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A055215 A path-counting array, read by rows: T(i,j)=number of paths from (0,0) to (i-j,j) using steps (1 unit right and 1 unit up) or (1 unit right and 2 units up). 1

%I

%S 1,1,1,1,1,1,1,1,2,1,1,1,2,2,1,1,1,2,3,2,1,1,1,2,3,4,2,1,1,1,2,3,5,4,

%T 2,1,1,1,2,3,5,7,4,2,1,1,1,2,3,5,8,8,4,2,1,1,1,2,3,5,8,12,8,4,2,1,1,1,

%U 2,3,5,8,13,15,8,4,2,1,1,1,2,3,5,8,13,20,16

%N A path-counting array, read by rows: T(i,j)=number of paths from (0,0) to (i-j,j) using steps (1 unit right and 1 unit up) or (1 unit right and 2 units up).

%C If m >= 1 and n >= 2, then T(m+n-1,m) is the number of strings (s(1),s(2),...,s(n)) of nonnegative integers satisfying s(n)=m and 1<=s(k)-s(k-1)<=2 for k=2,3,...,n.

%H C. Kimberling, <a href="https://www.fq.math.ca/Scanned/40-4/kimberling.pdf">Path-counting and Fibonacci numbers</a>, Fib. Quart. 40 (4) (2002) 328-338, Example 1D.

%F T(i, 0)=T(i, i)=1 for i >= 0; T(i, 1)=1 for i >= 1; T(i, j)=T(i-2, j-1)+T(i-3, j-2) for 2<=j<=i-1, i >= 3.

%e 7=T(8,5) counts these strings: 0135, 0235, 0245, 1235, 1245, 1345, 2345.

%e Rows: {1}; {1,1}; {1,1,1}; {1,1,2,1}; {1,1,2,2,1}; ...

%Y T(2n, n)=A000045(n+1), the Fibonacci numbers.

%K nonn,tabl,walk

%O 1,9

%A _Clark Kimberling_, May 07 2000

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Last modified August 4 15:50 EDT 2020. Contains 336202 sequences. (Running on oeis4.)