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A059779
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A Lucas triangle: T(m,n), m >= n >= 0.
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0
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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
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OFFSET
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0,1
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COMMENTS
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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)
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LINKS
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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.
Šána Josef, Lucas Triangle, The Fibonacci Quarterly, Vol. 21, No. 3 (1983), pp. 192-195.
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FORMULA
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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.
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EXAMPLE
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Triangle starts:
2;
1,1;
3,2,3;
4,3,3,4;
...
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MAPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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