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 1-5 of 5 results.

A090171 Triangle read by rows, related to Pascal's triangle read mod 2, starting with 0, 1, 0.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0
Offset: 0

Views

Author

N. J. A. Sloane, Jan 19 2004

Keywords

Examples

			Triangle begins:
0
1 0
1 1 0
0 1 0 0
1 1 1 1 0
...

Links

Crossrefs

Cf. A007318, A083093.
Cf. A090172, A090173, A090174, A091533, A091562, A205575 (same recurrence).
a(n, k) = A090174(n-1, k), k

Formula

T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-2, k) + T(n-2, k-1) + T(n-2, k-2) for n >= 2, k >= 0, with initial conditions specified by first two rows.

Extensions

Edited and extended by Christian G. Bower, Jan 20 2004

A090172 Triangle read by rows, related to Pascal's triangle read mod 2.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0
Offset: 0

Views

Author

N. J. A. Sloane, Jan 19 2004

Keywords

Examples

			1; 0,0; 1,1,1; 1,0,0,1; 0,1,1,1,0; ...

Links

Crossrefs

Row sums: A091563. Cf. A007318, A083093, A090171-A090174.

Formula

a(n, k) = A091562(n, k) mod 2.

Extensions

Edited and extended by Christian G. Bower, Jan 20 2004

A090173 Triangle read by rows, related to Pascal's triangle read mod 2, starting with 0, 0, 1.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1
Offset: 0

Views

Author

N. J. A. Sloane, Jan 19 2004

Keywords

Examples

			Triangle begins
  0;
  0,1;
  0,1,1;
  0,0,1,0;
  0,1,1,1,1;
  ...

Links

Crossrefs

Cf. A007318, A083093.
Cf. A090171, A090172, A090174, A091533, A091562, A205575 (same recurrence).
a(n, k) = A090174(n-1, k-1), k>0, 0 otherwise.

Formula

T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-2, k) + T(n-2, k-1) + T(n-2, k-2) for n >= 2, k >= 0, with initial conditions specified by first two rows.

Extensions

Edited and extended by Christian G. Bower, Jan 20 2004

A205575 Triangle read by rows, related to Pascal's triangle, starting with rows 1; 1,0.

Original entry on oeis.org

1, 1, 0, 2, 2, 1, 3, 5, 4, 1, 5, 12, 14, 8, 2, 8, 25, 38, 32, 15, 3, 13, 50, 94, 104, 71, 28, 5, 21, 96, 215, 293, 260, 149, 51, 8, 34, 180, 468, 756, 822, 612, 304, 92, 13, 55, 331, 980, 1828, 2346, 2136, 1376, 604, 164, 21
Offset: 0

Views

Author

Philippe Deléham, Jan 29 2012

Keywords

Comments

Antidiagonal sums are in A052980, row sums are in A046717.
Similar to A091533 and to A091562. Triangle satisfying the same recurrence as A091533 and A091562, but with the initial values T(0,0) = 1, T(0,1) = 1, T(1,1) = 0.

Examples

			Triangle begins :
1
1, 0
2, 2, 1
3, 5, 4, 1
5, 12, 14, 8, 2
8, 25, 38, 32, 15, 3
13, 50, 94, 104, 71, 28, 5

Crossrefs

Cf. Column 0: A000045, Diagonals : A000045, A029907, A036681.
Cf. A090171, A090172, A090173, A090174, A091533, A091562 (same recurrence).

Programs

  • PARI
    T(n,k) = {if(n<0, return(0)); if (n==0, if (k<0, return(0)); if (k==0, return(1))); if (n==1, if (k<0, return(0)); if (k==0, return(1)); if (k==1, return(0))); T(n-1,k)+T(n-1,k-1)+T(n-2,k)+T(n-2,k-1)+T(n-2,k-2);} \\ Michel Marcus, Oct 27 2021

Formula

T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) + T(n-2,k-2) for n>=2, k>=0, with initial conditions specified by first two rows. T(0,0) = 1, T(1,0) = 1, T(1,1) = 0.

Extensions

a(46), a(48) corrected by Georg Fischer, Oct 27 2021

A228815 Symmetric triangle, read by rows, related to Fibonacci numbers.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 5, 5, 2, 3, 10, 14, 10, 3, 5, 20, 36, 36, 20, 5, 8, 38, 83, 106, 83, 38, 8, 13, 71, 182, 281, 281, 182, 71, 13, 21, 130, 382, 690, 834, 690, 382, 130, 21, 34, 235, 778, 1606, 2268, 2268, 1606, 778, 235, 34, 55, 420, 1546, 3586, 5780, 6750
Offset: 0

Views

Author

Philippe Deléham, Oct 30 2013

Keywords

Comments

Triangles satisfying the same recurrence: A091533, A091562, A185081, A205575, A209137, A209138.

Examples

			Triangle begins :
0
1, 1
1, 2, 1
2, 5, 5, 2
3, 10, 14, 10, 3
5, 20, 36, 36, 20, 5
8, 38, 83, 106, 83, 38, 8
13, 71, 182, 281, 281, 182, 71, 13
21, 130, 382, 690, 834, 690, 382, 130, 21
34, 235, 778, 1606, 2268, 2268, 1606, 778, 235, 34
55, 420, 1546, 3586, 5780, 6750, 5780, 3586, 1546, 420, 55

Crossrefs

Cf. A000045 (1st column), A001629 (2nd column), A008998, A152011, A261055 (3rd column).

Formula

G.f.: x*(1+y)/(1-x-x*y-x^2-x^2*y-x^2*y^2).
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = 0, T(1,0) = T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n.
Sum_{k = 0..n} T(n,k)*x^k = A000045(n), 2*A015518(n), 3*A015524(n), 4*A200069(n) for x = 0, 1, 2, 3 respectively.
Sum_{k = 0..floor(n/2)} T(n-k,k) = A008998(n+1).
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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