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 1-3 of 3 results.

A247049 Rectangular array read upwards by columns: T = T(n,k) = number of paths from (0,0) to (n,k), where 0 >= k <= 2, consisting of segments given by the vectors (1,1), (1,2), (1,-1).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 5, 5, 5, 8, 8, 8, 13, 13, 13, 21, 21, 21, 34, 34, 34, 55, 55, 55, 89, 89, 89, 144, 144, 144, 233, 233, 233, 377, 377, 377, 610, 610, 610, 987, 987, 987, 1597, 1597, 1597, 2584, 2584, 2584, 4181, 4181, 4181
Offset: 0

Views

Author

Clark Kimberling, Sep 11 2014

Keywords

Comments

Also, T(n,k) = number of strings s(0)..s(n) of integers such that s(0) = 0, s(n) = k, and if i > 0, then s(i) is in {0,1,2} and s(i) - s(i-1) is in {1,2,-1}. Every row of T is a Fibonacci sequence (A000045), as is the sequence of column sums.
This is a 3-rowed array read upwards by columns. - N. J. A. Sloane, Sep 14 2014

Examples

			First 10 columns:
0 .. 1 .. 1 .. 2 .. 3 .. 5 .. 8 .. 13 .. 21 .. 34
0 .. 1 .. 1 .. 2 .. 3 .. 5 .. 8 .. 13 .. 21 .. 34
1 .. 0 .. 1 .. 1 .. 2 .. 3 .. 5 .. 8 ... 13 .. 21
T(4,1) counts these 3 paths, given as vector sums applied to (0,0):
(1,2) + (1,-1) + (1,1) + (1,-1);
(1,1) + (1,-1) + (1,2) + (1,-1);
(1,2) + (1,-1) + (1,-1) + (1,1).
Partial sums of second components in each vector sum give the 3 integer strings described in Comments:  (0,2,1,2,1), (0,1,0,2,1), (0,2,1,0,1).

Links

Crossrefs

Cf. A000045, A247050, A247051, A247309, A247310, A247311.

Programs

  • Mathematica
    t[0, 0] = 1; t[0, 1] = 0; t[0, 2] = 0; t[n_, 0] := t[n, 0] = t[n - 1, 1]; t[n_, 1] := t[n, 1] = t[n - 1, 0] + t[n - 1, 2]; t[n_, 2] := t[n, 2] = t[n - 1, 0] + t[n - 1, 1]; TableForm[ Reverse[Transpose[Table[t[n, k], {n, 0, 12}, {k, 0, 2}]]]] (* array *)
u = Flatten[Table[t[n, k], {n, 0, 20}, {k, 0, 2}]] (* sequence *)

Formula

Let F = A000045, the Fibonacci numbers. Then (row 0, the bottom row) = F(n-1) for n >= 0; (row 1, the middle row) = F(n) for n >=0; (row 2, the top row) = (row 1).
Conjectures from Chai Wah Wu, Apr 16 2025: (Start)
a(n) = a(n-3) + a(n-6) for n > 5.
G.f.: (-x^5 - x^4 + x^3 - 1)/(x^6 + x^3 - 1). (End)

A247309 Rectangular array read upwards by columns: T = T(n,k) = number of paths from (0,1) to (n,k), where 0 <= k <= 2, consisting of segments given by the vectors (1,1), (1,0), (1,-1), (1,-2).

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 2, 3, 3, 5, 8, 8, 13, 21, 21, 34, 55, 55, 89, 144, 144, 233, 377, 377, 610, 987, 987, 1597, 2584, 2584, 4181, 6765, 6765, 10946, 17711, 17711, 28657, 46368, 46368, 75025, 121393, 121393, 196418, 317811, 317811, 514229, 832040, 832040
Offset: 0

Views

Author

Clark Kimberling, Sep 12 2014

Keywords

Comments

Every member of T is a Fibonacci number, and the sum of the numbers in column n is A000045(2n+2).

Examples

			First 10 columns:
0 .. 1 .. 3 .. 8 .. 21 .. 55 .. 144 .. 377 .. 987 ... 2584
0 .. 1 .. 3 .. 8 .. 21 .. 55 .. 144 .. 377 .. 987 ... 2584
1 .. 1 .. 2 .. 5 .. 13 .. 34 .. 89 ... 233 .. 610 ... 1597
T(2,2) counts these 3 paths, given as vector sums applied to (0,0):
(1,2) + (1,0); (1,1) + (1,1); (1,0) + (1,2).

Links

Crossrefs

Cf. A247049, A247310, A000045.

Programs

  • Mathematica
    t[0, 0] = 1; t[0, 1] = 0; t[0, 2] = 0; t[1, 2] = 1;
t[n_, 0] := t[n, 0] = t[n - 1, 0] + t[n - 1, 1];
t[n_, 1] := t[n, 1] = t[n - 1, 0] + t[n - 1, 1] + t[n - 1, 2];
t[n_, 2] := t[n, 2] = t[n - 1, 0] + t[n - 1, 1] + t[n - 1, 2]
TableForm[Reverse[Transpose[Table[t[n, k], {n, 0, 12}, {k, 0, 2}]]]] (*  array *)
Flatten[Table[t[n, k], {n, 0, 20}, {k, 0, 2}]] (*  A247309 *)

Formula

Let F = A000045 (Fibonacci numbers); then
(row 0, the bottom row) = (F(2n)), n >= 0;
(row 1, the middle row) = (F(2n)), n >= 0;
(row 2, the top row) = (F(2n-1)), n >= 0.
(n-th column sum) = (F(2n+2)), n >= 0.

A247311 Rectangular array read upwards by columns: T = T(n,k) = number of paths from (0,1) to (n,k), where 0 <= k <= 2, consisting of segments given by the vectors (1,1), (1,0), (1,-1).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 2, 2, 1, 4, 5, 3, 9, 12, 8, 21, 29, 20, 50, 70, 49, 120, 169, 119, 289, 408, 288, 697, 985, 696, 1682, 2378, 1681, 4060, 5741, 4059, 9801, 13860, 9800, 23661, 33461, 23660, 57122, 80782, 57121, 137904, 195025, 137903, 332929, 470832, 332928
Offset: 0

Views

Author

Clark Kimberling, Sep 12 2014

Keywords

Comments

Also, T(n,k) = number of strings s(0)..s(n) of integers such that s(0) = 0, s(n) = k, and for 0 < i <= n, s(i) is in {0,1,2}, and s(i) - s(i-1) is in {-1,0,1}.
(row 0, the bottom row): A024537;
(row 1, the middle row): A000129;
(row 2, the top row): A048739;
(n-th column sum): A000129.

Examples

			First 10 columns:
  0 .. 0 .. 1 .. 3 .. 8 ... 20 .. 49 .. 119 .. 288 .. 696
  0 .. 1 .. 2 .. 5 .. 12 .. 29 .. 70 .. 169 .. 408 .. 985
  1 .. 1 .. 2 .. 4 .. 9 ... 21 .. 50 .. 120 .. 289 .. 697
T(3,2) counts these 3 paths, given as vector sums applied to (0,0):
  (1,1) + (1,1) + (1,0); (1,1) + (1,0) + (1,1); (1,0) + (1,1) + (1,1).

Links

Crossrefs

Cf. A247049, A247309, A247310, A000129, A024537, A048739.

Programs

  • Mathematica
    t[0, 0] = 1; t[0, 1] = 0; t[0, 2] = 0; t[1, 2] = 0;
t[n_, 0] := t[n, 0] = t[n - 1, 0] + t[n - 1, 1];
t[n_, 1] := t[n, 1] = t[n - 1, 0] + t[n - 1, 1] + t[n - 1, 2];
t[n_, 2] := t[n, 2] = t[n - 1, 1] + t[n - 1, 2]
TableForm[Reverse[Transpose[Table[t[n, k], {n, 0, 12}, {k, 0, 2}]]]] (*  array *)
Flatten[Table[t[n, k], {n, 0, 20}, {k, 0, 2}]] (* A247311 *)
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.