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A Lucas triangle: T(m,n), m >= n >= 0.
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%I #21 Oct 07 2023 13:13:52

%S 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,

%T 21,24,23,23,24,21,29,47,34,39,37,38,37,39,34,47,76,55,63,60,61,61,60,

%U 63,55,76,123,89,102,97,99,98,99,97,102,89,123,199,144,165,157,160,159

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

%C From _Amiram Eldar_, May 15 2023: (Start)

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

%C 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.

%C 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)

%H Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8. <a href="https://www.fq.math.ca/pascal.html">English translation</a>, by Richard C. Bollinger, The Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 28.

%H Alexander Engstrom, <a href="https://arxiv.org/abs/1412.8460">Graph colouring and the total Betti number</a>, arXiv preprint, arXiv:1412.8460 [math.CO], 2014.

%H Šána Josef, <a href="https://www.fq.math.ca/Scanned/21-3/josef.pdf">Lucas Triangle</a>, The Fibonacci Quarterly, Vol. 21, No. 3 (1983), pp. 192-195.

%H Thomas Koshy, <a href="https://www.jstor.org/stable/40378286">91.01 The central elements in Josef's triangle</a>, The Mathematical Gazette, Vol. 91, No. 520 (2007), pp. 63-68.

%F 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.

%e Triangle starts:

%e 2;

%e 1,1;

%e 3,2,3;

%e 4,3,3,4;

%e ...

%p 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

%t 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 *)

%Y Cf. A000045, A000032, A127546.

%K nonn,easy,tabl

%O 0,1

%A _N. J. A. Sloane_, Feb 22 2001

%E More terms from _James A. Sellers_, Feb 22 2001