A082870 Tribonacci array.

1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 4, 6, 2, 1, 5, 10, 7, 1, 1, 6, 15, 16, 6, 1, 7, 21, 30, 19, 3, 1, 8, 28, 50, 45, 16, 1, 1, 9, 36, 77, 90, 51, 10, 1, 10, 45, 112, 161, 126, 45, 4, 1, 11, 55, 156, 266, 266, 141, 30, 1, 1, 12, 66, 210, 414, 504, 357, 126, 15, 1, 13, 78, 275, 615, 882

Offset: 0

Author

Gary W. Adamson, May 24 2003

Keywords

• nonn
• tabf

Comments

Row sums are tribonacci numbers.

From Gary W. Adamson, Nov 15 2016: (Start)

With an alternative format:

1, 0, 0, 0, 0, 0, 0, ...

1, 1, 1, 0, 0, 0, 0, ...

1, 2, 3, 2, 1, 0, 0, ...

1, 3, 6, 7, 6, 3, 1, ...

... (where the k-th row is (1 + x + x^2)^k), let q(x) = (r(x) * r(x^3) * r(x^9) * r(x^27) * ...). Then q(x) is the binomial sequence beginning (1, k, ...). Example: (1, 3, 6, 10, ...) = q(x) with r(x) = (1, 3, 6, 7, 3, 1, 0, 0, 0). (End)

Examples

			Triangle begins:
  1,
  1,
  1,  1,
  1,  2,  1,
  1,  3,  3,
  1,  4,  6,  2,
  1,  5, 10,  7,  1,
  1,  6, 15, 16,  6,
		

References

• Thomas Koshy, <"Fibonacci and Lucas Numbers with Applications">, Wiley, 2001; Chapter 47: Tribonacci Polynomials: ("In 1973, V.E. Hoggat, Jr. and M. Bicknell generalized Fibonacci polynomials to Tribonacci polynomials tx(x)"); Table 47.1, page 534: "Tribonacci Array".

Links

• Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened

Crossrefs

A082601 is a better version. Cf. A000073, A078802.

Cf. A004396 (row lengths).

Programs

Haskell

a082870 n k = a082870_tabf !! n !! k
a082870_row n = a082870_tabf !! n
a082870_tabf = map (takeWhile (> 0)) a082601_tabl
-- Reinhard Zumkeller, Apr 13 2014

Formula

G.f.: x/(1 - x - x^2*y - x^3*y^2). - Vladeta Jovovic, May 30 2003

Extensions

More terms from Vladeta Jovovic, May 30 2003