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瞊ç·ã4æ¬ã§æšªç·ãnæ¬ã§ã¯(A088305)
gp > a(n)=(((3+sqrt(5))/2)^(n+1)-((3-sqrt(5))/2)^(n+1))/sqrt(5)
gp > for(n=1,10,print(n";"round(a(n))))
1;3
2;8
3;21
4;55
5;144
6;377
7;987
8;2584
9;6765
10;17711
瞊ç·ã5æ¬ã§æšªç·ãnæ¬ã§ã¯(A261547)
gp > b(n)=(3^(n+1)-1)/2
gp > for(n=1,10,print(n";"b(n)))
1;4
2;13
3;40
4;121
5;364
6;1093
7;3280
8;9841
9;29524
10;88573
瞊ç·ã6æ¬ã§æšªç·ãnæ¬ã§ã¯(A005021)
c(n)={S=[];}for(i=0,n,for(j=0,n-i,for(k=0,n-i-j,\
S=concat(S,[binomial(i+j,j)*binomial(j+k,k)*binomial(n+1-j,n-(i+j+k))]))));vecsum(S)
gp > for(n=1,10,print(n";"c(n)))
1;5
2;19
3;66
4;221
5;728
6;2380
7;7753
8;25213
9;81927
10;266110
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P_6ãšåŒã°ããéïŒçŽç·äžïŒç¹AãBãCãDãEãF ã䞊ãã§ãããïŒããAããåºçºãã
2*n+5(æ©)ã«ãŠFã®å°ç¹ã«å°çããé
æ©ã®ã³ãŒã¹ãäœéãã§ãããïŒ ããã«åããšããã
ãã£ãŠæšªæ£2æ¬ã§ã¯2*2+5=9æ©ã§é²ãå®äŸãæ§æãããš
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]
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瞊æ£ã5æ¬ã§ããæã¯
P_5ãšåŒã°ããéïŒçŽç·äž5ç¹AãBãCãDãE ã䞊ãã§ãããïŒããAããåºçºãã
2*n+4(æ©)ã«ãŠEã®å°ç¹ã«å°çããé
æ©ã®ã³ãŒã¹ãäœéãã§ãããïŒ
ãšäžã®ãã¿ãŒã³ãåèã«
ä»åºŠã¯2*2+4=8æ©ã§é²ã¿
1;[A, B, A, B, A, B, C, D, E]
2;[A, B, A, B, C, B, C, D, E]
3;[A, B, A, B, C, D, C, D, E]
4;[A, B, A, B, C, D, E, D, E]
5;[A, B, C, B, A, B, C, D, E]
6;[A, B, C, B, C, B, C, D, E]
7;[A, B, C, B, C, D, C, D, E]
8;[A, B, C, B, C, D, E, D, E]
9;[A, B, C, D, C, B, C, D, E]
10;[A, B, C, D, C, D, C, D, E]
11;[A, B, C, D, C, D, E, D, E]
12;[A, B, C, D, E, D, C, D, E]
13;[A, B, C, D, E, D, E, D, E]
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ã ãã瞊æ£4æ¬ã®ãšãã¯
P_4ãšåŒã°ããéïŒçŽç·äž4ç¹AãBãCãD ã䞊ãã§ãããïŒããAããåºçºãã
2*n+3(æ©)ã«ãŠDã®å°ç¹ã«å°çããé
æ©ã®ã³ãŒã¹ãäœéãã§ãããïŒ
ã§åŠçãããæšªæ£2æ¬ã§ã¯7æ©ã§é²ã¿
1;[A, B, A, B, A, B, C, D]
2;[A, B, A, B, C, B, C, D]
3;[A, B, A, B, C, D, C, D]
4;[A, B, C, B, A, B, C, D]
5;[A, B, C, B, C, B, C, D]
6;[A, B, C, B, C, D, C, D]
7;[A, B, C, D, C, B, C, D]
8;[A, B, C, D, C, D, C, D]
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gp > F(n)={S=[];}for(i=0,n,for(j=0,n-i,for(k=0,n-i-j,for(l=0,n-i-j-k,for(m=0,n-i-j-k-l,
W=binomial(i+j,j)*binomial(j+k,k)*binomial(k+l,l)*binomial(l+m,m)*binomial(n+1-(j+k+l),n-(i+j+k+l+m));
S=concat(S,[W]))))));vecsum(S)
gp > for(n=1,12,print(n";"F(n)))
1;7
2;34
3;143
4;560
5;2108
6;7752
7;28101
8;100947
9;360526
10;1282735
11;4552624
12;16131656
OEISã§æ€çŽ¢ãããšA005023ããããããŸããã
åŸã£ãŠæ±ããã¹ãå€ã¯16131656(éã)ã§ã¯ïŒ
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ãã¿ã ããã§çžŠç·ãnæ¬(nâ§3)ã§æšªç·ãkæ¬ã§ã®äœãæ¹Tn(k)ãæŒžååŒã§æ§æãããš
T3(k)=if(k==1,2,2*memorize(T3,k-1))
T4(k)=if(k==1,3,k==2,8,3*memorize(T4,k-1)-binomial(2,2)*memorize(T4,k-2))
T5(k)=if(k==1,4,k==2,13,4*memorize(T5,k-1)-binomial(3,2)*memorize(T5,k-2))
T6(k)=if(k==1,5,k==2,19,k==3,66,5*memorize(T6,k-1)-binomial(4,2)*memorize(T6,k-2)+binomial(3,3)*memorize(T6,k-3))
T7(k)=if(k==1,6,k==2,26,k==3,100,6*memorize(T7,k-1)-binomial(5,2)*memorize(T7,k-2)+binomial(4,3)*memorize(T7,k-3))
T8(k)=if(k==1,7,k==2,34,k==3,143,k==4,560,7*memorize(T8,k-1)-binomial(6,2)*memorize(T8,k-2)+binomial(5,3)*memorize(T8,k-3)-binomial(4,4)*memorize(T8,k-4))
T9(k)=if(k==1,8,k==2,43,k==3,196,k==4,820,8*memorize(T9,k-1)-binomial(7,2)*memorize(T9,k-2)+binomial(6,3)*memorize(T9,k-3)-binomial(5,4)*memorize(T9,k-4))
T10(k)=if(k==1,9,k==2,53,k==3,260,k==4,1156,k==5,4845,9*memorize(T10,k-1)-binomial(8,2)*memorize(T10,k-2)+binomial(7,3)*memorize(T10,k-3)-binomial(6,4)*memorize(T10,k-4)+binomial(5,5)*memorize(T10,k-5))
T11(k)=if(k==1,10,k==2,64,k==3,336,k==4,1581,k==5,6954,10*memorize(T11,k-1)-binomial(9,2)*memorize(T11,k-2)+binomial(8,3)*memorize(T11,k-3)-binomial(7,4)*memorize(T11,k-4)+binomial(6,5)*memorize(T11,k-5))
*ã¹ããŒãã¢ãããèšãããã¡ã¢åããŠåŠçããŠããŸãã
çžŠã®æ¬æ°ãå€ããªããšåæå€ãããã€ãéããªããšãããªãã®ã§ãã®èŸºãé¢åãïŒ
çžŠã®æ¬æ°
-3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11(æ¬)
æšªã®æ¬æ°;ã§èŠãŠäžããã
1;2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
2;4 | 8 | 13 | 19 | 26 | 34 | 43 | 53 | 64
3;8 | 21 | 40 | 66 | 100 | 143 | 196 | 260 | 336
4;16 | 55 | 121 | 221 | 364 | 560 | 820 | 1156 | 1581
5;32 | 144 | 364 | 728 | 1288 | 2108 | 3264 | 4845 | 6954
6;64 | 377 | 1093 | 2380 | 4488 | 7752 | 12597 | 19551 | 29260
7;128 | 987 | 3280 | 7753 | 15504 | 28101 | 47652 | 76912 | 119416
8;256 | 2584 | 9841 | 25213 | 53296 | 100947 | 177859 | 297275 | 476905
9;512 | 6765 | 29524 | 81927 | 182688 | 360526 | 657800 | 1134705 | 1874730
10;1024 | 17711 | 88573 | 266110 | 625184 | 1282735 | 2417416 | 4292145 | 7283640
11;2048 | 46368 | 265720 | 864201 | 2137408 | 4552624 | 8844448 | 16128061 | 28048800
12;4096 | 121393 | 797161 | 2806272 | 7303360 | 16131656 | 32256553 | 60304951 | 107286661
13;8192 | 317811 | 2391484 | 9112264 | 24946816 | 57099056 | 117378336 | 224660626 | 408239530
14;16384 | 832040 | 7174453 | 29587889 | 85196928 | 201962057 | 426440955 | 834641671 | 1547129284
15;32768 | 2178309 | 21523360 | 96072133 | 290926848 | 714012495 | 1547491404 | 3094322026 | 5844716616
16;65536 | 5702887 | 64570081 | 311945595 | 993379072 | 2523515514 | 5610955132 | 11453607152 | 22025185281
17;131072 | 14930352 | 193710244 | 1012883066 | 3391793664 | 8916942687 | 20332248992 | 42344301686 | 82836630954
18;262144 | 39088169 | 581130733 | 3288813893 | 11580678656 | 31504028992 | 73645557469 | 156404021389 | 311063682160
19;524288 | 102334155 | 1743392200 | 10678716664 | 39539651584 | 111295205284 | 266668876540 | 577291806894 | 1166646177136
20;1048576 | 267914296 | 5230176601 | 34673583028 | 134998297600 | 393151913464 | 965384509651 | 2129654436910 | 4371207361885
ãŸã äœã®æ€èšŒãããŸããã... Python ã§ããã°ã©ã ããŠæ°ããååžããããªããŸããã
1282735ã¯ç·æ°ã®çã§ããçããã®æ°å€ãšåã£ãŠãªããããªæ°ãããŸã...
1282735
[764877, 279584, 133631, 64604, 27257, 9481, 2693, 608]
[279584, 478114, 262307, 147260, 72988, 29809, 9980, 2693]
[133631, 262307, 365985, 252938, 153368, 75216, 29809, 9481]
[64604, 147260, 252938, 314118, 250202, 153368, 72988, 27257]
[27257, 72988, 153368, 250202, 314118, 252938, 147260, 64604]
[9481, 29809, 75216, 153368, 252938, 365985, 262307, 133631]
[2693, 9980, 29809, 72988, 147260, 262307, 478114, 279584]
[608, 2693, 9481, 27257, 64604, 133631, 279584, 764877]
ãéšããããŸãããïŒç·šéã®ãã¹ã¯ãŒããééã£ãŠå
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Pythonã§çµãã ãã®ã¯ãªããªãåŠçãçµãããªãã®ã§
Claudiã«ãé¡ãããŠJuliaã«æžãçŽããŠè²°ã£ãŠå®è¡ãããšããããã®æéã§çµæãåºãŸããã
--start---------------------------------------
8 12
--Ans-------------------------------------------
16131656
[9188341, 3508269, 1778834, 939616, 451633, 184261, 63000, 17702]
[3508269, 5568480, 3238722, 1961381, 1086206, 507952, 197646, 63000]
[1778834, 3238722, 4168532, 3074842, 2048283, 1130230, 507952, 184261]
[939616, 1961381, 3074842, 3550353, 3019342, 2048283, 1086206, 451633]
[451633, 1086206, 2048283, 3019342, 3550353, 3074842, 1961381, 939616]
[184261, 507952, 1130230, 2048283, 3074842, 4168532, 3238722, 1778834]
[63000, 197646, 507952, 1086206, 1961381, 3238722, 5568480, 3508269]
[17702, 63000, 184261, 451633, 939616, 1778834, 3508269, 9188341]
--end-------------------------------------------
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-----
#include <bits/stdc++.h>
using namespace std;
int main () {
int n, m;
cin >> n >> m;
assert(n>2);
vector<vector<vector<long long>>> a(n-1,vector<vector<long long>>(n,vector<long long>(n,0)));
for (int j=0; j<n; j++) {
a.at(0).at(j).at(j) = 1;
}
for (int loop=0; loop<m; loop++) {
vector<vector<vector<long long>>> next(n-1,vector<vector<long long>>(n,vector<long long>(n,0)));
for (int j=0; j<n; j++) {
for (int k=0; k<n; k++) {
next.at(0).at(j).at(k) = a.at(0).at(j).at(k) + a.at(1).at(j).at(k);
for (int i=1; i<n-2; i++) {
next.at(i).at(j).at(k) = next.at(i-1).at(j).at(k) + a.at(i+1).at(j).at(k);
}
next.at(n-2).at(j).at(k) = next.at(n-3).at(j).at(k);
}
}
for (int i=0; i<n-1; i++) {
for (int j=0; j<n; j++) {
swap (next.at(i).at(j).at(i),next.at(i).at(j).at(i+1));
}
}
swap (a,next);
}
long long total = 0LL;
for (int j=0; j<n; j++) {
for (int k=0; k<n; k++) {
long long sum = 0LL;
for (int i=0; i<n-1; i++) {
sum += a.at(i).at(j).at(k);
}
cout << sum;
if (k==n-1) {
total += sum;
cout << endl;
} else {
cout << " ";
}
}
}
cout << "total:" << total << endl;
return 0;
}
-----
åºåãµã³ãã«
8 12
9188341 3508269 1778834 939616 451633 184261 63000 17702
3508269 5568480 3238722 1961381 1086206 507952 197646 63000
1778834 3238722 4168532 3074842 2048283 1130230 507952 184261
939616 1961381 3074842 3550353 3019342 2048283 1086206 451633
451633 1086206 2048283 3019342 3550353 3074842 1961381 939616
184261 507952 1130230 2048283 3074842 4168532 3238722 1778834
63000 197646 507952 1086206 1961381 3238722 5568480 3508269
17702 63000 184261 451633 939616 1778834 3508269 9188341
total:16131656
5 30
69706010502882 64476946102498 61603155451959 58814544487309 54236041597325
64476946102498 62876755878626 61865352374327 60803099299213 58814544487309
61603155451959 61865352374327 61899682489401 61865352374327 61603155451959
58814544487309 60803099299213 61865352374327 62876755878626 64476946102498
54236041597325 58814544487309 61603155451959 64476946102498 69706010502882
total:308836698141973
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