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 1-4 of 4 results.

A129100 Triangle T, read by rows, where column n of T = column 0 of T^(2^n) for n>0, such that column 0 (A129092) equals the row sums of the prior row, starting with T(0,0)=1.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 6, 4, 1, 16, 24, 20, 8, 1, 69, 136, 136, 72, 16, 1, 430, 1162, 1360, 880, 272, 32, 1, 4137, 15702, 21204, 16032, 6240, 1056, 64, 1, 64436, 346768, 537748, 461992, 214336, 46784, 4160, 128, 1, 1676353, 12836904, 22891448, 21944520
Offset: 0

Views

Author

Paul D. Hanna, Mar 29 2007

Keywords

Comments

T(n,0) = A129092(n) = A030067(2^n - 1) for n>0 where A030067 is the Semi-Fibonacci numbers.

Examples

			Column 0 of row n equals A129092(n) = A030067(2^n-1) for n>=1,
where A030067 is the semi-Fibonacci numbers:
[(1), 1, (2), 1, 3, 2, (5), 1, 6, 3, 9, 2, 11, 5, (16), 1, ...],
which obey the recurrence:
A030067(n) = A030067(n/2) when n is even; and
A030067(n) = A030067(n-1) + A030067(n-2) when n is odd.
Triangle begins:
1;
1, 1;
2, 2, 1;
5, 6, 4, 1;
16, 24, 20, 8, 1;
69, 136, 136, 72, 16, 1;
430, 1162, 1360, 880, 272, 32, 1;
4137, 15702, 21204, 16032, 6240, 1056, 64, 1;
64436, 346768, 537748, 461992, 214336, 46784, 4160, 128, 1;
1676353, 12836904, 22891448, 21944520, 11720016, 3107456, 361856, 16512, 256, 1; ...
where columns shift left under matrix square, A129100^2, which starts:
1;
2, 1;
6, 4, 1;
24, 20, 8, 1;
136, 136, 72, 16, 1;
1162, 1360, 880, 272, 32, 1; ...
Inserting a left column of all 1's, yields matrix A129104:
1, 1;
1, 1, 1;
1, 2, 2, 1;
1, 5, 6, 4, 1;
1, 16, 24, 20, 8, 1;
1, 69, 136, 136, 72, 16, 1; ...
where row 0 of matrix power A129104^k forms row k of A129100,
as illustrated below.
For row 2: A129104^2 begins:
2, 2, 1;
3, 4, 3, 1;
6, 12, 12, 6, 1;
17, 54, 65, 42, 12, 1;
70, 362, 512, 400, 156, 24, 1;
431, 3708, 6223, 5656, 2744, 600, 48, 1; ...
and row 0 of A129104^2 equals row 2 of A129100: [2, 2, 1].
For row 3: A129104^3 begins:
5, 6, 4, 1;
11, 18, 16, 7, 1;
37, 88, 96, 56, 14, 1;
191, 672, 860, 609, 210, 28, 1;
1525, 8038, 11956, 9856, 4256, 812, 56, 1; ...
and row 0 of A129104^3 equals row 3 of A129100: [5, 6, 4, 1].
For row 4: A129104^4 begins:
16, 24, 20, 8, 1;
53, 112, 116, 64, 15, 1;
292, 890, 1088, 736, 240, 30, 1;
2571, 11350, 16056, 12664, 5185, 930, 60, 1; ...
and row 0 of A129104^4 equals row 4 of A129100: [16, 24, 20, 8, 1].

Crossrefs

Cf. A030067 (Semi-Fibonacci); A129092 (row sums=column 0), A129101 (column 1), A129102 (column 2), A129103 (column 3); variant: A129104.

Programs

  • PARI
    T(n,k)=local(A=Mat(1),B);for(m=1,n+1,B=matrix(m,m);for(r=1,m,for(c=1,r, if(r==c || r==1 || r==2,B[r,c]=1,if(c==1,B[r,1]=sum(i=1,r-1,A[r-1,i]), B[r,c]=(A^(2^(c-1)))[r-c+1,1])); )); A=B); return(A[n+1, k+1])

Formula

Row k = row 0 of matrix power A129104^k, where A129104 equals triangle A129100 with an additional leftmost column of all 1's.

A129102 Column 2 of triangle A129100; also equals column 0 of matrix power A129100^4.

Original entry on oeis.org

1, 4, 20, 136, 1360, 21204, 537748, 22891448, 1675538928, 214841466180, 48966357498452, 20069542925092392, 14932742187505262032, 20328765555338724571508, 50972424276515393704138196
Offset: 0

Views

Author

Paul D. Hanna, Mar 29 2007

Keywords

Crossrefs

Cf. A129100 (triangle); A129092 (column 0), A129101 (column 1), A129103 (column 3).

Programs

  • PARI
    a(n)=local(A=Mat(1),B);for(m=1,n+3,B=matrix(m,m);for(r=1,m,for(c=1,r, if(r==c || r==1 || r==2,B[r,c]=1,if(c==1,B[r,1]=sum(i=1,r-1,A[r-1,i]), B[r,c]=(A^(2^(c-1)))[r-c+1,1])); )); A=B); return(A[n+3,3])

A129103 Column 3 of triangle A129100; also equals column 0 of matrix power A129100^8.

Original entry on oeis.org

1, 8, 72, 880, 16032, 461992, 21944520, 1770483408, 248136807776, 61460157645704, 27267123945408968, 21902527945097597616, 32134030755805852610720, 86739696410985770930422760
Offset: 0

Views

Author

Paul D. Hanna, Mar 29 2007

Keywords

Crossrefs

Cf. A129100 (triangle); A129092 (column 0), A129101 (column 1), A129102 (column 2).

Programs

  • PARI
    a(n)=local(A=Mat(1),B);for(m=1,n+4,B=matrix(m,m);for(r=1,m,for(c=1,r, if(r==c || r==1 || r==2,B[r,c]=1,if(c==1,B[r,1]=sum(i=1,r-1,A[r-1,i]), B[r,c]=(A^(2^(c-1)))[r-c+1,1])); )); A=B); return(A[n+4,4])

A129104 Triangle T, read by rows, where row n (shifted left) of T equals row 0 of matrix power T^n for n>=0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 5, 6, 4, 1, 1, 16, 24, 20, 8, 1, 1, 69, 136, 136, 72, 16, 1, 1, 430, 1162, 1360, 880, 272, 32, 1, 1, 4137, 15702, 21204, 16032, 6240, 1056, 64, 1, 1, 64436, 346768, 537748, 461992, 214336, 46784, 4160, 128, 1, 1, 1676353, 12836904
Offset: 0

Views

Author

Paul D. Hanna, Apr 14 2007

Keywords

Comments

This irregular-shaped triangle T results from inserting a left column of all 1's into triangle A129100; curiously, column k of A129100 equals column 0 of matrix power A129100^(2^k), while row n of A129100 equals row 0 of matrix power T^n (T is this triangle).

Examples

			Triangle T begins:
1, 1;
1, 1, 1;
1, 2, 2, 1;
1, 5, 6, 4, 1;
1, 16, 24, 20, 8, 1;
1, 69, 136, 136, 72, 16, 1; ...
where row 0 of matrix power T^k forms row k of T shift left,
as illustrated below.
For row 2: the matrix square T^2 begins:
2, 2, 1;
3, 4, 3, 1;
6, 12, 12, 6, 1;
17, 54, 65, 42, 12, 1;
70, 362, 512, 400, 156, 24, 1;
431, 3708, 6223, 5656, 2744, 600, 48, 1; ...
and row 0 of T^2 equals row 2 of T shift left: [2, 2, 1].
For row 3: the matrix cube T^3 begins:
5, 6, 4, 1;
11, 18, 16, 7, 1;
37, 88, 96, 56, 14, 1;
191, 672, 860, 609, 210, 28, 1;
1525, 8038, 11956, 9856, 4256, 812, 56, 1; ...
and row 0 of T^3 equals row 3 of T shift left: [5, 6, 4, 1].
For row 4: T^4 begins:
16, 24, 20, 8, 1;
53, 112, 116, 64, 15, 1;
292, 890, 1088, 736, 240, 30, 1;
2571, 11350, 16056, 12664, 5185, 930, 60, 1; ...
and row 0 of T^4 equals row 4 of T shift left: [16, 24, 20, 8, 1].

Crossrefs

Cf. A030067 (Semi-Fibonacci); A129092 (column 1), A129101 (column 2), A129102 (column 3), A129103 (column 4); variant: A129100.

Programs

  • PARI
    T(n,k)=local(A=[1,1;1,1],B);for(m=1,n+1,B=matrix(m+1,m+1); for(r=1,m,for(c=1,r+1,if(r==c-1 || c==1,B[r,c]=1, B[r,c]=(A^(r-1))[1,c-1])));A=B); return(A[n+1, k+1])

Formula

Column 1: T(n,1) = A129092(n) = A030067(2^n - 1) for n>=1, where A030067 is the semi-Fibonacci numbers.
Showing 1-4 of 4 results.

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