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

0,7

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).

Links

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

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, ...

Example

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).

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 *)

Crossrefs

Cf. A247049, A247309, A000045.

Sequence in context: A259668 A261118 A063181 * A059220 A261630 A301503

Adjacent sequences: A247307 A247308 A247309 * A247311 A247312 A247313

Keyword

nonn,tabf,easy

Author

Clark Kimberling, Sep 12 2014

Status

approved