3 * (b(n-1) + c(n-1)) = ... = 3^(n-3) * (b(3) + c(3)) = 3^(n-3) * (a(3) + a(2)) = 4 * 3^(n-1)
a(1) = 4, a(2) = 12, a(3) = 24 ꡢn >= 3 Ω뤳Ȥȡa(2) = 4 * 3
a(5) = 4 * 3^4 - a(4) = 4 * 3^4 - (4 * 3^3 - a(3))
= 4 * 3^4 - 4 * 3^3 + a(3) = 4 * 3^4 - 4 * 3^3 + 4 * 3^2 - a(2)
= 4 * 3^4 - 4 * 3^3 + 4 * 3^2 - 4 * 3
Ǥ͡ϡ줤ʤ
ͥνȡ 520ʶ 11:49:52 MAIL:uchi@sco.bekkoame.ne.jp 25110
ޥ
#25108
>ξ⤽Ǥ֡פϰ̣ΤǤ礦

ʥ̣ʤǤm(__)m

>cos k(q/p)=cos (2t/p)С ʢqk = 2tȤ
>䡡=cos (ps+r)/p2Сʢ2tpdzäs;rȤ
>ѤǤ (^^;

ۥȤǤ͡ߥǤ

>Ȥʤ뤬rϹ⡹p-1̤ǤΤǡ̷⡣
>⡹ p ̤ꡩcos(2) ϽƹͤƤΤǤ͡äȡ

pqϸߤǤ....ȤͤƤޤk=pξ⤢Ǥorz

ʤߤˤβˡǤܳѤθ̵¤¸ߤ뤳ȤˤʤäơθˡפäƤҥȤäΤDz򤱤ޤǤϤв򤱤Ϥޤʤ...(^^;;
PowerBook G4 519ڡ 22:17:44 MAIL:masaru-y@sansu.org HomePage:25111
⡼ޥ
櫓Ƥäޤ
ȤϴñǤϤʤäΤǤ͡

Ȥǡͭ̾

500߶̤äƤʤnͤ1000߻
äƤʤnͤι2nͤʽ֤
ĤäƤʤǤ롣դϳƿͤ500
ħ롣
ΤȤĤäƤʤդϤŬ
ͤʻפޤˡ˽򳫻ϤСĤ
­뤳Ȥʤ褦ˤǤ뤳Ȥ򼨤
519ڡ 23:31:23 MAIL:kennji72001@yahoo.co.jp 25112
ޥ
Ǥ͡ˤޤ...
520ʶ 0:13:33 25113
b4kumatyan
ԥ󥯤ȤƸꤷơϡȰ㤦̤ꡣϡȰ㤦̤ꡢȤƥޤǹͤȡ
ޤ᣸̤ꡣ
ԥ󥯤ˤʤäƥʥϣ̤ꡣ
äơԥ󥯤ΤȤդΤ
ݣ᣶̤ꡣ

¾οˤĤƤƱͤ
ᣲ̤ꡣ

ťä
520ʶ 0:35:07 25114
̵
ۿѥĴ٤ơ˿ƤϤǹͤޤ
ʥˤ˿A,B,C,DƤϤ롣
򴹤ƱˤʤϽ㤨Сʥˤˤ(ABA)(BAB)ƱȤ롣
ۿѥͤȤA˸ꤷ,BB,BCȹͤȴñǤ롣

ɤԲġ

ξ(ABCAB)(ABACB)(ABABC)(ABCBC)(ABCAC)Σ̤ꡣ
4̤ꡣοʬ36̤ʤΤǡ465=120̤ꤢ뤳Ȥˤʤ롣

ξ(ABCDB)(ABDAC)(ABDBC)(ABADC)(ABCDC)Σ̤ꡣ
1̤ꡣοʬ412̤ʤΤǡ1125=120̤ꤢ뤳Ȥˤʤ롣

äȤ߹碌Ϲ碌240̤ꤢ뤳Ȥˤʤ롣
520ʶ 7:54:15 25115
̾õ
ʬ
ξοL(n)= 4*3^(n-1)
ĤξοO(n)
ξԤδطϡ
n = 1,2ΤȤL(n)= O(n)
n >2 ΤȤL(n)= O(n)+O(n-1)

äơk>=1 ΤȤ
O(2k)= {i= 1 to k}L(2i)- {i= 1 to k-1}L(2i+1)
L(x)ͤơO(2k)= 3^2k+3
O(2k+1)= {i= 1 to k}L(2i+1)-{i= 1 to k}L(2i)
ʤȤСO(5)=L(5)-L(4)+L(3)-L(2)Ǥ
L(x)ͤơO(2k+1)= 3^(2k+1)-3

ޤO(n)ϡ
L(n)= O(n)+O(n-1)
L(n)=3L(n-1)顢
n>=4ΤȤ
O(n)+O(n-1)=3O(n-1)+3O(n-2)
äơ
O(n)=2O(n-1)+3O(n-2)
520ʶ 8:20:24 25116
uchinyan
#25116#25105(ˡ2)#25110(ˡ4)դƱǤ͡
äˡL(n) = b(n) + c(n), O(n) = a(n) Ǥ͡
ͥνȡ 520ʶ 11:40:51 MAIL:uchi@sco.bekkoame.ne.jp 25117
uchinyan
#25112
ŪǼˡǾǤޤ͡
ˤإԥåȤǤ㤦Τʡ
#25113
βˤΤǤǤͤ
ͥνȡ 520ʶ 13:08:30 MAIL:uchi@sco.bekkoame.ne.jp 25118
ޥ
#25118
ȸäƤ⤵ˤΤޤޤǤϤޤ󤱤ɡޤݤϻ褦ʤΤäȡ
PowerBook G4 520ʶ 13:33:20 MAIL:masaru-y@sansu.org HomePage:25119
uchinyan
#25117Ѥߤޤ󡢸­餺ǤΤǤʤ
#25116#25105(ˡ2)#25110(ˡ4)դƱǤ͡
äˡL(n) = b(n) + c(n), O(n) = a(n) Ǥ͡
#25116ɤ֤ƤߤơL(n)λȤޤʤȴޤ
äˡO(n) a(n)ľܤ˵L(n)ͳǵưפǤ
O(n)L(n)θɽΤ⼫Ǥ
ٶˤʤޤ
ͥνȡ 520ʶ 14:38:17 MAIL:uchi@sco.bekkoame.ne.jp 25120
⡼ޥ
#25118
ºݤˤɤ餪ħФθĤʬǤ礦
520ʶ 16:01:26 MAIL:kennji72001@yahoo.co.jp 25121
ϥ饮㡼ƥ
ץǤǤޤ
ԤƤޤ
̶彣 520ʶ 17:04:41 HomePage:ĩ25123
uchinyan
#25121ɤʡ (^^;ǼŪ˲ǽʤΤʡ
ιͤ񤤤ƤޤξǤϡǰʤ顢ä̵Τ褦ʵޤ͡
(#25112Τ򽸤ξ)
n = 1 ϡ500߶̤äƤͤϤФΤǡ餫
n-1 ʲǡȲꤷޤ
500߶̤äƤͤŬ˰Ǥοͤ鳫ϤȤޤοͤ 1 ֹĤƤޤ
⤷ǺǸޤǤޤСʤǤ
ޤʤä硢1000߻äƤƤĤ꤬ʧʤʤǽοͤ k+1 ܤοͤȤޤ
ȡ1 k ǤϼդλäƤ500߶̤ 0 İʾǡk Ǥ礦 0 ĤˤʤäƤޤ
ΤȤ1 k οͤϡ500߶̤äƤͤ1000߻äƤͤȤƱǤ
Ǥʤȡk Ǥ礦 0 Ĥˤʤޤ
ǡk >= 2 ǡ 1 k οͤĤο͡k+1, ..., 2n-1, 2n ϡ2n ͤ꾯ʤ
500߶̤äƤͤ1000߻äƤͤȤƱˤʤäƤޤ
äơĤοͤ˵Ǽˡβ꤬ŬѤǤơĤοͤˤϡɬդħޤ
ξγϤͤ A Ȥޤ
ȡk+1 ΰ2n A (A 2n ξ)Ǥϡդ500߶̤ 1 İʾǤ
ơ1 k οͤᤷ 2n ͤξ硢 A 鿷ħ򳫻Ϥޤ
ȡA 2n ޤǤϽƱʤ1 k Ϥ⤽ʤ
k+1 ֤Ǥդ 1 İʾ500߶̤äƤΤʤϡƱʤ٤Ƥޤޤ
()
ξǤϡޤʤä硢ڤӤ 1 k ꤬ǤˤʤΤǡϤͤꤹΤ̵Ǥ͡
¾ΤޤФΤǤ
ͥνȡ 520ʶ 17:31:21 MAIL:uchi@sco.bekkoame.ne.jp 25124
⡼ޥ
#25124uchinyan󡢤꤬Ȥޤ
줫äɤޤƤޡ
520ʶ 19:23:51 MAIL:kennji72001@yahoo.co.jp 25125
Ŵƻ˥ޥʥޥǡ棱
סΥǤ
ޤɤΤ褦ʬɤ뤫ϡʣˡʣˤΣ̤ꡣ
ʣˤξ
ʬɤϡʥˡʥˡʥˡʥˡʥˤΣ̤ꡣ
Ȥʬɤϡʬ;ɤФΤǡ᣶̤
ʬΤȤˡ򣴤ĤΤǡᣱ̤
ʣˤξ
ޤʬϣ̤ꡣ
ǡĤꣴĤȤϡ¤Ǥ뤳Ȥ˵Ť
äơ٤ʤ褦ˣɤ뤿ˤϡʺü飲ܡˡʱü飲ܡˤΣ̤ꡣ
ʬλϡʬˣ̤ꡣʬ˻ĤΣ̤ꡣäơᣱ̤ꡣᣱ̤
ΣĤ碌ơܣᣲ̤ꡣ

ȤڭҤǤʥޥǡˡ 520ʶ 21:44:16 25126
оϩ
ˣɤξ
󤷤Ȥʤϡ4̤ꡣ
ޤ󤷤Ȥʤɤ5̤ꡣ
Ȥϡ3C2=3̤ꡣ
ޤ¾ʬοɤ2̤ʤΤǡ
4*5*3*2=120̤

ˣɤξ
Ȥ4̤ꡣ
Ȥʬλ5̤ꡣ
Ĥʬοɤ3*2*1=6̤ʤΤǡ
4*5*6=120̤

ʾꡢ120+120=240̤

ǤͤƱˡǤ(^^;
γ 520ʶ 21:58:43 MAIL:pctakada@mail.goo.ne.jp HomePage:Ĥ줭25127
LEN
#25112
500߽ͭԡ+1ɡ
1000߽ͭԡ-1ɡ
ؤǤդγϰ֤i=0Ȥf(0)=0)ײ˿ʤȤ
ͭԤȤˡi1󥯥Ȥ
500߽ͭԤʤСf(i)=f(i-1)+1,
1000߽ͭԤʤСf(i)=f(i-1)-1Ȥ롣
ϰ֤ȡf(2n)=0=f(0)ˤʤ롣
δؿf(i)ϡiν0Ƚʹ2nˤơͤȺǾͤġ

äơf(i)Ǿˤʤ֤ϤС
­뤳ȤϤޤ

ȿײ˿ʤȤϡײΤȤf(i)ˤʤ֤
f*(i)Ǿˤʤޤ
f*(i)ϰ֤i=2nȤ(f*(2n)=0)ȿײ˿ʤȤ
ͭԤȤˡi1ǥȤ
500߽ͭԤʤСf*(i)=f*(i-1)+1,
1000߽ͭԤʤСf*(i)=f*(i-1)-1Ȥ롣
ΤȤf(i)+f*(i)=0
521ڡ 3:25:24 25128
uchinyan
#25128
LEN󡢶̣ɤޤĺޤǤ
⡼ޥ󤵤󡢤βˡʤСħ򳫻ϤͤǤǤΰ̣ǡβˡͥƤȻפޤ
äȺ٤­Ƥ⤷ޤ

#25112
500߽ͭԡ+1ɡ
1000߽ͭԡ-1ɡ
ؤǤդγϰ֤i=0Ȥf(0)=0)ײ˿ʤȤ
ͭԤȤˡi1󥯥Ȥ
ϡi 򡢼դħ򳫻Ϥξ֡ȹͤ褦˻פޤ
i = 0 ϡħ򳫻Ϥξ֤֡Ǥ͡i = k ϡk ܤοͤħξ֡Ǥ
500߽ͭԤʤСf(i)=f(i-1)+1,
1000߽ͭԤʤСf(i)=f(i-1)-1Ȥ롣
䳫ϰ֤ȡf(2n)=0=f(0)ˤʤ롣
ֳϰ֤ȡפϡħ򤷽ä֤ǤϡסȤȤǤ
δؿf(i)ϡiν0Ƚʹ2nˤơͤȺǾͤġ
ϡf(i) ͭ¸ĤνʤΤǡ餫
äơf(i)Ǿˤʤ֤ϤС
Τˤϡf(i)Ǿˤʤ֤μοͤϤСפʡ
פˡf(m) = Ǿ Ȥơm+1 鳫Ϥ롢ȤȤǤ
­뤳ȤϤޤ
٤Ƥ i Ф f(i) >= Ǿ = f(m)ʤΤǡg(i) = f(i+m) - f(m) ȤСg(0) = 0 ǡg(i) >= 0 Ǥ
ʤײ꤫ȿײ꤫ϡܼȤϴطʤǤ͡

ħ򳫻ϤͤꤹˤϡͿ줿ºݤ¤ӤФơf(i) ׻ƺǾͤФ櫓Ǥ
ǾͤͿ i դǤʤ⤢ޤξˤϤΤɤȤäƤ OK Ǥ
f(i) μˡ괺ɤοͤ鳫Ϥ뤫Ǥޤ
Ǥϡf(0) = 0 ͿΥʤΤǡŪˤȼʿ԰ưʤΤǡɸǤ˲᤮
ɤ f ѤƤ⡢ǾͤƱǡбͤ⡢ɸΥθСƱǤ
פϡa 500̤äƤ뤳Ȥ [a,+] Ƚ񤯤Ȥơ[a,+], [b,-], [c,-], [d,+], [e,-], [f,+] ȤʤäƤȤơ
a Ϥȡf(i) ϡ0 (= f(0)), 1, 0, -1, 0, -1, 0 ʤΤǡǾͤͿΤ i = 3 or 5 Ǥϡc e ǡ
b Ϥȡf(i) ϡ0 (= f(0)), -1, -2, -1, -2, -1, 0 ʤΤǡǾͤͿΤ i = 2 or 4 ǤϡϤꡢc e Ǥ
ĤޤꡢɤǷ׻Ƥ⡢Ϥͤϡd f Ǥޤ
ǤޤΤǤϡ
ͥνȡ 521ڡ 11:50:41 MAIL:uchi@sco.bekkoame.ne.jp 25129
⡼ޥ
#25129,#25128:
LENuchinyan󡢲ˡ꤬Ȥޤ
錄ϿǤǤʤơ
ľŪˡ500-1000ߤϢ³ˤʤȤ򼡡˾äƹԤȡǸ1Ȥ500-1000κĤΤǡλĤä500ߤοͤ齸᤿餤˻פΤǤǤ礦
ߤʤƱȤäƤ⡦
521ڡ 12:51:38 MAIL:kennji72001@yahoo.co.jp 25130
uchinyan
#25130
ľŪˡ500-1000ߤϢ³ˤʤȤ򼡡˾äƹԤȡǸ1Ȥ500-1000κĤΤǡ
䤽λĤä500ߤοͤ齸᤿餤˻פΤǤǤ礦
ΤˤʡࡢǤ͡Τϻä (^^;
ߤʤƱȤäƤ⡦
βǽϤޤͥƤȻפޤ
ʤLENβˡǤϡǽʿְͤ٤ˤ٤ơǤͥƤȻפޤ
󡢥⡼ޥ󤵤ˡǤǽǤä򤤤ʤȤʤʤȻפޤ
ˤ衢βˡ֥Ǥ͡
ࡢޤޤԤ­ޤ󡥡ܤڤǤ ^^/
ͥνȡ 521ڡ 13:28:31 MAIL:uchi@sco.bekkoame.ne.jp 25131
ƣ
򤽤줾죱Ȥȡ򣱡򣲤ɤä硢⤦Ȥäϣ̤ꡤ⤦Ȥä⣱̤Ƿף̤ꡣȥȤ߹碌ϡȣȣȣȣȣȣȣȣȣȣȣȣΣ̤ʤΤǡߣᣲ̤ꡡߤĤ֤ťäơäȤȤޤ
521ڡ 13:18:01 25132
⡼ޥ
#25129,#25128:
LENuchinyan󡢲ˡ꤬Ȥޤ
錄ϿǤǤʤơ
ľŪˡ500-1000ߤϢ³ˤʤȤ򼡡˾äƹԤȡǸ1Ȥ500-1000κĤΤǡλĤä500ߤοͤ齸᤿餤˻פΤǤǤ礦
ߤʤƱȤäƤ⡦
521ڡ 13:39:12 MAIL:kennji72001@yahoo.co.jp 25133
ޥ
ڤɡǤɤΤ餻Ǥ

γƽŽäƤäʥå˱Ƥ⤬ġˤűѤƤߤޤۤɤˤʤ櫓ǤϤʤ⤽ˤäƤΥڡŤʤäƤΤˤʤޤơʤޤϹΥڡǤϤʤڤȤϻפޤ

# WILLCOMξҲȤƤޤϤϤäƤʤ־˹פǤΤǡ(^^;;
PowerBook G4 521ڡ 23:03:24 MAIL:masaru-y@sansu.org HomePage:25134
uchinyan
ꡢ⤦ġˡפĤޤ(ˡ3)䡢ҤܤβˡˤäȻƤޤʤ㤤ޤ

(ˡ5)
٤äʬο㤨ɤʬǽ̵뤷ƿ夫Ƥޤ
ɤʬ̵뤹С4^5 ̤Ǥ
줫顢ȥȥΤ褦ˡ٤äʬƱˤʤΤޤ
ϡ٤äʬζǤζȤäƤޤäΤͤФΤǡ5C1 * 4^4 ̤ꡣ
ȥȥΤ褦˻٤äΤ᤮Τǡ­ޤ
ϡƱͤ˹ͤơ٤ФΤǡ5C2 * 4^3 ̤ꡣ
ˡͤ٤äΤ褷ƤޤäΤǡޤ
ϡ򻰤٤ФΤǡ5C3 * 4^2 ̤ꡣ
ǸˡޤĤʬĤˤʤä᤮ƤΤǡ­ޤ
ϡߤĤ뤳ȤꡢޤǤȤäȰäơͤ硢5C4 * 4 ̤ꡢ­Ǥ­᤮ǡ
ޤ硢5C5 * 4 ̤ʤȥǤ
ǸʬߥǤɡ
4^5 - 5C1 * 4^4 + 5C2 * 4^3 - 5C3 * 4^2 + 5C4 * 4 - 5C5 * 4
= 4^5 - 5C1 * 4^4 + 5C2 * 4^3 - 5C3 * 4^2 + 5C4 * 4 - 1 - 3
= (4 - 1)^5 - 3 = 3^5 - 3
= 240 ̤
ˤʤޤ
ΰ̲ϡưפǤ餫ˡ
a(n) = 4^n - C(n,1) * 4^(n-1) + C(n,2) * 4^(n-2) - C(n,3) * 4^(n-3) + ...
+ (-1)^(n-2) * C(n,n-2) * 4^2 + (-1)^(n-1) * C(n,n-1) * 4 + (-1)^n * C(n,n) * 4
= (4 - 1)^n - (-1)^n + (-1)^n * 4
= 3^n + (-1)^n * 3
ˡʤޤ
ͥνȡ 522 22:38:59 MAIL:uchi@sco.bekkoame.ne.jp 25135
N.Nishi
ƻ˥ѥޤ
524ʲС 5:48:44 25136