A106522 A Pascal type matrix based on the tribonacci numbers.

1, 1, 1, 2, 2, 1, 4, 4, 3, 1, 7, 8, 7, 4, 1, 13, 15, 15, 11, 5, 1, 24, 28, 30, 26, 16, 6, 1, 44, 52, 58, 56, 42, 22, 7, 1, 81, 96, 110, 114, 98, 64, 29, 8, 1, 149, 177, 206, 224, 212, 162, 93, 37, 9, 1, 274, 326, 383, 430, 436, 374, 255, 130, 46, 10, 1, 504, 600, 709, 813, 866, 810, 629, 385, 176, 56, 11, 1

Offset: 0

Author

Paul Barry, May 06 2005

Keywords

• easy
• nonn
• tabl

Comments

Row sums of A106522 mod 2 are A106524.

Examples

			Triangle begins:
   1;
   1,  1;
   2,  2,  1;
   4,  4,  3,  1;
   7,  8,  7,  4, 1;
  13, 13, 15, 11, 5, 1;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A000073, A001590 (row sums), A106523 (diagonal sums).

Programs

Mathematica

b[n_]:= b[n]= If[n<2, 0, If[n==2, 1, b[n-1] +b[n-2] +b[n-3]]]; (* A000073 *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0, b[n+2], T[n-1, k-1] +T[n-1, k]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 06 2021 *)

Sage

@CachedFunction
def b(n): return 0 if (n<2) else 1 if (n==2) else b(n-1) +b(n-2) +b(n-3)
def T(n,k):
    if (k<0 or k>n): return 0
    elif (k==0): return b(n+2)
    else: return T(n-1, k) + T(n-1, k-1)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Aug 06 2021

Formula

Riordan array (1/(1-x-x^2-x^3), x/(1-x)).

Number triangle T(n, 0) = A000073(n+2), T(n, k) = T(n-1, k-1) + T(n-1, k).

Sum_{k=0..n} T(n,k) = A001590(n+3).

Sum_{k=0..floor(n/2)} T(n-k, k) = A106523(n).