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 A055794 Array T read by rows: T(i,0)=1 for i >= 0; T(i,i)=0 for i=0,1,2,3; T(i,i)=0 for i >= 4; T(i,j)=T(i-1,j)+T(i-2,j-1) for 1<=j<=i-1. 5

%I

%S 1,1,1,1,2,1,1,3,2,1,1,4,4,2,0,1,5,7,4,1,0,1,6,11,8,3,0,0,1,7,16,15,7,

%T 1,0,0,1,8,22,26,15,4,0,0,0,1,9,29,42,30,11,1,0,0,0,1,10,37,64,56,26,

%U 5,0,0,0,0,1,11,46,93,98,56,16,1,0,0,0,0,1,12

%N Array T read by rows: T(i,0)=1 for i >= 0; T(i,i)=0 for i=0,1,2,3; T(i,i)=0 for i >= 4; T(i,j)=T(i-1,j)+T(i-2,j-1) for 1<=j<=i-1.

%C T(i+j,j)=number of strings (s(1),...,s(i+1)) of nonnegative integers s(k) such that 0<=s(k)-s(k-1)<=1 for k=2,3,...,i+1 and s(i+1)=j.

%C T(i+j,j)=number of compositions of j consisting of i parts, all of in {0,1}.

%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 1B.

%e Triangle begins:

%e 1;

%e 1,1;

%e 1,2,1;

%e 1,3,2,1;

%e 1,4,4,2,0;

%e ...

%e T(7,4) counts the strings 3334, 3344, 3444, 2234, 2334, 2344, 1234.

%e T(7,4) counts the compositions 001, 010, 100, 011, 101, 110, 111.

%Y Row sums: A000032 (Lucas numbers, 1, 2, 4, 7, 11, 18, ...).

%Y T(2n, n)=A000125(n) (Cake numbers, 1, 2, 4, 8, 15, 26, ...).

%Y T(2n+2, n)=A027660(n).

%K nonn,tabl

%O 0,5

%A _Clark Kimberling_, May 28 2000

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Last modified March 18 19:58 EDT 2019. Contains 321293 sequences. (Running on oeis4.)