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

0,1

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)

Links

Table of n, a(n) for n=0..71.

Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8. English translation, by Richard C. Bollinger, The 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.

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.

Example

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

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 m<n then RETURN(0) fi: T(m-1, n) + T(m-2, n): end:for m from 0 to 20 do for n from 0 to m do printf(`%d, `, T(m, n)) od: od: # James A. 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 *)

Crossrefs

Cf. A000045, A000032, A127546.

Sequence in context: A370666 A351466 A070036 * A291874 A049346 A227310

Adjacent sequences: A059776 A059777 A059778 * A059780 A059781 A059782

Keyword

nonn,easy,tabl,changed

Author

N. J. A. Sloane, Feb 22 2001

Extensions

More terms from James A. Sellers, Feb 22 2001

Status

approved