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 28 results. Next

A113346 Column 0 of triangle A113345, also equals column 0 of A113350.

Original entry on oeis.org

1, 2, 5, 19, 113, 966, 10958, 156700, 2727794, 56306696, 1350043965, 36979531549, 1141573025172, 39272377323693, 1491452150268436, 62027842189908231, 2805631215820328992, 137199563717151509077, 7215932308408758314447
Offset: 0

Views

Author

Paul D. Hanna, Nov 08 2005

Keywords

Comments

A113345 is the matrix square of A113340.

Crossrefs

Cf. A113340, A113345, A113347 (column 1), A113348 (column 2), A113349 (column 3); A113350.

Programs

  • PARI
    a(n)=local(A,B);A=matrix(1,1);A[1,1]=1;for(m=2,n+1,B=matrix(m,m); for(i=1,m, for(j=1,i,if(i<3 || j==i || j>m-1,B[i,j]=1,if(j==1, B[i,1]=1,B[i,j]=(A^(2*j-1))[i-j+1,1]));));A=B);(A^2)[n+1,1]

A113355 Triangle T, read by rows, equal to the matrix square of triangle A113350, where T transforms column k of T into column k+1 of T.

Original entry on oeis.org

1, 4, 1, 18, 8, 1, 112, 68, 12, 1, 965, 712, 150, 16, 1, 10957, 9270, 2184, 264, 20, 1, 156699, 147174, 37523, 4912, 410, 24, 1, 2727793, 2786270, 754171, 104476, 9280, 588, 28, 1, 56306695, 61662544, 17502145, 2531004, 235025, 15672, 798, 32, 1
Offset: 0

Views

Author

Paul D. Hanna, Nov 08 2005

Keywords

Comments

Also, T transforms column k of A113340^2 into column k+1 of A113340^2. Column 0: T(n,0) = A113356(n) = A113346(n+1) - 1, where A113346 equals column 0 of triangle A113345 (=A113340^2).

Examples

			Triangle T begins:
1;
4,1;
18,8,1;
112,68,12,1;
965,712,150,16,1;
10957,9270,2184,264,20,1;
156699,147174,37523,4912,410,24,1;
2727793,2786270,754171,104476,9280,588,28,1;
56306695,61662544,17502145,2531004,235025,15672,798,32,1; ...
where T transforms column k of T into column k+1:
at k=0, [Q^2]*[1,4,18,112,965,...] = [1,8,68,712,9270,...];
at k=1, [Q^2]*[1,8,68,712,9270,...] = [1,12,150,2184,37523,...].
		

Crossrefs

Cf. A113340, A113350, A113356 (column 0), A113357 (column 1), A113358 (column 2), A113359 (column 3); A091351.

Programs

  • PARI
    T(n,k)=local(A,B);A=matrix(1,1);A[1,1]=1;for(m=2,n+1,B=matrix(m,m); for(i=1,m, for(j=1,i,if(i<3 || j==i || j>m-1,B[i,j]=1,if(j==1, B[i,1]=1,B[i,j]=(A^(2*j-1))[i-j+1,1]));));A=B); (matrix(#A,#A,r,c,if(r>=c,(A^(2*c))[r-c+1,1]))^2)[n+1,k+1]

Formula

T(n, k) = sum_{j=0..n-k} T(n-k, j)*T(j+k-1, k-1) for n>=k>0 with T(n, 0) = A113346(n+1) - 1, for n>=0.

A113347 Column 1 of triangle A113345, also equals column 0 of A113350^3.

Original entry on oeis.org

1, 6, 39, 327, 3556, 48659, 812462, 16136404, 373415239, 9900007028, 296557405704, 9921937128500, 367181525916035, 14906571298831661, 659191947156441025, 31558799717042019635, 1626968083690674214906
Offset: 0

Views

Author

Paul D. Hanna, Nov 08 2005

Keywords

Comments

A113345 is the matrix square of A113340.

Crossrefs

Cf. A113340, A113345, A113346 (column 0), A113348 (column 2), A113349 (column 3); A113350.

Programs

  • PARI
    a(n)=local(A,B);A=matrix(1,1);A[1,1]=1;for(m=2,n+2,B=matrix(m,m); for(i=1,m, for(j=1,i,if(i<3 || j==i || j>m-1,B[i,j]=1,if(j==1, B[i,1]=1,B[i,j]=(A^(2*j-1))[i-j+1,1]));));A=B);(A^2)[n+2,2]

A113348 Column 2 of triangle A113345, also equals column 0 of A113350^5.

Original entry on oeis.org

1, 10, 105, 1315, 19875, 357860, 7547602, 183518246, 5072961513, 157525315615, 5438681986872, 206954207984234, 8613936431369952, 389602050945939891, 19038814387466399303, 1000152089409979423044, 56229083214210734799693
Offset: 0

Views

Author

Paul D. Hanna, Nov 08 2005

Keywords

Comments

A113345 is the matrix square of A113340.

Crossrefs

Cf. A113340, A113345, A113346 (column 0), A113347 (column 1), A113349 (column 3); A113350.

Programs

  • PARI
    a(n)=local(A,B);A=matrix(1,1);A[1,1]=1;for(m=2,n+3,B=matrix(m,m); for(i=1,m, for(j=1,i,if(i<3 || j==i || j>m-1,B[i,j]=1,if(j==1, B[i,1]=1,B[i,j]=(A^(2*j-1))[i-j+1,1]));));A=B);(A^2)[n+3,3]

A113349 Column 3 of triangle A113345, also equals column 0 of A113350^7.

Original entry on oeis.org

1, 14, 203, 3367, 64750, 1435497, 36312626, 1036877170, 33086963196, 1169366274321, 45412092740791, 1924418011638535, 88445828358934074, 4384910640997110602, 233384463606862044134, 13278878088344760573344
Offset: 0

Views

Author

Paul D. Hanna, Nov 08 2005

Keywords

Comments

A113345 is the matrix square of A113340.

Crossrefs

Cf. A113340, A113345, A113346 (column 0), A113347 (column 1), A113348 (column 2); A113350.

Programs

  • PARI
    a(n)=local(A,B);A=matrix(1,1);A[1,1]=1;for(m=2,n+4,B=matrix(m,m); for(i=1,m, for(j=1,i,if(i<3 || j==i || j>m-1,B[i,j]=1,if(j==1, B[i,1]=1,B[i,j]=(A^(2*j-1))[i-j+1,1]));));A=B);(A^2)[n+4,4]

A113351 Column 1 of triangle A113350, also equals column 0 of A113340^4.

Original entry on oeis.org

1, 4, 22, 166, 1671, 21510, 341463, 6496923, 144856710, 3716746006, 108133664181, 3523951953981, 127331488458476, 5057405619797317, 219179491109978660, 10298771194610350427, 521765883689616772731, 28363341834962918344084
Offset: 0

Views

Author

Paul D. Hanna, Nov 08 2005

Keywords

Crossrefs

Cf. A113340, A113350, A113346 (column 0), A113352 (column 2), A113353 (column 3), A113354 (column 4).

Programs

  • PARI
    a(n)=local(A,B);A=matrix(1,1);A[1,1]=1;for(m=2,n+1,B=matrix(m,m); for(i=1,m, for(j=1,i,if(i<3 || j==i || j>m-1,B[i,j]=1,if(j==1, B[i,1]=1,B[i,j]=(A^(2*j-1))[i-j+1,1]));));A=B);(A^4)[n+1,1]

A113352 Column 2 of triangle A113350, also equals column 0 of A113340^6.

Original entry on oeis.org

1, 6, 51, 561, 7726, 129406, 2572892, 59525136, 1576414408, 47144606666, 1574176301192, 58126182666796, 2354183194432226, 103850488571214174, 4959424774062415780, 255032863135423053615, 14056171975717324065788
Offset: 0

Views

Author

Paul D. Hanna, Nov 08 2005

Keywords

Crossrefs

Cf. A113340, A113350, A113346 (column 0), A113351 (column 1), A113353 (column 3), A113354 (column 4).

Programs

  • PARI
    a(n)=local(A,B);A=matrix(1,1);A[1,1]=1;for(m=2,n+1,B=matrix(m,m); for(i=1,m, for(j=1,i,if(i<3 || j==i || j>m-1,B[i,j]=1,if(j==1, B[i,1]=1,B[i,j]=(A^(2*j-1))[i-j+1,1]));));A=B);(A^6)[n+1,1]

A113353 Column 3 of triangle A113350, also equals column 0 of A113340^8.

Original entry on oeis.org

1, 8, 92, 1324, 23010, 471724, 11198006, 303137822, 9239501237, 313689623136, 11755195387720, 482435619984340, 21537360479898341, 1039758627381190676, 54002918637393187687, 3003770445258147527669, 178205904321326719715551
Offset: 0

Views

Author

Paul D. Hanna, Nov 08 2005

Keywords

Crossrefs

Cf. A113340, A113350, A113346 (column 0), A113351 (column 1), A113352 (column 2), A113354 (column 4).

Programs

  • PARI
    a(n)=local(A,B);A=matrix(1,1);A[1,1]=1;for(m=2,n+1,B=matrix(m,m); for(i=1,m, for(j=1,i,if(i<3 || j==i || j>m-1,B[i,j]=1,if(j==1, B[i,1]=1,B[i,j]=(A^(2*j-1))[i-j+1,1]));));A=B);(A^8)[n+1,1]

A113354 Column 4 of triangle A113350, also equals column 0 of A113340^10.

Original entry on oeis.org

1, 10, 145, 2575, 53935, 1305070, 35924902, 1111107544, 38212134595, 1448428857802, 60055380515614, 2705916453572344, 131733359512179916, 6894586916062176161, 386203389346756538342, 23061963945353367894173
Offset: 0

Views

Author

Paul D. Hanna, Nov 08 2005

Keywords

Crossrefs

Cf. A113340, A113350, A113346 (column 0), A113351 (column 1), A113352 (column 2), A113353 (column 3).

Programs

  • PARI
    a(n)=local(A,B);A=matrix(1,1);A[1,1]=1;for(m=2,n+1,B=matrix(m,m); for(i=1,m, for(j=1,i,if(i<3 || j==i || j>m-1,B[i,j]=1,if(j==1, B[i,1]=1,B[i,j]=(A^(2*j-1))[i-j+1,1]));));A=B);(A^10)[n+1,1]

A113356 Column 0 of triangle A113355, which is the matrix square of A113350.

Original entry on oeis.org

1, 4, 18, 112, 965, 10957, 156699, 2727793, 56306695, 1350043964, 36979531548, 1141573025171, 39272377323692, 1491452150268435, 62027842189908230, 2805631215820328991, 137199563717151509076, 7215932308408758314446
Offset: 0

Views

Author

Paul D. Hanna, Nov 08 2005

Keywords

Crossrefs

Cf. A113340, A113350, A113355, A113357 (column 1), A113358 (column 2), A113359 (column 3).

Programs

  • PARI
    a(n)=local(A,B);A=matrix(1,1);A[1,1]=1;for(m=2,n+1,B=matrix(m,m); for(i=1,m, for(j=1,i,if(i<3 || j==i || j>m-1,B[i,j]=1,if(j==1, B[i,1]=1,B[i,j]=(A^(2*j-1))[i-j+1,1]));));A=B); (matrix(#A,#A,r,c,if(r>=c,(A^(2*c))[r-c+1,1]))^2)[n+1,1]

Formula

a(n) = A113346(n+1) - 1, where A113346 equals column 0 of triangle A113345 (=A113340^2).
Showing 1-10 of 28 results. Next