A247310 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), (1,2), (1,-1), (1,-2), where no segment is followed by a segment in the same direction.

1, 0, 0, 0, 1, 1, 2, 1, 0, 0, 2, 3, 5, 3, 0, 0, 5, 8, 13, 8, 0, 0, 13, 21, 34, 21, 0, 0, 34, 55, 89, 55, 0, 0, 89, 144, 233, 144, 0, 0, 233, 377, 610, 377, 0, 0, 610, 987, 1597, 987, 0, 0, 1597, 2584, 4181, 2584, 0, 0, 4181, 6765, 10946, 6765, 0, 0, 10946

Offset: 0

Author

Clark Kimberling, Sep 12 2014

Keywords

• nonn
• tabf
• easy

Comments

Also, T(n,k) = number of strings s(0)..s(n) of integers such that s(0) = 0, s(n) = k, s(i) is in {0,1,2} for i = 0..n, and for i > 0, s(i) - s(i-1) is in {1,2} if i is odd, and s(i) - s(i-1) is in {-1,-2} if i is even. Every row of T consists of Fibonacci numbers, and (sum of numbers in column n) = A000045(n+1).

Examples

			First 10 columns:
0 .. 1 .. 0 .. 3 .. 0 .. 8 .. 0 ... 21 .. 0 ... 55
0 .. 1 .. 1 .. 2 .. 3 .. 5 .. 8 ... 13 .. 21 .. 34
1 .. 0 .. 2 .. 0 .. 5 .. 0 .. 13 .. 0 ... 34 .. 0
T(3,2) counts these 3 paths, given as vector sums applied to (0,0):
(1,2) + (1,-1) + (1,1);
(1,1) + (1,-1) + (1,2);
(1,2) + (1,-2) + (1,2).
		

Links

• Clark Kimberling, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A247049, A247309, A000045.

Programs

Mathematica

t[0, 0] = 1; t[0, 1] = 0; t[0, 2] = 0;
t[1, 0] = 0; t[1, 1] = 1; t[1, 2] = 1;
t[2, 0] = 2; t[2, 1] = 1; t[2, 2] = 0;
t[n_, 0] := If[OddQ[n], 0 , t[n - 1, 1] + t[n - 1, 2]]
t[n_, 1] := If[OddQ[n], t[n - 1, 0] , t[n - 1, 2]]
t[n_, 2] := If[OddQ[n], t[n - 1, 0] + t[n - 1, 1], 0]
TableForm[Reverse[Transpose[Table[t[n, k], {n, 0, 12}, {k, 0, 2}]]]] (*  array *)
u = Flatten[Table[t[n, k], {n, 0, 25}, {k, 0, 2}]] (*  A247310 *)

Formula

Let F = A000045 (Fibonacci numbers); then

(row 0, the bottom row): F(1), 0 , F(3), 0 , F(5), 0, ...

(row 1, the middle row): F(0), F(1), F(2), F(3), F(4), F(5), ...

(row 2, the top row): 0, F(2), 0, F(4), 0, F(6) , 0, ...