OFFSET
1,4
LINKS
EXAMPLE
The first and second Fibonacci numbers are 1, 1, so the first and second rows of the triangle are 1; 1 1; respectively. The third row of the triangle begins and ends with the third Fibonacci number, 2 and the middle term is the sum of the contiguous two terms in the second row, i.e., 1 + 1 = 2, so the third row is 2 2 2.
Triangle begins:
1;
1, 1;
2, 2, 2;
3, 4, 4, 3;
5, 7, 8, 7, 5;
8, 12, 15, 15, 12, 8;
13, 20, 27, 30, 27, 20, 13;
21, 33, 47, 57, 57, 47, 33, 21;
34, 54, 80, 104, 114, 104, 80, 54, 34;
...
Formatted as a symmetric triangle:
1;
1, 1;
2, 2, 2;
3, 4, 4, 3;
5, 7, 8, 7, 5;
8, 12, 15, 15, 12, 8;
13, 20, 27, 30, 27, 20, 13;
21, 33, 47, 57, 57, 47, 33, 21;
34, 54, 80, 104, 114, 104, 80, 54, 34;
MATHEMATICA
T[n_, 1]:= Fibonacci[n]; T[n_, n_]:= Fibonacci[n]; T[n_, k_]:= T[n-1, k-1] + T[n-1, k]; Table[T[n, k], {n, 1, 12}, {k, 1, n}]//Flatten (* G. C. Greubel, Jul 12 2019 *)
PROG
(Haskell)
a074829 n k = a074829_tabl !! (n-1) !! (k-1)
a074829_row n = a074829_tabl !! (n-1)
a074829_tabl = map fst $ iterate
(\(u:_, vs) -> (vs, zipWith (+) ([u] ++ vs) (vs ++ [u]))) ([1], [1, 1])
-- Reinhard Zumkeller, Aug 15 2013
(PARI) T(n, k) = if(k==1 || k==n, fibonacci(n), T(n-1, k-1) + T(n-1, k));
for(n=1, 12, for(k=1, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Jul 12 2019
(Sage)
def T(n, k):
if (k==1 or k==n): return fibonacci(n)
else: return T(n-1, k-1) + T(n-1, k)
[[T(n, k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 12 2019
(GAP)
T:= function(n, k)
if k=1 then return Fibonacci(n);
elif k=n then return Fibonacci(n);
else return T(n-1, k-1) + T(n-1, k);
fi;
end;
Flat(List([1..15], n-> List([1..n], k-> T(n, k) ))); # G. C. Greubel, Jul 12 2019
CROSSREFS
KEYWORD
AUTHOR
Joseph L. Pe, Sep 30 2002
EXTENSIONS
More terms from Philippe Deléham, Sep 20 2006
Data error in 7th row fixed by Reinhard Zumkeller, Aug 15 2013
STATUS
approved