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A027023
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Tribonacci array: triangular array T read by rows: T(n,0)=1 for n >= 0, T(n,1) = T(n,2n) = 1 for n >= 1, T(n,2)=1 for n >= 2 and for n >= 3, T(n,k) = T(n-1,k-3) + T(n-1,k-2) + T(n-1,k-1) for 3 <= k <= 2n-1.
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30
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1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 5, 5, 1, 1, 1, 1, 3, 5, 9, 13, 11, 1, 1, 1, 1, 3, 5, 9, 17, 27, 33, 25, 1, 1, 1, 1, 3, 5, 9, 17, 31, 53, 77, 85, 59, 1, 1, 1, 1, 3, 5, 9, 17, 31, 57, 101, 161, 215, 221, 145, 1, 1, 1, 1, 3, 5, 9, 17, 31, 57, 105, 189, 319, 477, 597, 581, 367, 1
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OFFSET
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0,8
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COMMENTS
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The n-th row has 2n+1 terms.
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LINKS
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EXAMPLE
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The array begins:
1;
1, 1, 1;
1, 1, 1, 3, 1;
1, 1, 1, 3, 5, 5, 1;
1, 1, 1, 3, 5, 9, 13, 11, 1;
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MAPLE
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T:= proc(n, k) option remember;
if k<3 or k=2*n then 1
else T(n-1, k-3) + T(n-1, k-2) + T(n-1, k-1)
fi
end proc:
seq(seq(T(n, k), k=0..2*n), n=0..10); # G. C. Greubel, Nov 04 2019
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MATHEMATICA
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T[n_, 0] := 1; T[n_, 1] := 1; T[n_, k_]/; (k==2n) := 1 /; n >=1; T[n_, 2] := 1; T[n_, k_]/; (k <= 2n-1) := T[n, k]=T[n-1, k-3]+T[n-1, k-2]+T[n-1, k-1]
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PROG
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(PARI) {T(n, k) = if( k<0 || k>2*n, 0, if( k<3 || k==2*n, 1, T(n-1, k-3) + T(n-1, k-2) + T(n-1, k-1)))}; /* Michael Somos, Feb 14 2004 */
(Haskell)
a027023 n k = a027023_tabf !! (n-1) !! (k-1)
a027023_row n = a027023_tabf !! (n-1)
a027023_tabf = [1] : iterate f [1, 1, 1] where
f row = 1 : 1 : 1 :
zipWith3 (((+) .) . (+)) (drop 2 row) (tail row) row ++ [1]
(Sage)
def T(n, k):
if (k<3 or k==2*n): return 1
else: return T(n-1, k-3) + T(n-1, k-2) + T(n-1, k-1)
[[T(n, k) for k in (0..2*n)] for n in (0..10)] # G. C. Greubel, Nov 04 2019
(GAP)
T:= function(n, k)
if k<3 or k=2*n then return 1;
else return T(n-1, k-3) + T(n-1, k-2) + T(n-1, k-1);
fi;
end;
Flat(List([0..10], n-> List([0..2*n], k-> T(n, k) ))); # G. C. Greubel, Nov 04 2019
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CROSSREFS
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Columns are essentially constant with values from A000213 (tribonacci numbers).
Diagonals T(n, 2n-c) are A027050 (c=1), A027051 (c=2), A027027 (c=3), A027028 (c=4), A027029 (c=5), A027030 (c=6), A027031 (c=7), A027032 (c=8), A027033 (c=9), A027034 (c=10).
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KEYWORD
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nonn,tabf,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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