river_history
ɿ夲Ǥ줫ޤ롼ͤޤ
1121ڡ 0:11:23 41122
abcba@baLLjugglermoka
ޥܤʿԤĹϡ2ְդȤ߹碌ʣƤ硣

äơޥܤʿԤĹʤϡߣΥǤ7Ĥ2ĤʤΤǡ7C2=21̤ꡣ

̤ˣΡߣͤΥ(Ρ)ǤϡޥܤʿԤĹǤʤϡNC2(Ρ1)/2̤ꡣ
1121ڡ 0:21:20 41123

ºݤǤäƹͤޤĤˡ21ĸĤˤϹͤޤȤޤäƤʤȸǤ
1121ڡ 0:27:08 41124
algebra
г ܣܣܣܣᣲ
1121ڡ 0:29:09 41125
ʤ
33ξܤΤȤ6ȯơ22ξܤɲäʤͤơ1դ(2n+1)ΤȤn(n+11)/2

1121ڡ 1:29:01 41126

οޤ򻲹ͤ˲򤭤ޤ
1121ڡ 1:56:25 HomePage:41127
̴
οޤҥȡȤž䤷ƤȡǺ硣¨Ǥ͡ڤƱǤ
ٻΤ桡 1121ڡ 6:57:41 41128
ϥ饮㡼ƥ
Ϥ褦ޤ϶ǤǧڤǤ
1121ڡ 8:10:04 HomePage:湩ؤ˥41129
ǯ
ޥܤפ1010ȴ㤤Ƥޤ77ä
ԡ󶦤2^n1ξˤϡn(2^n1)ȤʤΤʡn=321
n=1ξ111ġĹϺޤ󤬡ˡn=2ξ336ġn=4ξ151560ġn>=5̤ǧ
ޥܤ2ĤƤ¤٤ơ2^(k-1)(k=1,2,...,n)κüФᱦعФ¤٤롣ΥޥܤϤ߽ФƲΥޥäʬϡΥޥ˽Ťͤ롣
̤algebraȰǤ͡

1121ڡ 17:04:31 41130
ʥޥ
Ϥ褦ޤ

(ε¿ʬȽƤޤm(__)m
#31136˵)

Ŀ֤ΤϤ줤ʷ
ξ濴˴ؤоΤΤȤʤȾˤĤޤ
Ĺ3ĤγѤ֤Ƥ褤Τǡ
̩ˤʤ褦ˡǾñ̤3ĤγѤ¿褦
гФ¤ٲοޤΤ褦оΤ֤ޤ









֤ꡢ֤ʤꡣ
Ȥϡˡ§ȡоΤȤȤä֤Ƥߤ
ɤ21֤ޤ
ȤԳΤʲˡǤ٤βǥåǤ

ŪʲˡͤޤϤޤʬʤǤ
狼äޤ񤭤ޤ

1122ʶ 1:46:24 41131
⡼ޥ
錄оΤȤ4x4γѤ濴˲žСĹϤǤʤ
4x4κϰ֤Ƥߤޤ ^^
oФǤ

xooo
oxxo
xxxo
oxox

žΤϡ3*4+2*4+1=21
4x4κ֤ǤɤˤϤʤޤ^^;
1121ڡ 10:24:33 41132

οޤäΤǤ餯λǤƤޤޤ
θ׻ǸĤˡפĤޤ˽񤭹dzΤޤ
ꤳߤʤιͤ򸫤ΤڤߤǤ
1121ڡ 10:31:18 41133
uchinyan
Ϥˤϡơϡ
ࡤϤ褯ʬ餺

ޤ֤ 17 ĤǡƤΤǡ˻Ժ 21 Ĥ餤ʤǧڡ
Ȥ𤱤ʤˡǤ

褯ʬʤΤǷǼĤɤǤߤޤԺʳˤϡ

#41123 7C2 = (7 * 6)/2 = 21
#41126 2n+1 ΤȤ n(n+11)/2 n = 3 Ǥ (3 * (3 + 11))/2 = (3 * 14)/2 = 21
#41126 2^n - 1 ΤȤ n(2^n - 1) n = 3 Ǥ 3 * (2^3 - 1) = 3 * 7 = 21

ࡤɤ줬Ȥ¾򤬤롩
ͥνȡ 1121ڡ 13:51:02 MAIL:uchi@sco.bekkoame.ne.jp 41134

̲˥ȥ饤Ƥߤޤ

󡢽ģԤγʻΤ1Ԥ
2սϡ
nC2n(n-1)/2
̤ǡʤ٤ԤǶʬ礬
ǤФ¿ȤȤˤʤޤʾά
1ʿѡ(m-1)/2n

ƹԤpĤޤp+1ĤθФ֤ȤС
pC2塡(n-1)/2㡡(p+1)C2
p(p-1) (n-1) p(p+1)
1/4äưʬ򤹤ȡ
(p-1/2)2塡n-3/4㡡(p+1/2)2
𡡡ᡡInt{1/2ܢ(n-3/4)}
Ȥʤޤ
IntʲڤΤơ

ƹpĤθФʤѥʸ2ĤȤ߹碌ˤϡ
n(n-1)/2-np(p-1)/2
̤ǡκ뤿ᡢ11ĤĸФ䤷Ƥޤ

Ф1䤹ԿqԤȤޤ
1ԤθФpĢp+1˸Ĥ䤹硢2ĤȤ߹碌
p̤Τǡ
qϾ嵭ʤѥpdzäˤʤޤ
;ڤΤơ
qᡡInt{n(n-1)/(2p)-n(p-1)/2}

嵭ѤФοιפϡ
np+q
Ȥʤޤ

ڤȡ
n p q Фο
3 2 0 6
4 2 1 9
5 2 2 12
6 2 4 16
7 3 0 21
8 3 1 25

100 10 45 1045
1121ڡ 20:46:31 41135
ʥޥ
Фϡ

#41131
ν񤭹ߤǤ
оΤǤ20Ĥˤʤʤ褦ʵΤ
ޤȤȤˤʤޤǤ礦()

줤ʷȻפäޤǤϤ褫äǤ
г˴ؤоΤˤΤʬ䤹ߤǤ









ȤǤФΤǤ(^_^;)
ηǼϤ˰쾣餻ơĤȰ̲ϻ϶ġ
ĿäƤΤʬʤǤ

ԥ塼򶵤㤤Ǥ͡
βˡ񤷤ǤǤ褦ؤƤߤޤ

1122ʶ 1:41:31 41136
pao
̵֤ˤʤޤ̲ʤ餳դ꤬ҥȤʤΤȻפޤJޥޤΤҥȤ˹ͤƤߤޤ
ޥܣᣳ
ޥܣܣ᣶
ޥܣܣ᣹
ޥܣܣᣱ
ޥܣܣܣᣱ
ޥܣܣܣܣᣲ
Τˤ褦Ǥ͡
֤Ȥ˸ڤƤߤʤޤ
1122ʶ 9:29:38 41137
uchinyan
#41135#41136#41137
̲ؤΥץ꤬Ȥޤ

伫Ȥϡλǡ#41136#41137Ʊ褦ʹͤǼοޤƤޤ

o o x x x o x
x o o x x x o
o x o o x x x
x o x o o x x
x x o x o o x
x x x o x o o
o x x x o x o

¿ʬפȻפޤ
¾ξƱͤΥץ#41137μФ뤳ȤϳǧƤޤ
Ĥޤꡤn * n ȤơΡǽϡ
n + (n-1) + (n-2) +
ʤΤǤ֡ġפΤȤ̯Ѥäȴ礱ꤷƤΤǤε§ʬʤ
ȤȤǤ

#41135ϡäݤɽˤʤޤ㤦ץǼȤǤ褦ʤΤǡ
֤򸫤ĤɤǤߤޤ͡
ͥνȡ 1122ʶ 12:51:39 MAIL:uchi@sco.bekkoame.ne.jp 41138
ǯ
#41130ˡϡ֤ФĹǤʤȡƤ֤ˤФ֤ʤȸˡǤäơݾڤˤϤʤäƤޤǤ
ޥܤ3,7ξ˰פˤʤ褦Ǥ
ޤ

1122ʶ 13:25:32 41139
⡼ޥ
錄Τ⤪ä褦ǡĤøʤȤν񤭹ߤǤߤޤ...^^; Orz

#41137 paoοͤǤ줤ο缭ŵϤСĸ˥ҥåȡ

A072567 Size of maximal set of points in an n X n (side length) grid such that no 4-set of it forms a rectangle with horizontal/vertical sides.

1, 3, 6, 9, 12, 16, 21, 24, 29, 34, 39, 45, 52 ޤǤʤ?

ΤȤȻפޤĤ̼ʤ͡?
1122ʶ 18:10:40 41140
ʥޥ
Фϡ
ͤιͻʤˤޤȤƤߤޤ
Ŧޤޤ
礭ʿޤǼ餷ޤm(__)m

šššŨ
šŨšš
šŨšŨ
šŨšŨ
šššŨ
šŨšš
šŨšŨ
šŨšš
šššš
šŨšŨ
šššš
šššššššššššššš
šš
šš
šššššššššš
šššššššššš
šššššššššš
šššššššššš
šššššššššš
šššššššššš
šššššššššš

ʴη֤ǤϤʤǤ礦
(ְ㤤ƼϤ)
ȡnǤ¤ꡢ
n(n-1)(n-2)(n-5)(n-6)(n-9)(n-10)(n-13)(n-14)ܡš
׻Τ(n-1)ĤΤȤκĿ⤷ʤ

ˤʤäƤۤȤߤޤm(__)m

1122ʶ 19:05:23 41141
ʥޥ
#41140

⡼ޥ󤵤ο̯ˤƤ褦ʡ
4Ǥʤ3ĤʤΤǤ礦ġ
ĥäƥꥢƤp(^-^)q

ɵ狼ޤ(>_<)
4Ĥˤäȷ֤55λͳѤǤƤޤȤ˺Ĥޤ
ޥߥǤǤǤȼ餷ޤ
uchinyanϤ⤦äȤλˤϵĤƤ餷
§ĤʤȤääƤΤǤ͡
1122ʶ 19:19:30 41142
fumio
Ϥ褦ޤ
򤫤äǤǤϤǤϡ
1123ڡ 6:03:29 41143
ʥޥ
Ϥ褦ޤ
īƬƯޤġϢꤹߤޤ
(>_<)Ԥ龯ؤӤޤ
ɮ

A1A2
ŨŨš
ššššššš
šššššš
a1šššššš



a2






гn٤(n-1)ȿ٤(n-2)ϸꤷơ
ޤΤ褦˲Фֳ֤֤A1, A2, ()
ֳ֤֤a1, a2()Ȥޤ
(A1A2A3, a1a2a3ĤȸǤ
nڤƤޤȤֳ֤礭ȤʤƤ
ĹǤʤȤ뤫⤷ʤǤ)

ξ硢a1=4(a1A1)ȸꤹ뤫
A1=4(A1a1)ȸꤹǡ礭θĿ
ȤʤˤʤȻפޤ(꤫ǤϤʤ)

κݡIJκƱˤʤĹ
ǤƤޤᡢʲξޤ
n8
A1A_kޤǤ¡(Ĥ)a1a_mޤǤ
(âk, mǤ)
ߤޤ󡢽13ȡȻפäΤǤ
Ĥͤйͤۤ
n礭ˤ뤳Ȥˤʤ褦ʵ
졢ǸǤʤʤäƤޤޤ(T_T)

a_pa_qޤǤ¡A_rA_sޤǤ
(pq, rsǤ٤Ǥ)
Ȥʤ뵤ޤ
âξǤĹnnΤʤ˼ޤȤΤߡ
(ʣƤ夲(*_*))

椬ˤʤäƤޤȤ
1ʳ¦μФӤޤ(Ǥ)

ߤ
n=6Ǥa1=416
n=7ǤA1=421
n=8ǤA1=4, a1=624
n=9Ǥa1=4, A1=629
n=10Ǥa1=4, A1=634
n=11ǤA1=4, a1=639
ȸꤷΤˤʤäƤȻפޤ
̲ɤn礭ʤȡʡ䤳ʤꤽǤ͡

ɤޤ⤷ƤʤǤ
狼ΤȤꤽäΤǡȤ餱Ȥϻפޤ
ХХͤùǤޤġ

1123ڡ 14:33:48 41144
Ф
Ҥ֤Ǥ
г¤٤ƤʿԤ˲ܰ뤫񤤤Ƥߤޤ
1123ڡ 10:48:55 41145
uchinyan
ȿʤߤޤ͡Ǥ뤳ȤΤˤʤäʤ

#41135
󡤰ǤȻפޤǡ֤١פͤ줽Ǥ͡
Ϥ공η׻ǤꡤޤθĿΥѥݾڤϤʤΤǡ
֤١פǤʤ褦Ǥ
ºݡn = 8 ΤȤ 25 ĤˤʤޤäƤߤƤ 24 ĤΥѥ󤷤ĤޤǤ
ɤΤȻפäƤΤǤ

#41140
⡼ޥ󤵤󡤸ڤӾ󶡤򤢤꤬Ȥޤ
ä褦ˡɤ餳Ǵְ㤤ʤ褦Ǥ͡
ȤȤϡϤꡤn = 8 24 ĤΤ褦Ǥ͡

#41141#41142
ʥޥޤ󡤤ⲿ餫ε§Ūưפ˸ĤΤǤϤʤȻפäΤǤ
ʤΤǰճʤȤĹǤƤꤷơñǤϤʤ褦Ǥ

#41144
ϡǻ֤򸫤ĤɤǤߤޤ͡

ˤ衤ñʥ롼ʤΤǡտĴ٤в餫ε§ϸĤꤽʵ⤹ΤǤ

ΥץǤΥѥ󤬺ĿͿ롤ȤȤϡӡɬפޤ

ǽ n * n г¤ӤϡɤιԤˤɤˤФĤʤĿ¤ӤǤ
ǡ˰ĸФɲäȡդޥܤľʿԤĹȾʬޤ
Ĺˤʤʤ褦˽缡Ф֤ƤΤΥץܼʤΤǤ
Ĥ̤ʤȤĹΤ񤷤櫓Ǥ
ȡȯΡɤιԤˤɤˤФĤʤĿ¤ӡϡгǤϤޤ
Υץʾϡ㤦ȯ¤ӤǤƱ褦ʵ§᤬äơɤƱĿʲ롤
ȤȤ򼨤ɬפޤ
äȤ⤳ϡͤƤϤޤ󤬡ԤؤDzȤʤΤΤޤ

ޤ̤ϤΤȤϸƤ褯гϤ륢ץDzʬʤΤʤ
äȤܤڤβˡФ줷ΤǤ⡼ޥ󤵤ξ󤫤餹ȡɤ񤷤Ǥ͡
ͥνȡ 1123ڡ 14:13:51 41146
ma-mu-ta
ν񤭹ߤǡ§ˤʤäΤǤĤƤߤޤ
ʤȤ˥åƤϼ¿Ф¤Ӥˡ֤ȤȤ򷫤֤ƤǤޤǤ

https://dl.dropboxusercontent.com/u/12482898/challe864-2.png

Jޥޤ˹碌ơгγʻθĿnȤƤޤ
n2213
n33216
n44329
n554312
n6654116
n77652121
n8876324
n99874129
n1010985234
n11111096339
n1212111074145
n13131211852152
n1414131296155

ŪJޥޤοޤƱ褦ʤȤˤʤޤA16a14Ǹꤷ褦˻פޤ
ʶĤ褦 n8n11 A14a16 Ȥȡ6776ĹǤޤ
ȡn369ǿʼФ󤬲ääƵ§ŪƤޤ
n7n13Ǥ§Ū˱γѤ1ġĤȤˤʤޤ
餬äȤ§ˤʤʤȤǤ礦

uchinyan󤬶Ĥ褦ˡ§ȯȺĿξǡ񤷤
⡼ޥ󤵤󤴾Ҳ缭ŵˤϡa(7)Mathematical Gems III MAAԡˤ
ѥȤƾҲ󼨡ˤ줿Ȥޤ⵹ʤ뤫ʡǤ

ɵn1457ϴְäƤ褦Ǥ
֤ʤΤǡ٤ʤ뤫⤷ޤ󤬸ǤľƤߤޤ
ɵ2ˡn1455Τ褦ǤεܤƤޤ
n121314κʬ(1)(2)(1)§Ū˽ФƤ褦Ǥ
缭ŵ˵ܤƤn13ǽäƤΤϤ§θΤǤȻפޤ
1124 2:51:42 41147
ʥޥ
#41147

ma-mu-ta󡢤ʤʬ䤹ľƤäƴǤ
äǤ͡
ȤϤۤ§ˡ§ɳ򤯤Ǥ礦ġ
n礭ʿΤȤ
(a_pa_qޤǤ)=(A_rA_sޤǤ)
Ȥʤ꤫nnĹǤƤޤΤӽ륱
ΤɤˤʤΤǤʬˤʤޤ
ͤΥ٥ι⤵
ϴФǤ
1123ڡ 22:22:20 41148

#41146

uchinyanۤͻ򸡾ĺ꤬Ȥޤ
ʡذ̲˥ȥ饤٤˼ԤʹǤ
κͤϽФƤ⡢ºݤˤ̵Ȥ
ꤨȤϻפäƤޤ
n=8᤯ФȤϡѤǤ
(25Ĥβ̵ȤϾǤޤ)

ϤۤȤˤĤ¿̤꤬Фƴޤ
ɸ 1123ڡ 22:29:06 41150
ʥޥ

Фϡ

nʬ礭ʿΤȤɤʤΤͤޤ
ƱΤĤȡĹǤƤޤΤǡ
#41144ǿ夲ޤA1, A2, Ĥa1, a2Ĥ򳫤ơ
ܤοľȤƹͤޤصгΰ֤֤ơ
α¦2A0¦1a0Ȥޤ
ɤ餫顢ޤñ˸ߤˤƤФ褤Τ
狼ʤǤΤ褦ʴȤɤǤ礦ġ
˿ľ2ĤΤĹ
Ťʤʤ褦Ƥ
a3a2a1a0A0A1A2A3

161041261222

г

㤨СĹ11ʤΤA2֤Ȥ
a2+a1+a0+A0=10+4+1+2=17=6+11=A1+(A2)
ʤΤǡ11ϽA212Ȥ롣
Ʊͤ19A3֤Ȥ뤬
a3+a2+a1+a0=16+10+4+1=31=12+19=A2+(A3)
ȤʤΤǡ22֤

ˤư֤ΤǤϤʤȡ
nnǻڤ뤳ȤǶ§ѥ
ƤΤǤϤʤȿ¬ޤ
񤷤ʤΤǤ͡¿ʬ
n̵¤ä˶ᤤΤǡޤΩޤ󤬡
;̤Ȥm(__)m

ɵгΰ֤1Ȥƺؤȿľ
äƤȡФ֤ΤܤȤ
Τޤбޤ


1124 8:06:30 41151
ʥޥ
§ŪʬˤĤƸڤƤߤޤ

n=7ǤϴФ̤ʤΤǤ
n=13նǤϾͻҤ㤦褦Ǥ

פαγѤȡγѤɽȡ

11 12 13
n=13Τߡ




12n=12, 13Τ
13n=13, 14Τ
14
15

ȤʤꡢޤǤȤޤѤäˤʤޤ
(Ȼפޤ(^^;))
ƤA1a1ͤȰ֤طƤޤ
n礭ʤȡ¿ʬA2, A3,ġa2, a3,
Ƚ֤˴طƤơ礭֤ˤä
§ʬٰۤʤΤˤʤǽ򴶤ޤ

⤷ޤǤäٹäƤС
̼򤿤ƤΤϤۤԲǽ⤷ʤ
§Ǥ뤫ġ

ǰΤ­硩ĿCn
C13=52C14=55C15=59C16=65C17=71
ȻϤʤޤߥϳиǤm(__)m

եƤäȿǤm(__)m

ɵma-mu-ta Ƥˤ褦ʤȤɵƤޤ͡
̣äƤʤΤǤƤˤʤǤ(^^;

ɵͤȿޤޤm(__)m
ʣǤĤǤäȤäȿŤĴ٤
ɤé失Τ

1124 21:19:39 41152
Ȼ
ץǣʥޥޤ#41144ˡäƤߤޤn=20ޤǡ
1, 3, 6, 9, 12, 16, 21, 24, 29, 34, 39, 45, 52, 54, 59, 65, 71, 77, 84, 91,
ͤˤʤޤn=1454ˤʤ褦Ǥ
#41135κͤȤκ빽礭Ǥ͡
1124 22:45:17 41153
ʥޥ
Ф

Ϣ˼Ϣꤹߤޤm(__)m
٤ְ㤤Ϥ뤫Ȼפޤ
n=16ΤȤκγѤ顢ؼФ³Ǥ
3ĤǧǤƤޤ󤬡
̡a2=10ǤȤ顢#41151οľƱˤϤʤޤ
§˸줿()Ǥ̴ط
õäƤϤޤɽŪˤȤɤޤäƤޤ

ետ夲ޤ
񤷤Ǥ

ɵ
#41154
Ȼ
꤬Ȥޤ
n=14ʥǤäȸľǤ̤Ǥ
ޤ
տ夲ޤm(__)m

1124 23:18:06 41154
ʥޥ
ˤϡ

ޤɤȻפΤǤ
ʤη̤Ǥޤ
äʤʿԤʰϢʤäƤơ
δ֤ûüȺζ˸ƤϾäƤȻפޤ
n̵ˤʤȡ#41151οľ꤫ΰΤߤˤʤǤ

ʲŪ


٤ƿľA(k), a(k)򸵤ˤƤޤA ,aˤĤƤ#41144 ˴ŤƤޤ
ޤгޤ
ˡX(k)={m=0to(k-1)}A(m)Y(k)={m=0to(k-1)}a(m)Ȥ
X(k)n, Y(k')nȤʤX1, X2, X(k)ܡY1, Y2, Y(k')ܤ
гʿԤޤ(γѤ00Ȥߤ)

üˤĤơ
ǤդX(p)ȤȡX(p)X(p+1)A(p)Ĥδ֤
n=X(p)+a(m) (m=1pޤǤΤ٤)顢
n1ĤˤĤ1Ĥġa(m-1)ġ
гʿԤ˸졢줾
n{z=0to(m-1)}a(z) Ϣ³Ƥ1ĤľäƤ
ñˤX(p)X(p+1)δ֤A(p)ĤΤʤ
pսꡢü졢
ơü˸ƤüäƤȤȤǤ

ʲΰˤϿ¬ޤޤƤޤ
X(p)+a(p)nX(p+1) ˤĤơ
ʤpսܤüˤĤƤϡ
X(p)+a(p) 󤫤1ĤA(p)-a(p) ĸ졢
n=X(p+1)ΤȤƾä롣
ʤnnǸ줿nX(p+1)-n Ϣ³Ƥä롣
âA(p)-a(p)䦲{z=0top}a(z)ξ礬ä顢
nnȯϡ{z=0top}a(z) Ϣ³Ƥ1ĤľäƤ

٤yؤǤդY(p)Ȥꡢ
A(m)a(m)羮طαդʤ
Ʊͤˤƹͤޤ


ʸϤǤϲ狼ʤǤ(>_<)
٤ʸ䡢˥롼뤬ȯ꤬
뤫⤷ޤ󤬡
ˤĿƳΤϻεǤ(T_T)

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ɵǤʸ­ޤ
νȤʤޤ鹬Ǥ

1126ʲС 1:11:30 41155
uchinyan
Ȼ󡤥ץǤΥå꤬Ȥޤ

ʥޥޤ󡤾ܺ٤ĴڤӸƤ򤢤꤬ȤޤơϫͤǤ
ľäơʤɤڤƤʤΤǤϤ겿餫ε§Ϥ롤ȤȤǤ͡
Τˤ줫ĿƳΤѤǤ

ɤΤȤʤ䡤ȤȤǤ礦
ͥνȡ 1125ʷ 13:23:14 MAIL:uchi@sco.bekkoame.ne.jp 41156
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uchinyan

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ľ꼫Τ̲Τ񤷤ʤΤ
򸵤ˤθƤ̤ǤĿޤé失ޤ

ʤĹ֤ҤȤĤβ˼ȤΤ
İǤ礦
ޤǥޥåȲᤴޤ(*^^*)

1125ʷ 15:28:24 41157