A074829 Triangle formed by Pascal's rule, except that the n-th row begins and ends with the n-th Fibonacci number.

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, 55, 88, 134, 184, 218, 218, 184, 134, 88, 55, 89, 143, 222, 318, 402, 436, 402, 318, 222, 143, 89

Offset: 1

Author

Joseph L. Pe, Sep 30 2002

Keywords

• easy
• nonn
• tabl

Examples

			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;
		

Links

• Reinhard Zumkeller, Rows n = 1..120 of table, flattened
• Hebert Pérez-Rosés, Asymptotic Analysis of Central Binomiacci Numbers, arXiv:2503.17462 [math.CO], 2025.
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

Some other Fibonacci-Pascal triangles: A027926, A036355, A037027, A105809, A108617, A109906, A111006, A114197, A162741, A228074.

Cf. A074878 (row sums).

Programs

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

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

Maple

A074829 := proc(n,k)
    option remember ;
    if k=1 or k=n then
        combinat[fibonacci](n) ;
    else
        procname(n-1,k-1)+procname(n-1,k) ;
    end if;
end proc:
seq(seq(A074829(n,k),k=1..n),n=1..12) ; # R. J. Mathar, Mar 31 2025

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 *)

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

Extensions

More terms from Philippe Deléham, Sep 20 2006

Data error in 7th row fixed by Reinhard Zumkeller, Aug 15 2013