ߤ
ȿġģߣ᣸
ģгģߣᣱ
ģгģߣᣲ
ģгģߣ᣸
ģгģߣᣱ

­ơܣܣܣܣ᣶̤ꡣ
32ڡ 0:08:24 26814
Taro
ɤ餳Τ褦Ǥ͡
http://www.research.att.com/~njas/sequences/A018808
32ڡ 0:09:28 26815
إ饯쥹ԡ
162=120
֤ʬޤ
42ݣˡߣʽIJ
ʣܣܣˡߣʤʤʤˡݣ᣶

ְ㤨ƣ̤äΤǽкǽΤۤäƤۤǤ
ʺǽΤۤΥ󥯴ְ㤨ǡ
軲̻ԡ 32ڡ 0:13:33 HomePage:ɥԥ26816
ݤ
162ݡ42ݣˡߣݡ32ݣˡߣ᣶
ǤʣΤޤ
Ǥ⺣٤ä
Ƹɡ 32ڡ 0:15:16 MAIL:hopes@mba.ocn.ne.jp HomePage:ȣϣУţӤ褤Ȥ٤Ϥ26817
DrK
(1,1)(1,2)(1,3)(2,3)(1,0)Ȥ߹碌ˤʤΤͤޤǽ(1,0)Ȥ߹碌ͤʤǼԤǤ
52+64+34+24+42=62Ǥ
٤Ͼγڱࡡ 32ڡ 0:16:32 MAIL:satoka@star.odn.ne.jp 26818
⡼ޥ
ΰۤʤΤ夲ޤ
#26814,#26818 ƱǤ
32ڡ 0:21:38 MAIL:kennji72001@yahoo.co.jp 26819
äѥȥ
1.5ܡܡܡ줾տäƣܡܡ줾ޥʥäƣܡ̵¤ܤġ
((2+6+3)*2+5)*2+4*2=62
ѥ򴹤ΤǥåäƤ
32ڡ 0:24:09 MAIL:k-kawakami@air.ocn.ne.jp 26820
ȥҥ
Ѥ4˿
7ܡ6ܡ5ܡ4ܡ22
ʾ嵭4˼Ϥ8˾嵭ȽʣʤΤ
8ܡ8ܡ6ܡ6ܡ4ܡ4ܡ2ܡ2ܡ40
4̤ľϾ嵭ľˤ٤ƴޤޤΤǡ
22+4062
̾Ų 32ڡ 0:59:10 26821

˿ޤ
32ڡ 8:41:01 26822

ˤĤ
顦νϡĤȣ£äʿԤ£ãĤꡢ¾ˣСãľˣѤ򡢣УѤĤʿԤˤʤ褦ˤȤ롣УѤ£ãĤѤ2ʬȤУѤĹ衣ġ᣷cm£á17cmȤ롣
(ġ᣷cm£á17cm̤οͤä⤷ޤ󡣿ޤĤƤޤ)

ǡGH ĹȾʳޤǤǡפФޤ
¾λŦˤ⤢褦ˡδطȤޤ

ꡢȾϻ餷ȯۤΤ褵䤦Ȼפޤ̵Ѥξϰڴطޤ󡣤⡢ߤˤȤꡢѤޤδФΤ6ǯ俷1ΤҤդǤ礦
32ڡ 10:19:24 26823
ϥ饮㡼ƥ
Ϥ褦ޤ

ǤΤǤݤȤ狼ޤ
̶彣 32ڡ 10:47:18 HomePage:ĩ26824
uchinyan
ϤˤϡñǤ͡줰餤ʤС񤷤 (^^;

16 Ĥǰľܿ 16C2 顢ʣΤޤʣϡξǤ
4 ľˤʤ 10 ܤܤȤ 4C2 ̤ꤢΤ 10 * 4C2 ܡ
3 ľˤʤ 4 ܤܤȤ 3C2 ̤ꤢΤ 4 * 3C2 ܡ
줾ľˤĤ 1 ܤĤϻĤʤȤʤΤǡ(10 + 4) ܤ­ޤ
ɡ16C2 - 10 * 4C2 - 4 * 3C2 + 10 + 4 = 120 - 10 * 6 - 4 * 3 + 10 + 4 = 62 ˤʤޤ

θ塢ǰΤˤȡƻ˿ƤߤޤषͤȤ̣Ǥϡ򤤤ʡ
鲣ľ a, b, c, d ̾Ĥޤơ֤˿ޤ
a ˤĤơa a ؤľͤޤ
餫 1 ܤǤ
b ˤĤơb b, a ؤľͤޤ
b ؤϡ餫 1 ܤǤ
a ؤϡb κü顢4, 4, 4, 4 Ȥʤޤäơ4 * 4 = 16 ܤǤ
碌ơ1 + 16 = 17 ܤǤ
c ˤĤơc c, b, a ؤľͤޤ
c ؤϡ餫 1 ܤǤ
b ؤϡc κü顢2, 2, 2, 2 Ȥʤޤäơ4 * 2 = 8 ܤǤ
a ؤϡc κü顢2, 2, 2, 2 Ȥʤޤäơ4 * 2 = 8 ܤǤ
碌ơ1 + 8 + 8 = 17 ܤǤ
d ˤĤơd d, c, b, a ؤľͤޤ
d ؤϡ餫 1 ܤǤ
c ؤϡd κü顢2, 2, 2, 2 Ȥʤޤäơ4 * 2 = 8 ܤǤ
b ؤϡd κü顢2, 2, 2, 2 Ȥʤޤäơ4 * 2 = 8 ܤǤ
a ؤϡd κü顢2, 3, 3, 2 Ȥʤޤäơ2 * (2 + 3) = 10 ܤǤ
碌ơ1 + 8 + 8 + 10 = 27 ܤǤ
ʾǤ٤ƤǤ
äơפϡ1 + 17 + 17 + 27 = 62 ˤʤޤ
ƻʿ夲ˡǤϡƱΤ缡ӽ롢ȤǤ͡
ӽϤʤ구§ŪʤΤǡΤ񤷤ʤǤ
ͥνȡ 32ڡ 16:20:33 MAIL:uchi@sco.bekkoame.ne.jp 26825
uchinyan
Ǽġɤߤޤ
ˡȤƤϡ礭ʬơ
ܤƿ夲롧#26814, #26818, #26819, #26820, #26825θȾ(ͤ)
餫ˤƿƤ#26821, #26822, #26825θȾ
ϡμˤäơºݤˤϤʤۤʤޤ
Τʣ#26816, #26817, #26825Ⱦ
λΤ褦Ǥ
ʤ̤ξȤơ#26815ޤʰ̼ˤʤȤϤ͡
աࡥ

(äɲ)
#26815Υΰ̼ϡ̼ a(n) Ȥȡ
a(n) = 1/2 * (f(n,1) - f(n.2))
f(n,k) = Sum{(n - |x|) * (n - |y|)}, - n < x < n, - n < y < n, (x,y) = k
ǤޤʤΤǤ(x,y) = k ϡx, y κ kȤȤΤ褦Ǥ
ˡ0 ϤˤġȲᤷơ(0,k) = (k,0) = k ȹͤΤǤ礦
βΤȤǤϡ
n = 4
f(4,1) = Sum{(4 - |x|) * (4 - |y|)}, - 4 < x < 4, - 4 < y < 4, (x,y) = 1
(x,y) = 1 Ȥʤ x >= 0 ͤȡ
x = 0, y = 1
x = 1, y = 0, 1, 2, 3
x = 2, y = 1, 3
x = 3, y = 1, 2
ǡx -> -x, y -> -y ʤɤǽʤȡ0 -> -0 ϰ̣ʤȡʤɤդȡ
f(4,1)
= 2 * {(4 - 0) * (4 - 1) + (4 - 1) * (4 - 0)}
+ 4 * {(4 - 1) * (4 - 1) + (4 - 1) * (4 - 2) + (4 - 1) * (4 - 3)
+ (4 - 2) * (4 - 1) + (4 - 2) * (4 - 3)
+ (4 - 3) * (4 - 1) + (4 - 3) * (4 - 2)}
= 2 * {12 + 12} + 4 * {(9 + 6 + 3) + (6 + 2) + (3 + 2)}
= 2 * 24 + 4 * 31
= 48 + 124
= 172
Ʊͤˤơ
f(4,2) = Sum{(4 - |x|) * (4 - |y|)}, - 4 < x < 4, - 4 < y < 4, (x,y) = 2
x = 0, y = 2
x = 2, y = 0, 2
f(4,2)
= 2 * {(4 - 0) * (4 - 2) + (4 - 2) * (4 - 0)} + 4 * {(4 - 2) * (4 - 2)}
= 2 * {8 + 8} + 4 * 4
= 32 + 16
= 48
ǡ
a(4) = 1/2 * (172 - 48) = 62
ȤʤäơΤˡȰפޤ͡
x, y κͤƤ뤳Ȥ顢ϡռƿ夲Ƥ褦˻פޤ
ͥνȡ 32ڡ 13:50:21 MAIL:uchi@sco.bekkoame.ne.jp 26826
uchinyan
#26823
ˤĤƤΥ顦β⡢꤬Ȥޤ
ʤۤɡȡ㤨С GH Ʊ褦ʹͤǡѤξǵޤޤ͡
ݥȤϡ#26807դν񤭹ߡ17^2 - 13^2 = 13^2 - 7^2 Ǥ礦

(äɲ)
ʤ轵ϡޥ뤵ɲþʤǤϡ
30 * CE = GH * GH + 221
ɤ褦ʤCE GH ޤޤɲþǡCE ڤ GH ξޤޤ
ΰ̣Ǥϡ顦ȤϡʤȻפä⤷ޤ
ͥνȡ 33ʶ 16:14:55 MAIL:uchi@sco.bekkoame.ne.jp 26827
L'A
夲ƽλ
32ڡ 13:17:38 26828
uchinyan
(ͤ򾯤ѹޤ)
#26826ɲʬ­
#26815Υΰ̼
a(n) = 1/2 * (f(n,1) - f(n.2))
f(n,k) = Sum{(n - |x|) * (n - |y|)}, - n < x < n, - n < y < n, (x,y) = k
ǤȤʤʬä褦˻פޤʤʤޤͤΤ褦Ǥ

ǤϡǡƱ̤ľνʣݥȤˤʤޤ
νǤΤ褦˹ͤƤ褦Ǥ㤨С
A-----B-----C-----D-----E-----F-----G-----H
ȤʤäƤȤޤΤȤϡɤ AH 1 ܤ̣ʤΤǤ
٤礦Ĥʤľʬ AB, BC, CD, DE, EF, FG, GH Ĺ 7
٤礦Ĥʤľʬ AC, BD, CE, DF, EG, FH Ĺ 6
ȹͤޤ٤礦Ĥʤľϡ٤礦ĤʤľޤǤΤǡ
(٤礦Ĥʤ)̣Τľϡ7 - 6 = 1 ˤʤޤ
ǡ٤礦ͤİʾʤľϡ٤礦Ĥʤľ ڤ ٤礦Ĥʤľ ˡ
ŪˤϴޤޤƤΤǡȤ껦ƹͤɬפʤȤˤʤޤ
ʤǡͤɬפͻޤ

ơιͤβˡ򡢷ܤƿƤޤ
򡢲 x y Ȥơy/x Ƚ񤭤ޤޤx y κg(x,y)Ƚ񤯤Ȥˤޤ
ȡx > 0, y >= 0 Фơ
0/1 = 0/2 = 0
٤礦Ĥʤľϡg(1,0) = 1 ǡ3 * 4 = 12 ܡ
٤礦Ĥʤľϡg(2,0) = 2 ǡ2 * 4 = 8 ܡ
ạ̇̄Τľϡ12 - 8 = 4 ܤˤʤޤ
ºݤΰϡλͤĤγľȤ 3 - 2 = 1 Ԥɬפޤ
ǽŪܿϤ¤ʤΤǡɡ弰˰פޤʲƱͤǤ
1/1 = 2/2 = 1
٤礦Ĥʤľϡg(1,1) = 1 ǡ3 * 3 = 9 ܡ
٤礦Ĥʤľϡg(2,2) = 2 ǡ2 * 2 = 4 ܡ
ạ̇̄Τľϡ9 - 4 = 5 ܤˤʤޤ
2/1 = 4/2 = 2
٤礦Ĥʤľϡg(2,1) = 1 ǡ2 * 3 = 6 ܡ
٤礦Ĥʤľϡg(4,2) = 2 ǡ0 ܡ
ạ̇̄Τľϡ6 - 0 = 6 ܤˤʤޤ
3/1 = 6/2 = 3
٤礦Ĥʤľϡg(3,1) = 1 ǡ1 * 3 = 3 ܡ
٤礦Ĥʤľϡg(6,2) = 2 ǡ0 ܡ
ạ̇̄Τľϡ3 - 0 = 3 ܤˤʤޤ
3/2 = 6/4 = 3/2
٤礦Ĥʤľϡg(3,2) = 1 ǡ1 * 2 = 2 ܡ
٤礦Ĥʤľϡg(6,4) = 2 ǡ0 ܡ
ạ̇̄Τľϡ2 - 0 = 2 ܤˤʤޤ
ʾǤ٤ƤǤºݤηϡx/y -> y/x, y -> -y ǽʤȡ0 -> -0 ̵̣ʤȤθȡ
1 * 2 * 4 + 2 * (1 * 5 + 2 * 6 + 2 * 3 + 2 * 2) = 8 + 2 * (5 + 12 + 6 + 4) = 8 + 54 = 62
Ȥʤäơޤ

嵭ξʬ x/y -> y/x ä#26826ɲʬ٤ȡбĤȤʬȻפޤ
ʾξܺ٤ϾĹˤʤΤǾάޤ̼Ƴ뤳ȡʤȤƳʤȡʬȻפޤ
ʤa(n) 1/2 ϡȤƤϰ̣ʤ x -> -x, y -> -y ƱѴǽʤȤ䴰ƤΤȻפޤ

ʤιͤĥȡ㤨С礦Ĥ̤ľܿ⡢Ʊͤ˷׻Ǥޤ
ޤäƤΤϡΤˤϡ٤礦Ĥʤ̣Τľ ʤΤǡ
줫顢٤礦Ĥʤ̣Τľ ФǤ
Ȥ
٤礦Ĥʤ̣Τľ ϡ٤礦ͤĤʤľ ٤礦Ĥʤľ Ф
Ȥˤʤޤ
ۤɤεͽۤĤȻפޤƱˤơ
٤礦Ĥʤľ g(x,y) = 2 ξǡ1 * 2 * 8 + 2 * (1 * 4 + 2 * 0 + 2 * 0 + 2 * 0) = 24
٤礦ͤĤʤľ g(x,y) = 3 ξǡ1 * 2 * 4 + 2 * (1 * 1 + 2 * 0 + 2 * 0 + 2 * 0) = 10
ǡ٤礦Ĥʤ̣Τľ ϡ24 - 10 = 14 ܤȤʤΤǡ
礦Ĥ̤ľ ϡ62 - 14 = 48 ˤʤޤ
ϡο󥵥Ȥ id A018809
http://www.research.att.com/~njas/sequences/A018809
Ǥ͡
ͥνȡ 33ʶ 15:48:48 MAIL:uchi@sco.bekkoame.ne.jp 26829
⡼ޥ
ȤǤϤʤǤ
ǡưۤʤľοɽȡ

0 0 0 1
3 1 1 4
5 3 3 6
8 9 9 9

ǡ 62

§ޤ͡
33ʶ 0:29:36 MAIL:kennji72001@yahoo.co.jp 26830

褦ϣ̤ľȡ̤ľΤʤǡʣΤͤʤƤϤʤʤäΤǤ͡
33ʶ 9:14:39 26832
uchinyan
#26830
䲿§ޤ͡
ɤʤǤ礦
¿ʬΤǤ礦ɤϡ#26829ˤĤʤäƤΤǤϤʤʤ

¤ϡ䤬#26825θȾǽ񤭹ƻʿ夲βˡ򡢷ռʤ⡢֤˿ȤΤϡ
§򸫤ĤʤʡȤäǤ
Τˤ٤ε§b a ؤϤ٤ư롢ٲʤǰľˤΤäƤϹͤʤƤ褤
νоμ˴ؤƳʤκβǽľܿоΡʤɤʬޤ
ϡɤϷռƤ뤳ȤƱǡѴɽƤޤ
츫ʬ붯Ϥʵ§줽ˤʤäΤǡƻˡȤƾҲ𤷤ޤ
θǼĤɤǡ#26815ΰ̲ΤäΤǡޤϡβߤޤ
#26829ǤʤʤޤͤƤ褦Ǥ

ʤߤˡ#26826βʬ1ܤ2ܤϡܤƤ롢ɤϤʤǤϡ纹ޤ
ΰǡ3ܤβˡϡռʤƤۤȤɰŻǤǤƤޤΤǡΰ̣ǤϡͥƤȤޤ
ϡߣ(ʲ)üǡ٤ľηФξˤϡ1 ˸¤Ƥ뤫Ǥ
ߣξϡ3 ľˤʤ 1 2 1/2 λˤʤꡢռʤȤʤʤޤ
̤ n n ǤϤޤޤѤǤ
ΰ̣Ǥϡĥ꤬ꤽǡĿŪˤϡǽ餫鷹ܤƻʲˡͥƤȻפޤ
#26825ǡƻˡɲäΤϡϤǤ̲ռȤ礿ͳǤޤǤϼԤǤ

äȤ⡢ɤϾŪˤäư㤦Τǡ쳵ˤϸޤ͡
ͥνȡ 34ڡ 9:05:27 MAIL:uchi@sco.bekkoame.ne.jp 26833
Ҥβ
ääƴ֤ΤѤʡ
줬Ƥä衣狼롩δ򤷤
礤礤礤
33ʶ 19:46:21 MAIL:wurong.zhu@morganstanley.com 26834
⡼ޥ
ʤߤˡ5*5 ΤȤ







ס140 Ȥʤä

Ťʤˡʤ顢
25C2-(12*5C2+8*4C2+24*3C2)=(25*24-12*5*4-8*4*3-24*3)/2=96
96+(12+8+24)=140

Ūñ

꤬ 5X5 ä򤹤ΤѤä

33ʶ 21:09:08 MAIL:kennji72001@yahoo.co.jp 26835
⡼ޥ
5*5 ΤȤ2ľο 108 ˤʤʤȤ
http://www.research.att.com/~njas/sequences/A018809
餹ȡ
34ڡ 11:42:12 MAIL:kennji72001@yahoo.co.jp 26836
uchinyan
#26835, #26836
(󡢤ɤƤɽƤޤ١ĩޤä ^^/)
ޤޤäƤ⤷礦ʤ⤷ޤޤǤ⡢͡

#26835
#26825θȾο夲ǤȡݤʤΤǾάޤηɽ񤱤СߣޤǻȤơ

ޣáޣáޣáޣáޣááޣáޣáޣáޣáޣ
ܣáܣáܣáܣáܣááݣáݣáݣáݣáݣ
ܣáܣáܣáܣáܣááݣáݣáݣáݣáݣ
ܣáܣáܣáܣáܣááݣáݣáݣáݣáݣ
ܣáܣáܣáܣáܣááݣáݣáݣáݣáݣ
ܣáܣáܣáܣáܣááݣáݣáݣáݣáݣ

n = 2


->
2 + 0 + 4 = 6

n = 3



->
3 + 0 + 9 + 8 = 20

n = 4




->
4 + 0 + 16 + 16 + 26 = 62

n = 5





->
5 + 0 + 25 + 24 + 40 + 46 = 140

n = 6






->
6 + 0 + 36 + 36 + 60 + 70 + 98 = 306

ʤ񤷤ʤǤ
ФƤȤ줤¤֤ľܤ˵§ɤ߼ʤʡȻפäΤǤ

#26836
Ʊο夲ˡǤȡ






->
0 + 0 + 12 + 14 + 36 + 46 = 108

ϡäݤǤ͡
ͥνȡ 35 9:06:57 MAIL:uchi@sco.bekkoame.ne.jp 26837
粬ҹ
夲ޤʡ˲§ߤʤΤʤĴ٤Ƥߤ褦Ȼפޤʤߤ





äƴˤʤޤ
35 10:11:41 MAIL:toshi009@land.hokuriku.ne.jp 26838
N.Nishi
ʤ˥ѥ󤬤ʤ褦äΤľηΥѥʬƹͤޤ
35 14:23:40 26839
Ƥ
Ʊľηǡ
36ʷ 17:15:30 26840
Ϥ뤭
Ƥ褳ߣ
売ߣ
壵ߣߣߣ
壱ߣߣ
壱ߣߣ

󡢲ˡꤽͤ(^^
37ʲС 3:09:47 26841