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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).

%I A055215 #13 Jan 05 2025 19:51:36
%S A055215 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 A055215 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 A055215 2,3,5,8,13,15,8,4,2,1,1,1,2,3,5,8,13,20,16
%N 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).
%C A055215 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 A055215 C. Kimberling, <a href="https://web.archive.org/web/2024*/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 A055215 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 A055215 7=T(8,5) counts these strings: 0135, 0235, 0245, 1235, 1245, 1345, 2345.
%e A055215 Rows: {1}; {1,1}; {1,1,1}; {1,1,2,1}; {1,1,2,2,1}; ...
%Y A055215 T(2n, n)=A000045(n+1), the Fibonacci numbers.
%K A055215 nonn,tabl,walk
%O A055215 1,9
%A A055215 _Clark Kimberling_, May 07 2000