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A247050 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).

%I A247050 #17 Apr 17 2025 09:52:24
%S A247050 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,
%T A247050 33,56,55,56,88,89,88,145,144,145,232,233,232,378,377,378,609,610,609,
%U A247050 988,987,988,1596,1597,1596,2585,2584,2585,4180,4181,4180
%N A247050 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).
%C A247050 Also, T(n,k) = number of strings s(0)..s(n) of integers such that s(0) = 1, 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).
%C A247050 This is a 3-rowed array read upwards by columns. - _N. J. A. Sloane_, Sep 14 2014
%H A247050 Clark Kimberling, <a href="/A247050/b247050.txt">Table of n, a(n) for n = 0..1000</a>
%F A247050 Let F = A000045, the Fibonacci numbers.  Then (row 0, the bottom row) = F(n-1) - (-1)^n for n >= 0; (row 1, the middle row) = F(n) + (-1)^n for n >=0; (row 2, the top row) = F.
%F A247050 Conjectures from _Chai Wah Wu_, Apr 16 2025: (Start)
%F A247050 a(n) = 2*a(n-6) + a(n-9) for n > 8.
%F A247050 G.f.: (-x^8 - x^5 - x^3 - x)/(x^9 + 2*x^6 - 1). (End)
%e A247050 First 10 columns:
%e A247050 0 .. 1 .. 1 .. 2 .. 3 .. 5 .. 8 .. 13 .. 21 .. 34
%e A247050 1 .. 0 .. 2 .. 1 .. 4 .. 4 .. 9 .. 12 .. 22 .. 33
%e A247050 0 .. 1 .. 0 .. 2 .. 1 .. 4 .. 4 .. 9 ... 12 .. 22
%e A247050 T(4,1) counts these 4 paths, given as vector sums applied to (0,1):
%e A247050 (1,1) + (1,-1) + (1,1) + (1,-1);
%e A247050 (1,-1) + (1,1) + (1,1) + (1,-1);
%e A247050 (1,1) + (1,-1) + (1,-1) + (1,1);
%e A247050 (1,-1) + (1,1) + (1,-1) + (1,1)
%e A247050 Partial sums of second components in each vector sum give the 4 integer strings described in Comments: (1,2,1,2,1), (1,0,1,2,1), (1,2,1,0,1), (1,0,1,0,1).
%t A247050 t[0, 0] = 0; t[0, 1] = 1; 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 *)
%t A247050 u = Flatten[Table[t[n, k], {n, 0, 20}, {k, 0, 2}]] (* sequence *)
%Y A247050 Cf. A000045, A247049, A247050.
%K A247050 nonn,tabf,easy
%O A247050 0,8
%A A247050 _Clark Kimberling_, Sep 11 2014