A059779 A Lucas triangle: T(m,n), m >= n >= 0.

2, 1, 1, 3, 2, 3, 4, 3, 3, 4, 7, 5, 6, 5, 7, 11, 8, 9, 9, 8, 11, 18, 13, 15, 14, 15, 13, 18, 29, 21, 24, 23, 23, 24, 21, 29, 47, 34, 39, 37, 38, 37, 39, 34, 47, 76, 55, 63, 60, 61, 61, 60, 63, 55, 76, 123, 89, 102, 97, 99, 98, 99, 97, 102, 89, 123, 199, 144, 165, 157, 160, 159

Offset: 0

Author

N. J. A. Sloane, Feb 22 2001

Keywords

• nonn
• easy
• tabl

Comments

From Amiram Eldar, May 15 2023: (Start)

Named "Lucas triangle" by Josef (1983), and "Josef's triangle" by Koshy (2007).

The rows of the triangle are the antidiagonals of the array in which the 0th row is T(0, k) = Lucas(k) = A000032(k), the 1st row is T(1, k) = Fibonacci(k+2) = A000045(k+2), and each subsequent row is the sum of the previous 2 rows.

The central elements in the even rows are in A127546, starting from the 2nd row, i.e., the central element of the k-th row, for even k >= 2, is A127546(k/2-1). (End)

Examples

			Triangle starts:
  2;
  1,1;
  3,2,3;
  4,3,3,4;
  ...
		

References

• Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.

Links

• Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 28.
• Alexander Engstrom, Graph colouring and the total Betti number, arXiv preprint, arXiv:1412.8460 [math.CO], 2014.
• Šána Josef, Lucas Triangle, The Fibonacci Quarterly, Vol. 21, No. 3 (1983), pp. 192-195.
• Thomas Koshy, 91.01 The central elements in Josef's triangle, The Mathematical Gazette, Vol. 91, No. 520 (2007), pp. 63-68.

Crossrefs

Cf. A000045, A000032, A127546.

Programs

Maple

T := proc(m, n) option remember: if m=0 and n=0 then RETURN(2) fi: if m=1 and n=0 then RETURN(1) fi: if m=1 and n=1 then RETURN(1) fi: if m=2 and n=1 then RETURN(2) fi: if m<=n+1 then RETURN(T(m, m-n)) fi: if mJames Sellers, Feb 22 2001

Mathematica

T[0, k_] := T[0, k] = LucasL[k]; T[1, k_] := T[1, k] = Fibonacci[k + 2]; T[n_, k_] := T[n, k] = T[n - 1, k] + T[n - 2, k]; Table[T[k, n - k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, May 15 2023 *)

Formula

T(m, n) = T(m-1, n) + T(m-2, n); T(0, 0)=2, T(1, 0)=1, T(1, 1)=1, T(2, 1)=2.

Extensions

More terms from James Sellers, Feb 22 2001