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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).

%I A247309 #11 Sep 14 2014 14:26:57
%S A247309 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,
%T A247309 987,987,1597,2584,2584,4181,6765,6765,10946,17711,17711,28657,46368,
%U A247309 46368,75025,121393,121393,196418,317811,317811,514229,832040,832040
%N 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).
%C A247309 Every member of T is a Fibonacci number, and the sum of the numbers in column n is A000045(2n+2).
%H A247309 Clark Kimberling, <a href="/A247309/b247309.txt">Table of n, a(n) for n = 0..1000</a>
%F A247309 Let F = A000045 (Fibonacci numbers); then
%F A247309 (row 0, the bottom row) = (F(2n)), n >= 0;
%F A247309 (row 1, the middle row) = (F(2n)), n >= 0;
%F A247309 (row 2, the top row) = (F(2n-1)), n >= 0.
%F A247309 (n-th column sum) = (F(2n+2)), n >= 0.
%e A247309 First 10 columns:
%e A247309 0 .. 1 .. 3 .. 8 .. 21 .. 55 .. 144 .. 377 .. 987 ... 2584
%e A247309 0 .. 1 .. 3 .. 8 .. 21 .. 55 .. 144 .. 377 .. 987 ... 2584
%e A247309 1 .. 1 .. 2 .. 5 .. 13 .. 34 .. 89 ... 233 .. 610 ... 1597
%e A247309 T(2,2) counts these 3 paths, given as vector sums applied to (0,0):
%e A247309 (1,2) + (1,0); (1,1) + (1,1); (1,0) + (1,2).
%t A247309 t[0, 0] = 1; t[0, 1] = 0; t[0, 2] = 0; t[1, 2] = 1;
%t A247309 t[n_, 0] := t[n, 0] = t[n - 1, 0] + t[n - 1, 1];
%t A247309 t[n_, 1] := t[n, 1] = t[n - 1, 0] + t[n - 1, 1] + t[n - 1, 2];
%t A247309 t[n_, 2] := t[n, 2] = t[n - 1, 0] + t[n - 1, 1] + t[n - 1, 2]
%t A247309 TableForm[Reverse[Transpose[Table[t[n, k], {n, 0, 12}, {k, 0, 2}]]]] (*  array *)
%t A247309 Flatten[Table[t[n, k], {n, 0, 20}, {k, 0, 2}]] (*  A247309 *)
%Y A247309 Cf.  A247049, A247310, A000045.
%K A247309 nonn,tabf,easy
%O A247309 0,7
%A A247309 _Clark Kimberling_, Sep 12 2014