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)

A247051 Rectangular array read upwards by columns: T = T(n,k) = number of paths from (0,2) 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

0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 1, 2, 1, 2, 1, 4, 3, 4, 4, 5, 4, 9, 8, 9, 12, 13, 12, 22, 21, 22, 33, 34, 33, 56, 55, 56, 88, 89, 88, 145, 144, 145, 232, 233, 232, 378, 377, 378, 609, 610, 609, 988, 987, 988, 1596, 1597, 1596, 2585, 2584, 2585, 4180, 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) = 2, 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}. The column sums form the Fibonacci sequence (A000045).
This is a 3-rowed array read upwards by columns. - N. J. A. Sloane, Sep 14 2014

Examples

			First 10 columns:
1 .. 0 .. 1 .. 1 .. 2 .. 3 .. 5 .. 8 .. 13 .. 21
0 .. 1 .. 0 .. 2 .. 1 .. 4 .. 4 .. 9 .. 12 .. 22
0 .. 0 .. 1 .. 0 .. 2 .. 1 .. 4 .. 4 .. 9 ... 12
T(3,1) counts these 2 paths, given as vector sums applied to (0,2): (1,-1) + (1,1) + (1,-1) and (1,-1) + (1,-1) + (1,1).
Partial sums of second components in each vector sum give the 2 integer strings described in Comments:  (2,1,2,1), (2,1,0,1).

Links

Crossrefs

Cf. A000045, A247049, A247050.

Programs

  • Mathematica
    t[0, 0] = 0; t[0, 1] = 0; t[0, 2] = 1; 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 *)
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-2) + (-1)^n for n >= 1; (row 1, the middle row) = F(n-1) - (-1)^n for n >=0; (row 2, the top row) = F(n+1) for n >= 1.

A247302 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), (2,1), (1,-1).

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 2, 1, 2, 2, 2, 2, 4, 4, 4, 8, 6, 8, 12, 12, 12, 24, 20, 24, 40, 36, 40, 72, 64, 72, 128, 112, 128, 224, 200, 224, 400, 352, 400, 704, 624, 704, 1248, 1104, 1248, 2208, 1952, 2208, 3904, 3456, 3904, 6912, 6112, 6912, 12224, 10816, 12224
Offset: 0

Views

Author

Clark Kimberling, Sep 11 2014

Keywords

Examples

			First 10 columns:
0 .. 1 .. 1 .. 2 .. 4 .. 6 .. 12 .. 20 .. 36 .. 64
1 .. 0 .. 2 .. 2 .. 4 .. 8 .. 12 .. 24 .. 40 .. 72
0 .. 1 .. 0 .. 2 .. 2 .. 4 .. 8 ... 12 .. 24 .. 40
T(4,1) counts these 4 paths, given as vector sums applied to (0,0):
(1,1) + (1,-1) + (1,1) + (1,-1);
(1,1) + (1,-1) + (1,-1) + (1,1);
(1,-1) + (1,1) + (1,-1) + (1,1);
(1,-1) + (1,1) + (1,1) + (1,-1).

Links

Crossrefs

Cf. A247050, A247301, A061275 (column sums).

Programs

  • Mathematica
    t[0, 0] = 0; t[0, 1] = 1; t[0, 2] = 0;
t[1, 0] = 1; t[1, 1] = 0; t[1, 2] = 1;
t[2, 0] = 0; t[2, 1] = 2; t[2, 2] = 1; 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, 0];
t[n_, 2] := t[n, 2] = t[n - 1, 1] + t[n - 2, 1];
TableForm[Reverse[Transpose[Table[t[n, k], {n, 0, 12}, {k, 0, 2}]]]]
Flatten[Table[t[n, k], {n, 0, 20}, {k, 0, 2}]]   (* A247302 *)

Formula

(row 0, the bottom row): r(n) = 2*r(n-2) + 2*r(n-3), with r(0) = 0, r(1) = 1, r(2) = 0;
(row 1, the middle row): r(n) = 2*r(n-2) + 2*r(n-3), with r(0) = 1, r(1) = 0, r(2) = 2;
(row 2, the top row): r(n) = 2*r(n-2) + 2*r(n-3), with r(0) = 0, r(1) = 1, r(2) = 1.
From Chai Wah Wu, Jan 24 2020: (Start)
a(n) = 2*a(n-6) + 2*a(n-9) for n > 8.
G.f.: (-x^8 - x^5 - x^3 - x)/(2*x^9 + 2*x^6 - 1). (End)
Showing 1-3 of 3 results.

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

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