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.

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A247352 Rectangular array read by upwards by columns: T = T(n,k) = number of paths from (0,1) to (n,k), where 0 <= k <= 3, consisting of segments given by the vectors (1,1), (1,2), (1,-1).

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 1, 0, 2, 2, 1, 2, 2, 3, 4, 2, 5, 8, 5, 5, 10, 12, 13, 10, 17, 28, 22, 17, 38, 49, 45, 38, 66, 100, 87, 66, 138, 191, 166, 138, 257, 370, 329, 257, 508, 724, 627, 508, 981, 1392, 1232, 981, 1900, 2721, 2373, 1900, 3702, 5254, 4621, 3702
Offset: 0

Views

Author

Clark Kimberling, Sep 15 2014

Keywords

Comments

Also, T(n,k) = number of strings s(0)..s(n) of integers such that s(0) = 1, s(n) = k, and for i > 0, s(i) is in {0,1,2,3} and s(i) - s(i-1) is in {-1,1,2} for 1 <= i <= n.

Examples

			First 10 columns:
0 .. 1 .. 1 .. 4 .. 5 .. 13 .. 22 .. 45 .. 87 ... 166
0 .. 1 .. 2 .. 3 .. 8 .. 12 .. 28 .. 49 .. 100 .. 191
1 .. 0 .. 2 .. 2 .. 5 .. 10 .. 17 .. 38 .. 66 ... 138
0 .. 1 .. 0 .. 2 .. 2 .. 5 ... 10 .. 17 .. 38 ... 66
T(5,0) counts these 5 paths, given as vector sums applied to (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)
(1,1) + (1,-1) + (1,-1) + (1,1) + (1,-1)
(1,-1) + (1,1) + (1,-1) + (1,1) + (1,-1)
Partial sums of second components in each vector sum give the 3 integer strings described in comments:
(1,2,3,2,1,0),
(1,2,1,2,1,0),
(1,0,1,2,1,0),
(1,2,1,0,1,0),
(1,0,1,0,1,0).

Links

Crossrefs

Cf. A247049, A247321, A247353, A247354, A247355.

Programs

  • Mathematica
    z = 50; t[0, 0] = 0; t[0, 1] = 1; t[0, 2] = 0; t[0, 3] = 0;
t[1, 3] = 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] + t[n - 1, 3]
t[n_, 3] := t[n, 3] = t[n - 1, 1] + t[n - 1, 2]
u = Flatten[Table[t[n, k], {n, 0, z}, {k, 0, 3}]] (* A247352 *)
TableForm[Reverse[Transpose[Table[t[n, k], {n, 0, 12}, {k, 0, 3}]]]] (* array  *)
v = Map[Total, u1]  (* A247353 *)
Table[t[n, 0], {n, 0, z}]   (* row 0, A247354*)
Table[t[n, 1], {n, 0, z}]   (* row 1, cf. row 0 *)
Table[t[n, 2], {n, 0, z}]   (* row 2, A247355 *)
Table[t[n, 3], {n, 0, z}]   (* row 3, A247325 *)

Formula

The four rows and column sums all empirically satisfy the linear recurrence r(n) = 3*r(n-2) + 2*r(n-3) - r(n-4), with g.f. of the form p(x)/q(x), where q(x) = 1 - 3 x^2 - 2 x^3 + x^4. Initial terms and p(x) follow:
(row 0, the bottom row): 0,1,0,2; x - x^3
(row 1): 1,0,0,2; 1 - x^2
(row 2): 0,1,2,3; x +2*x^2
(row 3): 0,1,1,4; x + x^2 + x^3
(n-th column sum) = 1,3,5,11; 1 + 3*x + 2*x^2.

A247354 Number of paths from (0,1) to (n,0), with vertices (i,k) satisfying 0 <= k <= 3, consisting of segments given by the vectors (1,1), (1,2), (1,-1).

Original entry on oeis.org

0, 1, 0, 2, 2, 5, 10, 17, 38, 66, 138, 257, 508, 981, 1900, 3702, 7154, 13925, 26966, 52381, 101594, 197150, 382578, 742257, 1440440, 2794777, 5423256, 10522954, 20418882, 39620597, 76879298, 149176601, 289460206, 561667802, 1089854522, 2114747217
Offset: 0

Views

Author

Clark Kimberling, Sep 15 2014

Keywords

Comments

Also, a(n) = number of strings s(0)..s(n) of integers such that s(0) = 1, s(n) = 0, and for i > 0, s(i) is in {0,1,2,3} and s(i) - s(i-1) is in {-1,1,2} for 1 <= i <= n; also, a(n) = row 0 (the bottom row) of the array at A247352, and a(n+1) = row 1 of the same array.

Examples

			a(5) counts these 5 paths, each represented by a vector sum applied to (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)
(1,1) + (1,-1) + (1,-1) + (1,1) + (1,-1)
(1,-1) + (1,1) + (1,-1) + (1,1) + (1,-1)

Links

Crossrefs

Cf. A247352, A247353, A247355.

Programs

  • Mathematica
    z = 50; t[0, 0] = 0; t[0, 1] = 1; t[0, 2] = 0; t[0, 3] = 0;
t[1, 3] = 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] + t[n - 1, 3]
t[n_, 3] := t[n, 3] = t[n - 1, 1] + t[n - 1, 2]
Table[t[n, 0], {n, 0, z}]   (* A247354*)

Formula

Empirically, a(n) = 3*a(n-2) + 2*a(n-3) - a(n-4) and g.f. = (x - x^3)/(1 - 3 x^2 - 2 x^3 + x^4).

A247355 Number of paths from (0,1) to (n,2), with vertices (i,k) satisfying 0 <= k <= 3, consisting of segments given by the vectors (1,1), (1,2), (1,-1).

Original entry on oeis.org

0, 1, 2, 3, 8, 12, 28, 49, 100, 191, 370, 724, 1392, 2721, 5254, 10223, 19812, 38456, 74628, 144769, 280984, 545107, 1057862, 2052520, 3982816, 7728177, 14995626, 29097643, 56460416, 109556004, 212580908, 412491201, 800394316, 1553079415, 3013584442
Offset: 0

Views

Author

Clark Kimberling, Sep 15 2014

Keywords

Comments

Also, a(n) = number of strings s(0)..s(n) of integers such that s(0) = 1, s(n) = 2, and for i > 0, s(i) is in {0,1,2,3} and s(i) - s(i-1) is in {-1,1,2} for 1 <= i <= n; also, a(n) = row 2 of the array at A247352.

Examples

			a(3) counts these 3 paths, each represented by a vector sum applied to (0,1):
(1,1) + (1,1) + (1,-1);
(1,1) + (1,-1) + (1,1);
(1,-1) + (1,1) + (1,1).

Links

Crossrefs

Cf. A247352, A247353, A247354.

Programs

  • Mathematica
    z = 50; t[0, 0] = 0; t[0, 1] = 1; t[0, 2] = 0; t[0, 3] = 0;
t[1, 3] = 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] + t[n - 1, 3];
t[n_, 3] := t[n, 3] = t[n - 1, 1] + t[n - 1, 2];
Table[t[n, 2], {n, 0, z}]   (* A247355 *)

Formula

Empirically, a(n) = 3*a(n-2) + 2*a(n-3) - a(n-4) and g.f. = (x + 2 x^3)/(1 - 3 x^2 - 2 x^3 + x^4).
Showing 1-3 of 3 results.

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