A027023 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.

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

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• tabf
• nice

Comments

The n-th row has 2n+1 terms.

Examples

			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;
		

Links

• R. J. Mathar, Table of n, a(n) for n = 0..1000, replaces Zumkeller's file for new offset.

Crossrefs

Columns are essentially constant with values from A000213 (tribonacci numbers).

Diagonals T(n, n+c) are A027024 (c=2), A027025 (c=3), A027026 (c=4).

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).

Many other sequences are derived from this one: see A027035 A027036 A027037 A027038 A027039 A027040 A027041 A027042 A027043 A027044 A027045 and A027046 A027047 A027048 A027049.

Other arrays of this type: A027052, A027082, A027113.

Cf. A027907.

Programs

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

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]
-- Reinhard Zumkeller, Jul 06 2014

Maple

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

Mathematica

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]

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 */

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

Extensions

Edited by N. J. A. Sloane and Ralf Stephan, Feb 13 2004

Offset corrected to 0. - R. J. Mathar, Jun 24 2020