A082793 A tribonacci triangle in which the top two northeast and southeast diagonals consist of tribonacci numbers.

1, 1, 1, 2, 1, 2, 4, 2, 2, 4, 7, 4, 4, 4, 7, 13, 7, 8, 8, 7, 13, 24, 13, 14, 16, 14, 13, 24, 44, 24, 26, 28, 28, 26, 24, 44, 81, 44, 48, 52, 49, 52, 48, 44, 81, 149, 81, 88, 96, 91, 91, 96, 88, 81, 149, 274, 149, 162, 176, 168, 169, 168, 176, 162, 149, 274

Offset

1,4

Comments

Uses a Hosoya-like format except that the latter has the Fibonacci recursion. This triangle uses the tribonacci recursion such that every interior number can be obtained by adding the 3 previous numbers, on its diagonal.

References

Thomas Koshy, <"Fibonacci and Lucas Numbers with Applications">John Wiley and Sons, 2001, Chapter 15, pages 187-195, "Hosoya's Triangle".

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Formula

T(n, j) = T(n-1, j) + T(n-2, j) + T(n-3, j) for j < n.

T(n, n) = T(n-1, n-1) + T(n-2, n-2) + T(n-3, n-3).

G.f.: x*y/((1 - x - x^2 - x^3)*(1 - y*x - y^2*x^2 - y^3*x^3)). - Andrew Howroyd, Sep 24 2025

Example

Triangle begins:
   1;
   1,  1;
   2,  1,  2;
   4,  2,  2,  4;
   7,  4,  4,  4,  7;
  13,  7,  8,  8,  7, 13;
  24, 13, 14, 16, 14, 13, 24;
  44, 24, 26, 28, 28, 26, 24, 44;
  ...
T(7,3) = 14 = (8 + 4 + 2) = T(6,3) + T(5,3) + T(4,3).

Prog

(PARI) T(n)={my(v=vector(n)); for(n=1, n, v[n]=vector(n, k, if(k!=n, v[n-1][k] + if(k<n-1, v[n-2][k]) + if(k<n-2, v[n-3][k]), if(n<3, 1, v[n-1][1] + v[n-2][1] + if(n>3, v[n-3][1]))))); v} \\ Andrew Howroyd, Sep 24 2025
{ my(A=T(10)); for(i=1, #A, print(A[i])) }
(PARI) T(n)={[Vecrev(p) | p<-Vec(1/((1 - x - x^2 - x^3)*(1 - y*x - y^2*x^2 - y^3*x^3)) + O(x^n))]} \\ Andrew Howroyd, Sep 24 2025

Crossrefs

Cf. A000073, tribonacci numbers, A058071, Hosoya's triangle.

Sequence in context: A361656 A270594 A270706 * A114929 A247321 A152251

Adjacent sequences: A082790 A082791 A082792 * A082794 A082795 A082796

Keyword

nonn,tabl

Author

Gary W. Adamson, May 24 2003

Extensions

a(37) onwards from Andrew Howroyd, Sep 24 2025

Status

approved