2nCnãš2nCn/4^nã®ååéšåããšãŠãé¢çœãé¢ä¿æ§ãæç«ããŠããããšãããããŸããã
ããã
2nCnã®æŽæ°ãçŽ å æ°å解ã§2^r0*p1^r1*p2^r2*p3^r3(p1,p2,p3,ã¯2以å€ã®å¥çŽ æ°)
ãšãªã£ãŠãããšã
2nCn/4^nã®ååéšåã¯p1^r1*p2^r2*p3^r3ãšãã£ããäžã®çŽ å æ°2^r0ã®éšåãæãèœã¡ã
ãã®ãçŸããããšã«ãªãã
ããã2ã§ã®ææ°r0ã¯ïœã2é²æ³ã§è¡šããæã®1ã®äœ¿çšåæ°(=hammingweight(n))ã察å¿ããŠããã
(確èª)
{2nCn/4^n}ã®æ°åã®æ§å
gp > for(n=1,20,print1(binomial(2*n,n)/4^n","))
1/2,3/8,5/16,35/128,63/256,231/1024,429/2048,6435/32768,12155/65536,46189/262144,
88179/524288,676039/4194304,1300075/8388608,5014575/33554432,9694845/67108864,
300540195/2147483648,583401555/4294967296,2268783825/17179869184,
4418157975/34359738368,34461632205/274877906944,
ãããã£ãŠãã®ååéšåã¯
1,3,5,35,63,231,429,6435,12155,46189,
88179,676039,1300075,5014575,9694845,
300540195,583401555,2268783825,
4418157975,34461632205,
ããã§2nCnã®å€ãã2^r0=2^hammingweight(n)ãåãé€ãæäœã§
2nCnã®å€(ãã€ããªãŒè¡šç€ºãå³ã«hammingweight(n)ã ãã·ããããã)
gp > for(n=1,20,print1(binomial(2*n,n)>>hammingweight(n)","))
1,3,5,35,63,231,429,6435,12155,46189,
88179,676039,1300075,5014575,9694845,
300540195,583401555,2268783825,
4418157975,34461632205,
ããšã§äžèŽãããããããšã«ãªãã
æŽã«é©ããããšã¯ããã®æ°åã1/â(1-x)ã§ã®ãã€ã©ãŒå±éåŒ
gp > taylor(1/sqrt(1-x),x)
%84 = 1 + 1/2*x + 3/8*x^2 + 5/16*x^3 + 35/128*x^4 + 63/256*x^5
+ 231/1024*x^6 + 429/2048*x^7 + 6435/32768*x^8 + 12155/65536*x^9 + 46189/262144*x^10 + 88179/524288*x^11 + 676039/4194304*x^12 + 1300075/8388608*x^13 + 5014575/33554432*x^14 + 9694845/67108864*x^15 + 300540195/2147483648*x^16 + 583401555/4294967296*x^17 + 2268783825/17179869184*x^18 + 4418157975/34359738368*x^19 + 34461632205/274877906944*x^20 +
O(x^21)
ã§ã®åä¿æ°ã®ååã«åºçŸããŠããŸããšããæã£ãŠãããªãç¹ãããæã€ããšã§ããã
No.1961GAI2024幎6æ14æ¥ 08:10
èªæã§ãªããäºçµã®æ°ã«ã€ããŠã
äŸãïŒïŒïŒïŒïŒïŒïŒãšïŒïŒïŒïŒïŒïŒïŒ
ïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒã2ÃïŒÃïŒïŒïŒÃïŒÃïŒ
ãæãç«ã€ã
ä»ã«åãšç©ãæãç«ã€çµãäºæ¡ãäžæ¡ããããŸããïŒ
No.1923ks2024幎6æ1æ¥ 09:50
ãããã§ããããããªæãã§ãããå®éæ¢çŽ¢ãããšãããã§ããããŸãã
äŸãã°
10+16+39=12+13+40=65
10*16*39=12*13*40=6240
100+108+119=102+105+120=327
100*108*119=102*105*120=1285200
No.1924ãããã2024幎6æ1æ¥ 13:02
ããããããããã€ãããããšãããããŸãã
ïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒ
ïŒã»ïŒã»ïŒïŒïŒïŒã»ïŒã»ïŒïŒïŒïŒ
ïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒ
ïŒã»ïŒã»ïŒïŒïŒã»ïŒã»ïŒïŒïŒïŒïŒ
ïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒ
ïŒïœ¥ïŒïŒã»ïŒïŒïŒã»ïŒã»ïŒïŒïŒïŒïŒïŒ
ã©ã®ããã«ãæ±ãããããã®ãåãããªããŠã
äžã±ã¿ãåã±ã¿ããããã§ããã
åŸãçŽ å æ°ããïŒïŒïŒïŒïŒïŒïŒãïŒïŒãïŒïŒïŒããããŸããã
ïŒïŒãèå³ã¯å°œããŸãããå
šãŠã®çŽ å æ°ã«ã€ããŠãããããå«ã
çµãããããã§ããã
No.1936ks2024幎6æ3æ¥ 12:33
ïŒåçµ
ïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒ
ïŒïœ¥ïŒã»ïŒã»ïŒïŒïŒã»ïŒã»ïŒã»ïŒïŒïŒïŒ
ïŒåçµ
ïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒ
ïŒã»ïŒã»ïŒã»ïŒã»ïŒïŒïŒã»ïŒã»ïŒã»ïŒã»ïŒïŒïŒïŒ
1ã足ããŠããã°ãäœåã®çµã§ããäœãããã§ããã
äž¡çµã«åãæ°ã䜿ããªãããšæ¡ä»¶ãå€ããã°ã©ããªãã§ããããïŒ
6,7,8åã®çµãäœããŸããïŒ
No.1937ks2024幎6æ5æ¥ 10:11
åæ°ãã4åãããå¢ãããŠããããšããèããŠã¿ãŠãåçŽãªè§£ããããæ¡ä»¶ãå€ããŠããïŒåçµãšïŒåçµãç¹ãã§ããã°ãïŒåçµãã§ãããã§ããã
ïŒåçµã¯ãã©ãããããããã§ããããïŒ
æãŠããªããç¶ãåé¡ã§ãããé£ãããªããŸãã
ããããããŠããããªãå®çã«åºäŒããŸããã
çŽ æ°ã®åã§ãçå·®ã«ãªã£ãŠãããã®
é·ãïŒã®ãã®ã7ïŒ37ïŒ67ïŒ97ïŒ127ïŒ157ãçå·®ã30
ïŒïŒïŒïŒå¹Žã§ãé·ã26ãæé·
ã«ãé¢ããããGreen-Taoã2004
çŽ æ°ã®ã¿ããæ§æãããä»»æã®é·ãã®çå·®æ°åãååšããã
å
·äœçã«ã¯èŠããªãããã©ãååšãããæ°åŠã®ååãã§ãã
No.1938ks2024幎6æ6æ¥ 10:04
5åçµã¯2å+3åã§ããã®ã§ã¯ïŒ
No.1939ãããã2024幎6æ6æ¥ 12:33
> "ks"ãããæžãããŸãã:
> çŽ æ°ã®åã§ãçå·®ã«ãªã£ãŠãããã®
> é·ãïŒã®ãã®ã7ïŒ37ïŒ67ïŒ97ïŒ127ïŒ157ãçå·®ã30
ãé¢çœãã£ãã®ã§ããã®å
ãæ¢ããŠã¿ãã
7åé£ç¶
[7, 157, 307, 457, 607, 757, 907]
[47, 257, 467, 677, 887, 1097, 1307]
[53, 1103, 2153, 3203, 4253, 5303, 6353]
8åé£ç¶
[61, 9931, 19801, 29671, 39541, 49411, 59281, 69151]
[73, 5953, 11833, 17713, 23593, 29473, 35353, 41233]
[103, 4723, 9343, 13963, 18583, 23203, 27823, 32443]
[199, 9439, 18679, 27919, 37159, 46399, 55639, 64879]
9åé£ç¶
[17, 6947, 13877, 20807, 27737, 34667, 41597, 48527, 55457]
[137, 8117, 16097, 24077, 32057, 40037, 48017, 55997, 63977]
10åé£ç¶
[199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089]
[443, 32783, 65123, 97463, 129803, 162143, 194483, 226823, 259163, 291503]
11åé£ç¶
[1619, 3413489, 6825359, 10237229, 13649099, 17060969, 20472839, 23884709, 27296579, 30708449, 34120319]
[3617, 213827, 424037, 634247, 844457, 1054667, 1264877, 1475087, 1685297, 1895507, 2105717]
12åé£ç¶
[18439, 33291679, 66564919, 99838159, 133111399, 166384639, 199657879, 232931119, 266204359, 299477599, 332750839, 366024079]
13åé£ç¶
[4943, 65003, 125063, 185123, 245183, 305243, 365303, 425363, 485423, 545483, 605543, 665603, 725663]
æ¢ãç¯å²ãäºæ³ãã€ããªãã®ã§ãé©åœãªç¯å²ã§ãã£ãŠããŸãã®ã§ãèŠèœãšããŠãããã®ããããšã¯æãããŸãã
14å以äžã«ææŠããŠããŸããããèªåã§èšå®ããç¯å²ã§ã¯æ¢ãåºãããšã¯åºæ¥ãŸããã§ããã
äœæ¹ãç¶ãåã³è£å
ããé¡ãããŸãã
No.1940GAI2024幎6æ6æ¥ 19:58
æ€çŽ¢ããŠããã
146141+54444390*k
äœãã0â€kâ€13
ãäŸç€ºãããŠããŸããã
No.1941Dengan kesaktian Indukmu2024幎6æ6æ¥ 22:58
OEISã«ããã°
ä»ã¿ã€ãã£ãŠããæãé·ãæ°åã¯æ¬¡ã®ãã®ã§ãã
A261152 - OEIS
https://oeis.org/A261152
No.1942Dengan kesaktian Indukmu2024幎6æ6æ¥ 23:18
> "Dengan kesaktian Indukmu"ãããæžãããŸãã:
> æ€çŽ¢ããŠããã
> 146141+54444390*k
> äœãã0â€kâ€13
> ãäŸç€ºãããŠããŸããã
ããããšãããããŸãã
次ãèŠã€ãããªãã¯ãã ããããªã«ãåé
ãé ãé¢ããŠãããšã¯ïŒ
ãªãåé
ã®æ°ã¯å¶æ°çªç®ã®çŽ æ°ã155åå ããå€ãšãªãããšã¯å¶ç¶ãªã®ã§ãããïŒ
gp > vector(155,i,prime(2*i))
%226 =
[3, 7, 13, 19, 29, 37, 43, 53, 61, 71, 79, 89, 101, 107, 113, 131, 139,
151, 163, 173, 181, 193, 199, 223, 229, 239, 251, 263, 271, 281, 293, 311,
317, 337, 349, 359, 373, 383, 397, 409, 421, 433, 443, 457, 463, 479, 491,
503, 521, 541, 557, 569, 577, 593, 601, 613, 619, 641, 647, 659, 673, 683,
701, 719, 733, 743, 757, 769, 787, 809, 821, 827, 839, 857, 863, 881, 887,
911, 929, 941, 953, 971, 983, 997, 1013, 1021, 1033, 1049, 1061, 1069, 1091,
1097, 1109, 1123, 1151, 1163, 1181, 1193, 1213, 1223, 1231, 1249, 1277, 1283,
1291, 1301, 1307, 1321, 1361, 1373, 1399, 1423, 1429, 1439, 1451, 1459, 1481,
1487, 1493, 1511, 1531, 1549, 1559, 1571, 1583, 1601, 1609, 1619, 1627, 1657,
1667, 1693, 1699, 1721, 1733, 1747, 1759, 1783, 1789, 1811, 1831, 1861, 1871,
1877, 1889, 1907, 1931, 1949, 1973, 1987, 1997, 2003, 2017, 2029, 2053]
gp > vecsum(%)
%227 = 146141
ç§ãåŸã§èª¿ã¹ãŠã¿ãã26åé£ç¶ã¯A204189,A261140,A317163,A317164,A317255,A317259,A317914ãèŠã€ãã£ãŠããããã§ããã
ãŸã2019幎4æã«æ°ãã«çºèŠãããé£ç¶27åã®ãã®ãA327760ã«èŒã£ãŠããŸããã
ããã£ãŠå¶ç¶ç¯å²ãããã°æ°ããé·ãã®çå·®æ°åçŽ æ°ãçºèŠã§ããããç¥ããŸãããã
No.1943GAI2024幎6æ7æ¥ 06:18
ãé°æ§ã§ãäºåã®çµããèŠã€ãããŸããã
1ïŒ4ïŒ8ïŒ9ïŒ10ïŒ2ïŒ3ïŒ5ïŒïŒïŒ16
1ã»4ã»8ã»9ã»10ïŒ2ã»3ã»5ã»6ã»16
ä»ã«ãããããšã¯æããŸããã
No.1956ks2024幎6æ10æ¥ 12:12
2,3,5,7,11,13,17ã®7åã®çŽ æ°ãèŸæžåŒã«äžŠã¹çŽããš
11,13,17,2,3,5,7
ãšãªãã®ã§ãããæ¹ããŠ
p1=11,p2=13,p3=17,p4=2,p5=3,p6=5,p7=7
ãšçªå·ãæ¯ãä»ãããš
å
šéšã§7åã®çŽ æ°ã§ã¯p7=7
ãªãçŸè±¡ãçºçããã
ããã§
çŽ æ°2ããå§ãçŽ æ°ãå
šéšã§kåéã
ãããèŸæžåŒé åºã«ããŠçªå·ãæ¯ãä»ã
p1,p2,,pk
ãšããæ
pk=kãªãããšãèµ·ããïœãããšïŒã€ã»ã©çºèŠããŠã»ããã
No.1922GAI2024幎6æ1æ¥ 07:09
ãããšïŒã€ãã97ãš997ã§æ£ãããã°ããã®æ¬¡ã¯999âŠ(9ã139å)âŠ9997ã€ãŸã10^140-3ããªïŒ
ïŒããããããããããå°ãããã®ãããããïŒ
No.1925ãããã2024幎6æ1æ¥ 14:31
ãã®3ã€ãæ¢ããåŸã¯ããè«ŠããŠããŸããã
ã©ãããŠç¬¬4ã®ãšãã§ããªããã®ãèŠã€ããããã®ãïŒïŒïŒ
No.1926GAI2024幎6æ1æ¥ 17:21
ãããããŠ
99999999999999997(=10^17-3)ãããã®ã§ããããïŒ
9999997(=10^990-3)ãããã§ããïŒ
https://stdkmd.net/nrr/9/99997.htm#primeãåèã«ãªããŸããã
No.1927GAI2024幎6æ2æ¥ 05:44
> "ãããã"ãããæžãããŸãã:
> ãããšïŒã€ãã97ãš997ã§æ£ãããã°ããã®æ¬¡ã¯999âŠ(9ã138å)âŠ9997ã€ãŸã10^140-3ããªïŒ
> ïŒããããããããããå°ãããã®ãããããïŒ
ããã¯
999âŠ(9ã139å)âŠ9997ã§ããã§ãããïŒ
ãã¿ãŸãã现ããããšã§
No.1928GAI2024幎6æ2æ¥ 06:57
> 999âŠ(9ã139å)âŠ9997ã§ããã§ãããïŒ
ãã£ãããéãã§ããåçŽã«ééããŸããã
ããã¯å
èšäºãä¿®æ£ããŸããã
> 99999999999999997(=10^17-3)ãããã®ã§ããããïŒ
ããã¯NGã§ãã
99999999999999997 çªç®ã®çŽ æ°ã¯
4185296581467695521 ã§ããã
ãã以äžã®çŽ æ°ã§
999999999999999989
ããããŸãã®ã§
99999999999999997ã¯æåŸã«ãªããŸããã
999âŠ(9ã139å)âŠ9997 (140æ¡) ã®å Žåã¯ã
999âŠ(9ã139å)âŠ9997 çªç®ã®çŽ æ°ã¯
32*********************** (143æ¡) ãšããæ°ã§ããã
999âŠ(9ã139å)âŠ999 7x (141æ¡)
999âŠ(9ã139å)âŠ999 8x (141æ¡)
999âŠ(9ã139å)âŠ999 9x (141æ¡)
999âŠ(9ã139å)âŠ999 7xx (142æ¡)
999âŠ(9ã139å)âŠ999 8xx (142æ¡)
999âŠ(9ã139å)âŠ999 9xx (142æ¡)
ããã¹ãŠåææ°ã§ããããšã確èªã§ããããã
999âŠ(9ã139å)âŠ9997 ã¯æåŸã«ãªããæ¡ä»¶ãæºãããŠããŸãã
9999997(=10^990-3)ã®å Žåã¯ã
9999997çªç®ã®çŽ æ°ã
çŽ2.29Ã10^993ã§ããããã
9999997x
9999998x
9999999x
9999997xx
9999998xx
9999999xx
9999997xxx
9999998xxx
9999999xxx
ããã¹ãŠåææ°ã§ããããšã確èªã§ããã°OKã«ãªããŸããã
察象ãçµæ§å€ãã®ã§çŽ æ°ãå«ãŸããŠããã®ã§ã¯ãªãããšããæ°ãããŸãã
ãããããããããå°ããã®ãããããããšããã®ã¯ã
999991
ãªã©ã§æ¡ä»¶ãæºãããã®ãããããã°ããšããæå³ã§ããã
æ«å°Ÿã1ã§ãããã確ççã«äœããšæããæªèª¿æ»ã§ãã
ããã ããªã調æ»ããããšã¯ã§ããŸããã
næ¡ã®æåŸã
999989 ãšã
999983 ãšã
999979
ã®ãããªå Žåãªã©èãããšå€§å€ãããªã®ã§èª¿æ»ã¯ãããŸããã
No.1929ãããã2024幎6æ2æ¥ 11:10
gp > for(i=70,99,if(isprime(10^2*(10^989-1)+i)==1,print1(10^2*(10^989-1)+i",")))
gp > for(i=700,999,if(isprime(10^3*(10^989-1)+i)==1,print1(10^3*(10^989-1)+i",")))
gp > for(i=7000,9999,if(isprime(10^4*(10^989-1)+i)==1,print1(10^4*(10^989-1)+i",")))
ã«å¯Ÿãäœã®åå¿ãèµ·ããªãã£ãã
äžå¿äžã®989æ¡ã®ãã®ã§ãã¡ãããšçŽ æ°ãšå€å®ããŠãããã
gp > isprime(10^990-3)
%53 = 1
å³ã¡
10^990-3çªç®ã«åºçŸããçŽ æ°(994æ¡ã®2ãé ã«ããçŽ æ°)
ããåã«ããçŽ æ°ã§10^990-3ãã倧ããªçŽ æ°ã¯äžåãååšããªããšå€å®ãããçµå±èŸæžåŒé åºã®æåŸã«äžŠã¶ã
ãã£ãŠæ±ããkã®å€ãšããŠ
10^140-3ã®æ¬¡ã«10^990-3ãèªããããã
ãã®å€å®ã§ããã®ã§ããããïŒ
ãªãå±ãªã次ã®çŽ æ°ãåºçŸãããã10^990-3ããåã«èŸæžåŒé åºã§ã¯äœçœ®ããŠãããããã§ãã
999(9ã989å)999199
999(9ã989å)999329
999(9ã989å)9993893
No.1933GAI2024幎6æ3æ¥ 06:59
ããã§ããã10^17-3ã§è©Šããšã¡ãããšçŽ æ°ã衚瀺ãããŸãã®ã§ãåé¡ãªããšæããŸãã
No.1935ãããã2024幎6æ3æ¥ 08:16
A005021ã瞊ç·ã6æ¬ã®æã暪ç·ãåèšnæ¬åŒããæã®ãã¿ã ããã®ãã¿ãŒã³æ°ãšããŠ
ãã®ãµã€ãã«ç¹ãã£ãŠãããã§ã¯Random Walksã®è§£èª¬ãšãªã£ãŠããããšã«èå³ãæã¡
ã©ããªå
容ãªã®ãèªãã§ã¿ããš
P_6ãšåŒã°ããéïŒçŽç·äž6ç¹A,B,C,D,E,Fã䞊ãã§ãããïŒ
ãAããåºçºã2*n+5(æ©)ã«ãŠFã®å°ç¹ã«å°çããé
æ©ã®ã³ãŒã¹ãäœéãã§ãããïŒ
ãšããããšãããã
n=1ãªãå
šéšã§7æ©ãªã®ã§ã次ã®5ã³ãŒã¹ããããšããã
1;[A, B, A, B, C, D, E, F]
2;[A, B, C, B, C, D, E, F]
3;[A, B, C, D, C, D, E, F]
4;[A, B, C, D, E, D, E, F]
5;[A, B, C, D, E, F, E, F]
ããã§n=2ãªãå
šéšã§9æ©ãªã®ã§ãå
šã³ãŒã¹ãæ§æããŠã¿ãã
1;[A, B, A, B, A, B, C, D, E, F]
2;[A, B, A, B, C, B, C, D, E, F]
3;[A, B, A, B, C, D, C, D, E, F]
4;[A, B, A, B, C, D, E, D, E, F]
5;[A, B, A, B, C, D, E, F, E, F]
6;[A, B, C, B, A, B, C, D, E, F]
7;[A, B, C, B, C, B, C, D, E, F]
8;[A, B, C, B, C, D, C, D, E, F]
9;[A, B, C, B, C, D, E, D, E, F]
10;[A, B, C, B, C, D, E, F, E, F]
11;[A, B, C, D, C, B, C, D, E, F]
12;[A, B, C, D, C, D, C, D, E, F]
13;[A, B, C, D, C, D, E, D, E, F]
14;[A, B, C, D, C, D, E, F, E, F]
15;[A, B, C, D, E, D, C, D, E, F]
16;[A, B, C, D, E, D, E, D, E, F]
17;[A, B, C, D, E, D, E, F, E, F]
18;[A, B, C, D, E, F, E, D, E, F]
19;[A, B, C, D, E, F, E, F, E, F]
ãã®æ§ã«ã€ãã¯n=3ã§ã®11æ©ã§ã®ã³ãŒã¹ã¥ãããããã°å
šéšã§66ã³ãŒã¹
åããn=4ã§ã®13æ©ã§ã®221ã³ãŒã¹
n=5ã§ã®15æ©ã§ã®728ã³ãŒã¹

ãšããã«èŒããããŠããæ°ã®ã³ãŒã¹ã次ã
ãšå€æãããšããããšã«ãªã£ãŠããæ§ã ã
ãŸããããã¿ã ãããé
ã£æãã®æ©ãæ¹ãšç¹ãã£ãŠãããšã¯å€¢ã«ãæããªãã£ãã(䌌ãŠãªãããªããïŒ)
No.1920GAI2024幎5æ29æ¥ 07:57
ç§ããã³ã¡ããã«å±ãããã®çªçµãèŠãŠããŠ
瞊ç·ã5æ¬ã暪ç·ã8æ¬ãããªããã¿ã ããã®å
šãã¿ãŒã³æ°ã9841éããã
äžã®ç· 1 | 2 | 3 | 4 | 5
äžã®ç·(1ã®äžãa,,5ã®äžãe)
a; 43.92 | 24.61 |16.53 |10.25 | 4.68
b; 24.61 |25.46 |22.06 |17.61 |10.25
c; 16.53 |22.06 |22.81 |22.06 |16.53
d; 10.25 |17.61 |22.06 |25.46 |24.61
e; 4.68 |10.25 |16.53 |24.61 |43.92
ã®è¡šãæ åã«åºããïŒéæ¢ç»é¢ã«ããŠã¡ã¢ãããïŒ
確ãã«çäžã«åœãããããã°ããããã¹ã¿ãŒãããã°ç¢ºçãé«ãã
ãã®ç¢ºçãã©ããããåºããã®ãè²ã
ææŠããŠããã®ã ãããªããªããã®å€ãæãããããªãã
ãŸã9841ã¯ã©ãããã©ãããŠç®åºãããã®ãªã®ãïŒ
No.1901GAI2024幎5æ26æ¥ 13:33
暪ç·ã®ç«äœäº€å·®ã¯ã¢ãªã§ãããïŒ
ããšãã°ã
â å·ŠããïŒæ¬ç®ã®çžŠç·ãšïŒæ¬ç®ã®çžŠç·ãšã®ããã ã«æšªç·ãïŒåã€ãªããããã ããïŒæ¬ç®ã®çžŠç·ãšãã®æšªç·ãšã¯ç«äœäº€å·®ã«ããã
â¡æšªç·ã©ããã§ç«äœäº€å·®ãããã
No.1902Dengan kesaktian Indukmu2024幎5æ26æ¥ 15:43
ç¹ã«ç«äœäº€å·®ã®ã³ã¡ã³ãã¯ç¡ãã£ãã®ã§ãéåžžã®ãã¿ã ã®æšªç·ã®åŒãæ¹ã§èãããã®ã ãšæããŸãã
No.1903GAI2024幎5æ26æ¥ 20:59
https://manabitimes.jp/math/1157
ã«ãããš
瞊ç·5æ¬ã暪ç·8æ¬ã§ã®ãã¿ã ããã®è¡ãå
ã®ç¢ºçã¯
P5=
[3/4 1/4 0 0 0]
[1/4 1/2 1/4 0 0]
[ 0 1/4 1/2 1/4 0]
[ 0 0 1/4 1/2 1/4]
[ 0 0 0 1/4 3/4]
ãã
P5^8=
[12155/32768 19449/65536 12393/65536 1581/16384 765/16384]
[19449/65536 8627/32768 3345/16384 9129/65536 1581/16384]
[12393/65536 3345/16384 6995/32768 3345/16384 12393/65536]
[ 1581/16384 9129/65536 3345/16384 8627/32768 19449/65536]
[ 765/16384 1581/16384 12393/65536 19449/65536 12155/32768]
ãããå°æ°ãžçŽã
=
[ 0.37094116 0.29676819 0.18910217 0.096496582 0.046691895]
[ 0.29676819 0.26327515 0.20416260 0.13929749 0.096496582]
[ 0.18910217 0.20416260 0.21347046 0.20416260 0.18910217]
[0.096496582 0.13929749 0.20416260 0.26327515 0.29676819]
[0.046691895 0.096496582 0.18910217 0.29676819 0.37094116]
ãšãªããã§ã¯ãªãããšæããã§ãããã»ã»ã»ïŒ
No.1904GAI2024幎5æ27æ¥ 07:12
ãã®ãµã€ãã§ã¯æšªç·ã®åŒãæ¹ãm^néããšèšã£ãŠããŸãã®ã§ã確çååžãéãã®ã§ã¯ãªãã§ããããã
äŸãã°
ââ¡â¡ââ¡â¡ââ¡â¡â
ââââ€â¡â¡ââ¡â¡â
ââ¡â¡ââ¡â¡ââââ€
ââ¡â¡ââââ€â¡â¡â
ââ¡â¡ââ¡â¡ââ¡â¡â
ãš
ââ¡â¡ââ¡â¡ââ¡â¡â
ââ¡â¡ââ¡â¡ââââ€
ââââ€â¡â¡ââ¡â¡â
ââ¡â¡ââââ€â¡â¡â
ââ¡â¡ââ¡â¡ââ¡â¡â
ãå¥ã®ãã®ãšèããŠããã®ã§ã¯ïŒ
# å
šéšãã¡ããšèªãã ããã§ã¯ãããŸããã®ã§ã
# ãããšãã¡ããããªããšãèšã£ãŠãããã容赊äžããã
No.1906ãããã2024幎5æ27æ¥ 11:38
ãã€ãã¿ãŒæ€çŽ¢ã§èª¿ã¹ãã
ãã³ã¡ããã®çªçµã§ãã¿ã ããã«ã€ããŠè§£èª¬ããå
çã岩æ倧åŠã®çå·¥åŠéšã®å±±äžå
ä¹
ææã§ãããšããããŸããã
OEIS ã®ãµã€ãã§ãã®å
çã®ååã§æ€çŽ¢ããããããããŸããã
https://oeis.org/A006245
A006245 ã®åèæç®ã«ä»¥äžããããããŠããŸããã
ã²ãšã€ã
Katsuhisa Yamanaka, Takashi Horiyama, Takeaki Uno and Kunihiro Wasa, Ladder-Lottery Realization, 30th Canadian Conference on Computational Geometry (CCCG 2018) Winnipeg.
ãµãã€ã
K. Yamanaka, S. Nakano, Y. Matsui, R. Uehara and K. Nakada, Efficient enumeration of all ladder lotteries and its application, Theoretical Computer Science, Vol. 411, pp. 1714-1722, 2010.
ãªã
ladder lotteries ãšã¯ã¢ããã¯ãžã®ããšã§ãã
No.1907Dengan kesaktian Indukmu2024幎5æ27æ¥ 14:27
9841éãã®èšç®ã¯
Σ[i=0ïœ8]Σ[j=0ïœ8-i](i+j)CjÃ9C(i+j+1)=9841
ãšããåŒã§åºããŸããã
iã¯2æ¬ç®ã®çžŠç·ãš3æ¬ç®ã®çžŠç·ã®éã«æã暪ç·ã®æ°ã
jã¯3æ¬ç®ã®çžŠç·ãš4æ¬ç®ã®çžŠç·ã®éã«æã暪ç·ã®æ°ã
(i+j)Cjã¯ã2æ¬ç®ã®çžŠç·ãš3æ¬ç®ã®çžŠç·ã®éã®æšªç·ããš
ã3æ¬ç®ã®çžŠç·ãš4æ¬ç®ã®çžŠç·ã®éã®æšªç·ãã®äœçœ®é¢ä¿ã®å Žåã®æ°ã
9C(i+j+1)ã¯æ®ãã®8-i-jæ¬ã®æšªç·ãã1æ¬ç®ã®çžŠç·ãš2æ¬ç®ã®çžŠç·ã®éããš
ã4æ¬ç®ã®çžŠç·ãš5æ¬ç®ã®çžŠç·ã®éãã«é
眮ããå Žåã®æ°ã§ãã
No.1908ãããã2024幎5æ27æ¥ 20:54
ãããããããåãã§ãã
No.1909Dengan kesaktian Indukmu2024幎5æ27æ¥ 21:30
ãã®æ°å€ãã¯ããåºãæ°åŒãèŠã€ãããªããŠããã¯ãªã§ãã
åæãããŠé ããŸããã
[i,j] [binomial(i+j,j)"*"binomial(9,8-(i+j))] [2ã€ã®ç©]
0,0 1*9 9
0,1 1*36 36
0,2 1*84 84
0,3 1*126 126
0,4 1*126 126
0,5 1*84 84
0,6 1*36 36
0,7 1*9 9
0,8 1*1 1
1,0 1*36 36
1,1 2*84 168
1,2 3*126 378
1,3 4*126 504
1,4 5*84 420
1,5 6*36 216
1,6 7*9 63
1,7 8*1 8
2,0 1*84 84
2,1 3*126 378
2,2 6*126 756
2,3 10*84 840
2,4 15*36 540
2,5 21*9 189
2,6 28*1 28
3,0 1*126 126
3,1 4*126 504
3,2 10*84 840
3,3 20*36 720
3,4 35*9 315
3,5 56*1 56
4,0 1*126 126
4,1 5*84 420
4,2 15*36 540
4,3 35*9 315
4,4 70*1 70
5,0 1*84 84
5,1 6*36 216
5,2 21*9 189
5,3 56*1 56
6,0 1*36 36
6,1 7*9 63
6,2 28*1 28
7,0 1*9 9
7,1 8*1 8
8,0 1*1 1
åèš 9841
ããã§
i,j=0,2 ã§ã1*84=84
ã®è§£éã
瞊ç·2,3çªç®ã«ã¯0æ¬,3,4çªç®ã«ã¯2æ¬åŒãããŠããã®ã§
1,2ãš4,5éã«ã¯åèš6æ¬ã®çžŠç·ãããã
ããã§
1,2çªéã;4,5çªé
6 ;0
5 ;1
4 ;2
3 ;3
2 ;4
1 ;5
0 ;6
æ¬ã®ç·ãããå Žåã«å¥ããã
ãšããã§2,3çªéã«ã¯i=0ããäžèšã®å·Šã®æ¬æ°ã¯äœã®å¶éããªãåŒãããšãåºæ¥ãã
äžæ¹j=2ããæ¢ã«3,4çªéã«ã¯2æ¬ã®æšªæ£ãåŒãããŠããã
ããã§äžèšã®å³ã®æ¬æ°ã®æšªæ£ãåŒãäœçœ®ã¯ããããéè€çµåããã
3H0=1
3H1=3
3H2=6
3H3=10
3H4=15
3H5=21
3H6=28
ããã®åèšã84ãšãªãã
ãªããšãããäžçºã§9C6=9C3=9*8*7/(3*2*1)=3*4*7=84ãªããã§ããã
åãã
i,j=0,3 ã§1*126=126ã¯
1,2çªéã;4,5çªé
5 ;0
4 ;1
3 ;2
2 ;3
1 ;4
0 ;5
äžèšã®å³ã®æ¬æ°ã®æšªæ£ãåŒãäœçœ®ã¯ããããéè€çµåããã
4H0=1
4H1=4
4H2=10
4H3=20
4H4=35
4H5=56
ãã®åèšã126
åããäžçºã§9C5=9C4=9*8*7*6/(4*3*2*1)=126
ãããªä»çµã¿ã§èšç®ãããŠãããã§ããïœ
No.1910GAI2024幎5æ28æ¥ 07:32
5æ¬ã®çžŠç·ã«næ¬ã®æšªç·ãåŒãå Žåã(3^(n+1)-1)/2 éãã§ããïŒ
No.1911DD++2024幎5æ28æ¥ 09:30
ïŒïŒæ¬ã®çžŠç·ã«næ¬ã®æšªç·ãåŒãå Žåã(3^(n+1)-1)/2 éãã§ããïŒ
ãããªæå¿«ãªåŒã«ãªããšã¯ïŒ
ã²ãã£ãšããŠæãå·Šã®çžŠç·ãšæãå³ã®çžŠç·ãšãåäžèŠããŠåèšïŒæ¬ã®çžŠç·ãšã¿ãªãããšã«ãã£ãŠå
šãŠã®çžŠç·ã«ã€ããŠå¯Ÿç§°ãšãããã¯ããã¯ã䜿ããšããããšãªã®ã§ããããïŒ
((4-1)^(n+1)-1)/2
-1ãã®ãã¡ã¯ã¿ãŒã®æå³ãåããŸãããâŠâŠ orz
No.1912Dengan kesaktian Indukmu2024幎5æ28æ¥ 09:55
(3^(n+1)-1)/2 éãã®å Žå
n=3ãªã40ãšãªãã
å¥ã§èª¿æ»ããã°ç¢ºãã«40éããšãªããŸããã
äŸã®
(i,j)=
(0,0)-->4
(0,1)-->6
(0,2)-->4
(0,3)-->1
(1,0)-->6
(1,1)-->8
(1,2)-->3
(2,0)-->4
(2,1)-->3
(3,0)-->1
ã§èš40éã
äœã瞊ç·mæ¬,暪ç·næ¬ã®ãã¿ã ããã§ã®äžè¬åŒãäœãããã§ããã
No.1913GAI2024幎5æ28æ¥ 11:05
äžè¬åŒãäœãããšæã£ãŠãããã瞊ç·ãå¥æ°ãšå¶æ°ã§ã¯æ§é ãå€ããæ§ã«
æããã®ã§çžŠç·ã4æ¬ã§æšªç·ãnæ¬ã§ããå Žåã®ãã¿ã ããã®çš®é¡ã調ã¹ãŠã¿ããã
n=1-->3
n=2-->8
n=3-->21
n=4-->55
n=5-->144
n=6-->377
ãããŸã§èª¿ã¹ãŠããã®æ°åã¯ãªããèŠãããšãããïŒ
ã§æ€çŽ¢ãããšãã£ããããæ°åfibo(n)ã§ã®
fibo(2*n+2)ã®éšåã察å¿ããŠããã
ãªã
ãã®ååèšæ°ã¯æ¬¡ã®çµåãé¢æ°nCr(=binomial(n,r))ã䜿ããš
gp > T(n,k)=binomial(n+k,2*k-1);
gp > for(n=1,10,S=[];for(k=1,n+1,S=concat(S,[T(n,k)]));print(n"=>"S";"vecsum(S)))
1=>[2, 1];3
2=>[3, 4, 1];8
3=>[4, 10, 6, 1];21
4=>[5, 20, 21, 8, 1];55
5=>[6, 35, 56, 36, 10, 1];144
6=>[7, 56, 126, 120, 55, 12, 1];377
7=>[8, 84, 252, 330, 220, 78, 14, 1];987
8=>[9, 120, 462, 792, 715, 364, 105, 16, 1];2584
9=>[10, 165, 792, 1716, 2002, 1365, 560, 136, 18, 1];6765
10=>[11, 220, 1287, 3432, 5005, 4368, 2380, 816, 171, 20, 1];17711
ã§æ±ããŠããããšã«ãªã£ãŠããã
ããã瞊ç·ã6æ¬ã®å Žåã¯æªèª¿æ»ãªã®ã§ãŸã äœãšãèšããªãã§ãã
No.1914GAI2024幎5æ28æ¥ 15:05
瞊ç·ã m æ¬ã§ããå Žåã
ã1 ãã m-1 ãŸã§ã®æ°ãéè€ãèš±ã㊠n å䞊ã¹ãããã ãçŽåã®æ°ãã 2 ã€ä»¥äžå°ãããªã£ãŠã¯ãããªãã
ã®äžŠã¹æ¹ã®ç·æ°ãšäžèŽããŸãã
ãªã®ã§ã
瞊ç·3æ¬
[1,1] * [[1,1],[1,1]]^(n-1) * t[1,1] = 2^n
瞊ç·4æ¬
[1,1,1] * [[1,1,0],[1,1,1],[1,1,1]]^(n-1) * t[1,1,1] = 1/â5 * ( ((3+â5)/2)^(n+1) - ((3-â5)/2)^(n+1) )
瞊ç·5æ¬
[1,1,1,1] * [[1,1,0,0],[1,1,1,0],[1,1,1,1],[1,1,1,1]]^(n-1) * t [1,1,1,1] = (3^(n+1)-1)/2
ãšåºããŸãã
瞊ç·6æ¬ã¯åºæå€ã綺éºã«åºãªãã®ã§é£ãããã
No.1915DD++2024幎5æ28æ¥ 17:24
ïŒæ¬ã®çžŠç·ãããå Žåã«ã€ããŠæšªç·ãnæ¬ã®å Žåã®æ§ææ°ã«ã€ããŠèª¿ã¹ãŠã¿ãŸããã
n=1-->5
n=2-->11
n=3-->40
n=4-->145
n=5-->525
n=6-->1900
n=7-->6875
n=8-->24875
ãããŸã§ã§OEISã®ãäžè©±ã«ãªããšn=1ãé€ããŠA136775ãããããããããã«ã¯
Number of multiplex juggling sequences of length n, base state <1,1> and hand capacity 2.
ãšããåãããããã説æãä»ããããŠããã
ãã ããã®æ°å€ã¯ãããããããæ§æãããŠããããã°ã©ã ãåèã«ãããŠããã
F(n)=for(i=0,n,for(j=0,n-i,for(k=0,n-i-j,\
print(i","j","k"=>"binomial(i+j,j)"*"binomial(j+k,k)"*"binomial(n-1,n-(i+j+k))"=>"\
binomial(i+j,j)*binomial(j+k,k)*binomial(n-1,n-(i+j+k))))))
ã§
2,3çªç®ã®éã«ãã暪ç·ã®æ°ãi
3,4çªç®ã®éã«ãã暪ç·ã®æ°ãj
4,5çªç®ã®éã«ãã暪ç·ã®æ°ãkæ¬
ãšããŠ
ãã®æšªç·ã®åãæ¹ãbinomial(i+j,j)*binomial(j+k,k)ã§èµ·ãããŠ
æ®ãã®æ¬æ°n-(i+j+k)ã1,2çªç®ãš5,6çªç®ã®éã«åããå Žåã®å¯èœæ§ãbinomial(n-1,n-(i+j+k))ãš
ããŠããããšã§äžæãåãããšã芳å¯ããŠã¿ãã
ããããã¹ãŠæãåãããããšã§ã(i,j,k)ã«å¯Ÿãããã¿ãŒã³æ°ãæ±ãŸã£ãŠããã®ã§ããã¹ãŠã®ç·åãã
n>=2ã§ã®æ°å€ãæ±ããŠè¡ããŸããã
n=6ã§1900ãšãªãçµéãäžã®è¡šã§ãã(F(6)ããã®è¡šç€º)
(i,j,k)
0,0,0= 1*1*0= 0
0,0,1= 1*1*1= 1
0,0,2= 1*1*5= 5
0,0,3= 1*1*10= 10
0,0,4= 1*1*10= 10
0,0,5= 1*1*5= 5
0,0,6= 1*1*1= 1
0,1,0= 1*1*1= 1
0,1,1= 1*2*5= 10
0,1,2= 1*3*10= 30
0,1,3= 1*4*10= 40
0,1,4= 1*5*5= 25
0,1,5= 1*6*1= 6
0,2,0= 1*1*5= 5
0,2,1= 1*3*10= 30
0,2,2= 1*6*10= 60
0,2,3= 1*10*5= 50
0,2,4= 1*15*1= 15
0,3,0= 1*1*10= 10
0,3,1= 1*4*10= 40
0,3,2= 1*10*5= 50
0,3,3= 1*20*1= 20
0,4,0= 1*1*10= 10
0,4,1= 1*5*5= 25
0,4,2= 1*15*1= 15
0,5,0= 1*1*5= 5
0,5,1= 1*6*1= 6
0,6,0= 1*1*1= 1
1,0,0= 1*1*1= 1
1,0,1= 1*1*5= 5
1,0,2= 1*1*10= 10
1,0,3= 1*1*10= 10
1,0,4= 1*1*5= 5
1,0,5= 1*1*1= 1
1,1,0= 2*1*5= 10
1,1,1= 2*2*10= 40
1,1,2= 2*3*10= 60
1,1,3= 2*4*5= 40
1,1,4= 2*5*1= 10
1,2,0= 3*1*10= 30
1,2,1= 3*3*10= 90
1,2,2= 3*6*5= 90
1,2,3= 3*10*1= 30
1,3,0= 4*1*10= 40
1,3,1= 4*4*5= 80
1,3,2= 4*10*1= 40
1,4,0= 5*1*5= 25
1,4,1= 5*5*1= 25
1,5,0= 6*1*1= 6
2,0,0= 1*1*5= 5
2,0,1= 1*1*10= 10
2,0,2= 1*1*10= 10
2,0,3= 1*1*5= 5
2,0,4= 1*1*1= 1
2,1,0= 3*1*10= 30
2,1,1= 3*2*10= 60
2,1,2= 3*3*5= 45
2,1,3= 3*4*1= 12
2,2,0= 6*1*10= 60
2,2,1= 6*3*5= 90
2,2,2= 6*6*1= 36
2,3,0= 10*1*5= 50
2,3,1= 10*4*1= 40
2,4,0= 15*1*1= 15
3,0,0= 1*1*10= 10
3,0,1= 1*1*10= 10
3,0,2= 1*1*5= 5
3,0,3= 1*1*1= 1
3,1,0= 4*1*10= 40
3,1,1= 4*2*5= 40
3,1,2= 4*3*1= 12
3,2,0= 10*1*5= 50
3,2,1= 10*3*1= 30
3,3,0= 20*1*1= 20
4,0,0= 1*1*10= 10
4,0,1= 1*1*5= 5
4,0,2= 1*1*1= 1
4,1,0= 5*1*5= 25
4,1,1= 5*2*1= 10
4,2,0= 15*1*1= 15
5,0,0= 1*1*5= 5
5,0,1= 1*1*1= 1
5,1,0= 6*1*1= 6
6,0,0= 1*1*1= 1
ãåèš 1900
ã¯ãŠããã¯äžã€ã®åŒã§äœããã®ãïŒ
No.1916GAI2024幎5æ28æ¥ 20:24
> æ®ãã®æ¬æ°n-(i+j+k)ã1,2çªç®ãš5,6çªç®ã®éã«åããå Žåã®å¯èœæ§ãbinomial(n-1,n-(i+j+k))
ããã¯
binomial(n+1-j,n-(i+j+k))
ã«ããªããšãããªããšæããŸã(äžå€®ã«äœ¿ã£ãjæ¬ã¯æ®ãæ¬æ°ã«åœ±é¿ããŸããé
眮ã«åœ±é¿ããŸãã)ã
ãã£ãŠn=1,2,3,âŠã«å¯Ÿããæ§ææ°ã¯
5,19,66,221,728,2380,7753,25213,81927,âŠ
ã®ããã«ãªããŸãã
ïŒn=2ãæäœæ¥ã§æ°ããŠã¿ããšã19ã§æ£ããããšãããããšæããŸããïŒ
ãããŠãã®æ°åã¯A005021ã«ããã挞ååŒã
a[1]=5, a[2]=19, a[3]=66, a[n+3]=5a[n+2]-6a[n+1]+a[n]
ãšæžãããŠããŸãã®ã§ãããã解ããŠäžè¬é
ã¯
a[n]=up^n+vq^n+wr^n
ãã ã
u={-(4â91)sin(arcsin(127â91/2366)/3)+7}/21
v={-(4â91)cos(arccos(-127â91/2366)/3)+7}/21
w={(4â91)cos(arccos(127â91/2366)/3)+7}/21
p={-(2â7)cos(arccos(-â7/14)/3)+5}/3
q={-(2â7)sin(arcsin(â7/14)/3)+5}/3
r={(2â7)cos(arccos(â7/14)/3)+5}/3
ãšããããŸãã
# u,v,wã¯49x^3-49x^2-105x+1=0ã®3解ãp,q,rã¯x^3-5x^2+6x-1=0ã®3解ã§ãã
No.1918ãããã2024幎5æ29æ¥ 00:38
ããããããããã®ææãåããŠæ¹ããŠ(æ°åã®ãã¿ãŒã³ã§ãããšåæã«æã£ãŠããŸãæªãç)
ïŒæ¬ã®çžŠç·ãããå Žåã«ã€ããŠæšªç·ãnæ¬ã®å Žåã®æ§ææ°ã«ã€ããŠèª¿ã¹ãŠã¿ãŸããã
n=1-->5
n=2-->19
n=3-->66
n=4-->221
n=5-->728
n=6-->2380
n=7-->7753
n=8-->25213
ãããŸã§ã§OEISã®ãäžè©±ã«ãªããšA005021ããããããã
ãããããããšn=3ã§éã£ãã®ã§
(i,j,k)
0,0,0= 1*1*4= 4
0,0,1= 1*1*6= 6
0,0,2= 1*1*4= 4
0,0,3= 1*1*1= 1
0,1,0= 1*1*3= 3
0,1,1= 1*2*3= 6
0,1,2= 1*3*1= 3
0,2,0= 1*1*2= 2
0,2,1= 1*3*1= 3
0,3,0= 1*1*1= 1
1,0,0= 1*1*6= 6
1,0,1= 1*1*4= 4
1,0,2= 1*1*1= 1
1,1,0= 2*1*3= 6
1,1,1= 2*2*1= 4
1,2,0= 3*1*1= 3
2,0,0= 1*1*4= 4
2,0,1= 1*1*1= 1
2,1,0= 3*1*1= 3
3,0,0= 1*1*1= 1
åèš; 66
ã§ãã§ãã¯ããŠã¿ãã®ã§ãããã©ãããŠã67ã«ã¯ãªããªãã®ã§ããã»ã»ã»?
ïŒããïŒä¿®æ£ããããã§ãããå®å¿ããŸãããïŒ
A005021ã®ã³ã¡ã³ãã¯Random walksãšãªã£ãŠããã®ã§ãšãŠãé©ããŠããŸãã
No.1919GAI2024幎5æ29æ¥ 05:43