cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 23 results. Next

A135881 Column 0 of triangle A135880.

Original entry on oeis.org

1, 1, 2, 6, 25, 138, 970, 8390, 86796, 1049546, 14563135, 228448504, 4002300038, 77523038603, 1646131568618, 38043008887356, 950967024783228, 25573831547118764, 736404945614783668, 22611026430036582671
Offset: 0

Views

Author

Paul D. Hanna, Dec 15 2007

Keywords

Comments

Amazingly, this sequence also equals column 0 of tables A135878 and A135879, which have unusual recurrences seemingly unrelated to triangle A135880.

Examples

			Equals column 0 of triangle P=A135880:
1;
1, 1;
2, 2, 1;
6, 7, 3, 1;
25, 34, 15, 4, 1;
138, 215, 99, 26, 5, 1;
970, 1698, 814, 216, 40, 6, 1;
8390, 16220, 8057, 2171, 400, 57, 7, 1; ...
where column k of P^2 equals column 0 of P^(2k+2)
such that column 0 of P^2 equals this sequence shift left.
Also equals column 0 of irregular triangle A135879:
1;
1,1;
2,2,1,1;
6,6,4,4,2,2,1;
25,25,19,19,13,13,9,5,5,3,1,1;
138,138,113,113,88,88,69,50,50,37,24,24,15,10,5,5,2,1; ...
which has a recurrence similar to that of triangle A135877
which generates the double factorials.

Links

Crossrefs

Cf. A135880, A135879, A135878; other columns: A135882, A135883, A135884.

Programs

  • PARI
    /* Generated as column 0 in triangle A135880: */ {a(n)=local(P=Mat(1),R,PShR);if(n==0,1,for(i=0,n, PShR=matrix(#P,#P, r,c, if(r>=c,if(r==c,1,if(c==1,0,P[r-1,c-1]))));R=P*PShR; R=matrix(#P+1, #P+1, r,c, if(r>=c, if(r<#P+1,R[r,c],if(c==1,(P^2)[ #P,1],(P^(2*c-1))[r-c+1,1])))); P=matrix(#R, #R, r,c, if(r>=c, if(r<#R,P[r,c], (R^c)[r-c+1,1]))));P[n+1,1])}
  • PARI
    /* Generated as column 0 in triangle A135879 (faster): */ {a(n)=local(A=[1],B);if(n>0,for(i=1,n,m=1;B=[]; for(j=1,#A,if(j+m-1==floor((m+2)^2/4)-1,m+=1;B=concat(B,0));B=concat(B,A[ j])); A=Vec(Polrev(Vec(Pol(B)/(1-x+O(x^#B)))))));A[1]}

Extensions

Typo in entries (false comma) corrected by N. J. A. Sloane, Jan 23 2008

A135894 Triangle R, read by rows, where column k of R equals column 0 of P^(2k+1) where P=A135880.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 6, 12, 5, 1, 25, 63, 30, 7, 1, 138, 421, 220, 56, 9, 1, 970, 3472, 1945, 525, 90, 11, 1, 8390, 34380, 20340, 5733, 1026, 132, 13, 1, 86796, 399463, 247066, 72030, 13305, 1771, 182, 15, 1, 1049546, 5344770, 3430936, 1028076, 194646, 26565, 2808
Offset: 0

Views

Author

Paul D. Hanna, Dec 15 2007

Keywords

Comments

Triangle P = A135880 is defined by: column k of P^2 equals column 0 of P^(2k+2) such that column 0 of P^2 equals column 0 of P shift left.

Examples

			Triangle R begins:
1;
1, 1;
2, 3, 1;
6, 12, 5, 1;
25, 63, 30, 7, 1;
138, 421, 220, 56, 9, 1;
970, 3472, 1945, 525, 90, 11, 1;
8390, 34380, 20340, 5733, 1026, 132, 13, 1;
86796, 399463, 247066, 72030, 13305, 1771, 182, 15, 1;
1049546, 5344770, 3430936, 1028076, 194646, 26565, 2808, 240, 17, 1;
14563135, 81097517, 53741404, 16477041, 3182778, 442948, 47801, 4185, 306, 19, 1; ...
where column k of R equals column 0 of P^(2k+1) for k>=0,
and P = A135880 begins:
1;
1, 1;
2, 2, 1;
6, 7, 3, 1;
25, 34, 15, 4, 1;
138, 215, 99, 26, 5, 1;
970, 1698, 814, 216, 40, 6, 1; ...
where column k of P equals column 0 of R^(k+1).
The matrix product P^-1*R = A135898 = P (shifted right one column);
the matrix product R^-1*P^2 = A135900 = R (shifted down one row).

Crossrefs

Cf. A135881 (column 0), A135889 (column 1); A135880 (P), A135885 (Q=P^2), A135895 (R^2), A135896 (R^3), A135897 (R^4); A135888 (P^3) A135892 (P^5); A135898 (P^-1*R), A135899 (P*R^-1*P), A135900 (R^-1*Q).

Programs

  • PARI
    {T(n,k)=local(P=Mat(1),R=Mat(1),PShR);if(n>0,for(i=0,n, PShR=matrix(#P,#P, r,c, if(r>=c,if(r==c,1,if(c==1,0,P[r-1,c-1]))));R=P*PShR; R=matrix(#P+1, #P+1, r,c, if(r>=c, if(r<#P+1,R[r,c], if(c==1,(P^2)[ #P,1],(P^(2*c-1))[r-c+1,1])))); P=matrix(#R, #R, r,c, if(r>=c, if(r<#R,P[r,c], (R^c)[r-c+1,1])))));R[n+1,k+1]}

Formula

Column k of R = column 0 of P^(2k+1) for k>=0 where triangle P = A135880; column 0 of R = column 0 of P; column 1 of R = column 0 of P^3; column 2 of R = column 0 of P^5. See more formulas relating triangles P, Q and R, in entry A135880.

Extensions

Typo in formula corrected by Paul D. Hanna, Mar 26 2010

A135885 Triangle Q, read by rows, where column k of Q equals column 0 of Q^(k+1) and Q is equal to the matrix square of integer triangle P = A135880 such that column 0 of Q equals column 0 of P shift left.

Original entry on oeis.org

1, 2, 1, 6, 4, 1, 25, 20, 6, 1, 138, 126, 42, 8, 1, 970, 980, 351, 72, 10, 1, 8390, 9186, 3470, 748, 110, 12, 1, 86796, 101492, 39968, 8936, 1365, 156, 14, 1, 1049546, 1296934, 528306, 121532, 19090, 2250, 210, 16, 1, 14563135, 18868652, 7906598
Offset: 0

Views

Author

Paul D. Hanna, Dec 15 2007

Keywords

Examples

			Triangle Q = P^2 begins:
1;
2, 1;
6, 4, 1;
25, 20, 6, 1;
138, 126, 42, 8, 1;
970, 980, 351, 72, 10, 1;
8390, 9186, 3470, 748, 110, 12, 1;
86796, 101492, 39968, 8936, 1365, 156, 14, 1;
1049546, 1296934, 528306, 121532, 19090, 2250, 210, 16, 1;
14563135, 18868652, 7906598, 1861416, 298830, 36028, 3451, 272, 18, 1;
228448504, 308478492, 132426050, 31785380, 5193982, 637390, 62230, 5016, 342, 20, 1; ...
where column k of Q equals column 0 of Q^(k+1) for k>=0.
Related triangle P = A135880 begins:
1;
1, 1;
2, 2, 1;
6, 7, 3, 1;
25, 34, 15, 4, 1;
138, 215, 99, 26, 5, 1;
970, 1698, 814, 216, 40, 6, 1; ...
where column k of Q equals column 0 of P^(2k+2)
such that column 0 of P^2 equals column 0 of P shift left.
The matrix product P*R^-1*P = A135899 = Q (shifted down one row),
where R = A135894 begins:
1;
1, 1;
2, 3, 1;
6, 12, 5, 1;
25, 63, 30, 7, 1;
138, 421, 220, 56, 9, 1;
970, 3472, 1945, 525, 90, 11, 1; ...
in which column k of R equals column 0 of P^(2k+1).

Crossrefs

Cf. columns: A135881, A135886, A135887; related tables: A135880 (P), A135894 (R), A135891 (Q^2), A135893 (Q^3); A135898 (P^-1*R), A135899 (P*R^-1*P), A135900 (R^-1*Q).

Programs

  • PARI
    {T(n,k)=local(P=Mat(1),R,PShR);if(n>0,for(i=0,n, PShR=matrix(#P,#P, r,c, if(r>=c,if(r==c,1,if(c==1,0,P[r-1,c-1]))));R=P*PShR; R=matrix(#P+1, #P+1, r,c, if(r>=c, if(r<#P+1,R[r,c], if(c==1,(P^2)[ #P,1],(P^(2*c-1))[r-c+1,1])))); P=matrix(#R, #R, r,c, if(r>=c, if(r<#R,P[r,c], (R^c)[r-c+1,1])))));(P^2)[n+1,k+1]}

Formula

See formulas relating triangles P, Q and R, in entry A135880.

A135888 Triangle, read by rows, equal to the matrix cube of triangle P = A135880.

Original entry on oeis.org

1, 3, 1, 12, 6, 1, 63, 39, 9, 1, 421, 300, 81, 12, 1, 3472, 2741, 816, 138, 15, 1, 34380, 29380, 9366, 1716, 210, 18, 1, 399463, 363922, 122148, 23647, 3105, 297, 21, 1, 5344770, 5135894, 1795481, 362116, 49880, 5088, 399, 24, 1, 81097517, 81557270
Offset: 0

Views

Author

Paul D. Hanna, Dec 15 2007

Keywords

Comments

Matrix square equals triangle A135893.

Examples

			Triangle P^3 begins:
1;
3, 1;
12, 6, 1;
63, 39, 9, 1;
421, 300, 81, 12, 1;
3472, 2741, 816, 138, 15, 1;
34380, 29380, 9366, 1716, 210, 18, 1;
399463, 363922, 122148, 23647, 3105, 297, 21, 1;
5344770, 5135894, 1795481, 362116, 49880, 5088, 399, 24, 1;
81097517, 81557270, 29478724, 6138746, 875935, 93306, 7770, 516, 27, 1;
where P = A135880 begins:
1;
1, 1;
2, 2, 1;
6, 7, 3, 1;
25, 34, 15, 4, 1;
138, 215, 99, 26, 5, 1;
970, 1698, 814, 216, 40, 6, 1; ...
where column k of P^2 equals column 0 of P^(2k+2)
such that column 0 of P^2 equals column 0 of P shift left.

Crossrefs

Cf. columns: A135889, A135890; related tables: A135880 (P), A135894 (R), A135893 (P^6).

Programs

  • PARI
    {T(n,k)=local(P=Mat(1),R,PShR);if(n>0,for(i=0,n, PShR=matrix(#P,#P, r,c, if(r>=c,if(r==c,1,if(c==1,0,P[r-1,c-1]))));R=P*PShR; R=matrix(#P+1, #P+1, r,c, if(r>=c, if(r<#P+1,R[r,c], if(c==1,(P^2)[ #P,1],(P^(2*c-1))[r-c+1,1])))); P=matrix(#R, #R, r,c, if(r>=c, if(r<#R,P[r,c], (R^c)[r-c+1,1])))));(P^3)[n+1,k+1]}

A135882 Column 1 of triangle A135880.

Original entry on oeis.org

1, 2, 7, 34, 215, 1698, 16220, 182714, 2378780, 35219202, 585245185, 10797322816, 219163958124, 4856832298391, 116735215192864, 3025759884533190, 84155831914971391, 2500599947944218716, 79072271422935678302
Offset: 0

Views

Author

Paul D. Hanna, Dec 15 2007

Keywords

Examples

			Equals column 1 of triangle P=A135880:
1;
1, 1;
2, 2, 1;
6, 7, 3, 1;
25, 34, 15, 4, 1;
138, 215, 99, 26, 5, 1;
970, 1698, 814, 216, 40, 6, 1; ...
where column k of P^2 equals column 0 of P^(2k+2)
such that column 0 of P^2 equals column 0 of P shift left.

Links

Crossrefs

Cf. A135880; other columns: A135881, A135883, A135884.

Programs

  • PARI
    {a(n)=local(P=Mat(1),R,PShR);if(n==0,1,for(i=0,n, PShR=matrix(#P,#P, r,c, if(r>=c,if(r==c,1,if(c==1,0,P[r-1,c-1]))));R=P*PShR; R=matrix(#P+1, #P+1, r,c, if(r>=c, if(r<#P+1,R[r,c], if(c==1,(P^2)[ #P,1],(P^(2*c-1))[r-c+1,1])))); P=matrix(#R, #R, r,c, if(r>=c, if(r<#R,P[r,c], (R^c)[r-c+1,1]))));P[n+2,2])}

Extensions

Error in entries (false comma) corrected by N. J. A. Sloane, Jan 23 2008

A135883 Column 2 of triangle A135880.

Original entry on oeis.org

1, 3, 15, 99, 814, 8057, 93627, 1252752, 19003467, 322722064, 6071897378, 125464556309, 2826120900315, 68954181763586, 1812280504183309, 51059994255961903, 1535575877864707548, 49107734497585814006
Offset: 0

Views

Author

Paul D. Hanna, Dec 15 2007

Keywords

Examples

			Equals column 2 of triangle P=A135880:
1;
1, 1;
2, 2, 1;
6, 7, 3, 1;
25, 34, 15, 4, 1;
138, 215, 99, 26, 5, 1;
970, 1698, 814, 216, 40, 6, 1;
8390, 16220, 8057, 2171, 400, 57, 7, 1; ...
where column k of P^2 equals column 0 of P^(2k+2)
such that column 0 of P^2 equals column 0 of P shift left.

Crossrefs

Cf. A135880; other columns: A135881, A135882, A135884.

Programs

  • PARI
    {a(n)=local(P=Mat(1),R,PShR);if(n==0,1,for(i=0,n+1, PShR=matrix(#P,#P, r,c, if(r>=c,if(r==c,1,if(c==1,0,P[r-1,c-1]))));R=P*PShR; R=matrix(#P+1, #P+1, r,c, if(r>=c, if(r<#P+1,R[r,c], if(c==1,(P^2)[ #P,1],(P^(2*c-1))[r-c+1,1])))); P=matrix(#R, #R, r,c, if(r>=c, if(r<#R,P[r,c], (R^c)[r-c+1,1]))));P[n+3,3])}

A135884 Column 3 of triangle A135880.

Original entry on oeis.org

1, 4, 26, 216, 2171, 25628, 348050, 5352788, 92056223, 1752149568, 36591725976, 832352590164, 20493399785598, 543168774618834, 15424012639825146, 467276557333020682, 15046702103550879196, 513273141160665106150
Offset: 0

Views

Author

Paul D. Hanna, Dec 15 2007

Keywords

Examples

			Equals column 3 of triangle P=A135880:
1;
1, 1;
2, 2, 1;
6, 7, 3, 1;
25, 34, 15, 4, 1;
138, 215, 99, 26, 5, 1;
970, 1698, 814, 216, 40, 6, 1;
8390, 16220, 8057, 2171, 400, 57, 7, 1; ...
where column k of P^2 equals column 0 of P^(2k+2)
such that column 0 of P^2 equals column 0 of P shift left.

Crossrefs

Cf. A135880; other columns: A135881, A135882, A135883.

Programs

  • PARI
    {a(n)=local(P=Mat(1),R,PShR);if(n==0,1,for(i=0,n+2, PShR=matrix(#P,#P, r,c, if(r>=c,if(r==c,1,if(c==1,0,P[r-1,c-1]))));R=P*PShR; R=matrix(#P+1, #P+1, r,c, if(r>=c, if(r<#P+1,R[r,c], if(c==1,(P^2)[ #P,1],(P^(2*c-1))[r-c+1,1])))); P=matrix(#R, #R, r,c, if(r>=c, if(r<#R,P[r,c], (R^c)[r-c+1,1]))));P[n+4,4])}

A135892 Triangle, read by rows, equal to P^5, where triangle P = A135880.

Original entry on oeis.org

1, 5, 1, 30, 10, 1, 220, 95, 15, 1, 1945, 990, 195, 20, 1, 20340, 11635, 2625, 330, 25, 1, 247066, 154450, 38270, 5440, 500, 30, 1, 3430936, 2302142, 611525, 94515, 9750, 705, 35, 1, 53741404, 38229214, 10721093, 1761940, 196500, 15870, 945, 40, 1
Offset: 0

Views

Author

Paul D. Hanna, Dec 15 2007

Keywords

Comments

Triangle P = A135880 is defined by: column k of P^2 equals column 0 of P^(2k+2) such that column 0 of P^2 equals column 0 of P shift left.

Examples

			Triangle P^5 begins:
1;
5, 1;
30, 10, 1;
220, 95, 15, 1;
1945, 990, 195, 20, 1;
20340, 11635, 2625, 330, 25, 1;
247066, 154450, 38270, 5440, 500, 30, 1;
3430936, 2302142, 611525, 94515, 9750, 705, 35, 1;
53741404, 38229214, 10721093, 1761940, 196500, 15870, 945, 40, 1;
938816814, 701685738, 205607124, 35429974, 4182295, 363820, 24115, 1220, 45, 1;
where P = A135880 begins:
1;
1, 1;
2, 2, 1;
6, 7, 3, 1;
25, 34, 15, 4, 1;
138, 215, 99, 26, 5, 1;
970, 1698, 814, 216, 40, 6, 1; ...
in which column k of P = column 0 of R^(k+1),
where R = A135894 begins:
1;
1, 1;
2, 3, 1;
6, 12, 5, 1;
25, 63, 30, 7, 1;
138, 421, 220, 56, 9, 1;
970, 3472, 1945, 525, 90, 11, 1; ...
in which column k of R equals column 0 of P^(2k+1).

Crossrefs

Cf. A135880 (P), A135894 (R), A135895 (R^2), A135896 (R^3), A135897 (R^4).

Programs

  • PARI
    {T(n,k)=local(P=Mat(1),R,PShR);if(n>0,for(i=0,n, PShR=matrix(#P,#P, r,c, if(r>=c,if(r==c,1,if(c==1,0,P[r-1,c-1]))));R=P*PShR; R=matrix(#P+1, #P+1, r,c, if(r>=c, if(r<#P+1,R[r,c], if(c==1,(P^2)[ #P,1],(P^(2*c-1))[r-c+1,1])))); P=matrix(#R, #R, r,c, if(r>=c, if(r<#R,P[r,c], (R^c)[r-c+1,1])))));(P^5)[n+1,k+1]}

Formula

Column k of P^5 = column 2 of R^(k+1) for k>=0 where triangle R = A135894; column 0 of P^5 = column 2 of R; column 1 of P^5 = column 2 of R^2; column 2 of P^5 = column 2 of R^3; column 3 of P^5 = column 2 of R^4.

A135893 Triangle, read by rows, equal to P^6, where triangle P = A135880; also equals Q^3 where Q = P^2 = A135885.

Original entry on oeis.org

1, 6, 1, 42, 12, 1, 351, 132, 18, 1, 3470, 1554, 270, 24, 1, 39968, 20260, 4089, 456, 30, 1, 528306, 294218, 65874, 8436, 690, 36, 1, 7906598, 4745522, 1147662, 161576, 15075, 972, 42, 1, 132426050, 84534154, 21710680, 3277148, 334390, 24486, 1302
Offset: 0

Views

Author

Paul D. Hanna, Dec 15 2007

Keywords

Comments

Triangle P = A135880 is defined by: column k of P^2 equals column 0 of P^(2k+2) such that column 0 of P^2 equals column 0 of P shift left.

Examples

			Triangle P^6 = Q^3 begins:
1;
6, 1;
42, 12, 1;
351, 132, 18, 1;
3470, 1554, 270, 24, 1;
39968, 20260, 4089, 456, 30, 1;
528306, 294218, 65874, 8436, 690, 36, 1;
7906598, 4745522, 1147662, 161576, 15075, 972, 42, 1;
132426050, 84534154, 21710680, 3277148, 334390, 24486, 1302, 48, 1;
2457643895, 1652665714, 445574768, 70977244, 7732100, 617100, 37149, 1680, 54, 1;
where P = A135880 begins:
1;
1, 1;
2, 2, 1;
6, 7, 3, 1;
25, 34, 15, 4, 1;
138, 215, 99, 26, 5, 1;
970, 1698, 814, 216, 40, 6, 1; ...
and Q = P^2 = A135885 begins:
1;
2, 1;
6, 4, 1;
25, 20, 6, 1;
138, 126, 42, 8, 1;
970, 980, 351, 72, 10, 1;
8390, 9186, 3470, 748, 110, 12, 1; ...
where column k of Q = column 0 of Q^(k+1).

Crossrefs

Cf. A135887 (column 0); A135880 (P), A135885 (Q=P^2), A135891 (Q^2).

Programs

  • PARI
    {T(n,k)=local(P=Mat(1),R,PShR);if(n>0,for(i=0,n, PShR=matrix(#P,#P, r,c, if(r>=c,if(r==c,1,if(c==1,0,P[r-1,c-1]))));R=P*PShR; R=matrix(#P+1, #P+1, r,c, if(r>=c, if(r<#P+1,R[r,c], if(c==1,(P^2)[ #P,1],(P^(2*c-1))[r-c+1,1])))); P=matrix(#R, #R, r,c, if(r>=c, if(r<#R,P[r,c], (R^c)[r-c+1,1])))));(P^6)[n+1,k+1]}

Formula

Column k of Q^3 = column 2 of Q^(k+1) for k>=0 where triangle Q = P^2 = A135885; column 0 of Q^3 = column 2 of Q; column 1 of Q^3 = column 2 of Q^2.

A135898 Triangle, read by rows equal to the matrix product P^-1*R, where P = A135880 and R = A135894; P^-1*R equals triangle P shifted right one column.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 6, 7, 3, 1, 0, 25, 34, 15, 4, 1, 0, 138, 215, 99, 26, 5, 1, 0, 970, 1698, 814, 216, 40, 6, 1, 0, 8390, 16220, 8057, 2171, 400, 57, 7, 1, 0, 86796, 182714, 93627, 25628, 4740, 666, 77, 8, 1, 0, 1049546, 2378780, 1252752, 348050, 64805
Offset: 0

Views

Author

Paul D. Hanna, Dec 15 2007

Keywords

Examples

			Triangle begins:
1;
0, 1;
0, 1, 1;
0, 2, 2, 1;
0, 6, 7, 3, 1;
0, 25, 34, 15, 4, 1;
0, 138, 215, 99, 26, 5, 1;
0, 970, 1698, 814, 216, 40, 6, 1;
0, 8390, 16220, 8057, 2171, 400, 57, 7, 1;
0, 86796, 182714, 93627, 25628, 4740, 666, 77, 8, 1; ...
This triangle equals matrix product P^-1*R,
which equals triangle P shifted right one column,
where P = A135880 begins:
1;
1, 1;
2, 2, 1;
6, 7, 3, 1;
25, 34, 15, 4, 1;
138, 215, 99, 26, 5, 1;
970, 1698, 814, 216, 40, 6, 1; ...
and Q = P^2 = A135885 begins:
1;
2, 1;
6, 4, 1;
25, 20, 6, 1;
138, 126, 42, 8, 1;
970, 980, 351, 72, 10, 1;
8390, 9186, 3470, 748, 110, 12, 1; ...
and R = A135894 begins:
1;
1, 1;
2, 3, 1;
6, 12, 5, 1;
25, 63, 30, 7, 1;
138, 421, 220, 56, 9, 1;
970, 3472, 1945, 525, 90, 11, 1; ...
where column k of R equals column 0 of P^(2k+1),
and column k of Q=P^2 equals column 0 of P^(2k+2), for k>=0.

Crossrefs

Cf. A135880 (P), A135885 (Q=P^2), A135894 (R); A135899 (P*R^-1*P), A135900 (R^-1*Q).

Programs

  • PARI
    {T(n,k)=local(P=Mat(1),R=Mat(1),PShR);if(n>0,for(i=0,n, PShR=matrix(#P,#P, r,c, if(r>=c,if(r==c,1,if(c==1,0,P[r-1,c-1]))));R=P*PShR; R=matrix(#P+1, #P+1, r,c, if(r>=c, if(r<#P+1,R[r,c], if(c==1,(P^2)[ #P,1],(P^(2*c-1))[r-c+1,1])))); P=matrix(#R, #R, r,c, if(r>=c, if(r<#R,P[r,c], (R^c)[r-c+1,1])))));(P^-1*R)[n+1,k+1]}
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