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A154957
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A symmetric (0,1)-triangle.
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5
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1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,1
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COMMENTS
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LINKS
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FORMULA
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T(n,k) = Sum_{j=0..n} [j<=k]*[j<=n-k]*(mod(j+1,2) - mod(j,2)).
T(2*n, n) - T(2*n, n+1) = (-1)^n.
T(2*n, n) = (n+1) mod 2.
Sum_{k=0..n} T(n, k) = A004524(n+3).
Sum_{k=0..floor(n/2)} T(n-k, k) = A154958(n) (diagonal sums).
T(n, n-k) = T(n, k).
Sum_{k=0..floor(n/2)} T(n, k) = floor((n+4)/4).
T(2*n+1, n) = (1+(-1)^n)/2. (End)
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EXAMPLE
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Triangle begins
1;
1, 1;
1, 0, 1;
1, 0, 0, 1;
1, 0, 1, 0, 1;
1, 0, 1, 1, 0, 1;
1, 0, 1, 0, 1, 0, 1;
1, 0, 1, 0, 0, 1, 0, 1;
1, 0, 1, 0, 1, 0, 1, 0, 1;
1, 0, 1, 0, 1, 1, 0, 1, 0, 1;
1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1;
1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1;
1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1;
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MATHEMATICA
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T[n_, k_]:= Sum[(Mod[j+1, 2] - Mod[j, 2]), {j, 0, Min[k, n-k]}];
Table[T[n, k], {n, 0, 20}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 07 2022 *)
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PROG
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(Sage)
def A154957(n, k): return sum( (j+1)%2 - j%2 for j in (0..min(k, n-k)) )
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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