A108617 Triangle read by rows: T(n,k) = T(n-1,k-1) + T(n-1,k) for 0 < k < n, T(n,0) = T(n,n) = n-th Fibonacci number.

0, 1, 1, 1, 2, 1, 2, 3, 3, 2, 3, 5, 6, 5, 3, 5, 8, 11, 11, 8, 5, 8, 13, 19, 22, 19, 13, 8, 13, 21, 32, 41, 41, 32, 21, 13, 21, 34, 53, 73, 82, 73, 53, 34, 21, 34, 55, 87, 126, 155, 155, 126, 87, 55, 34, 55, 89, 142, 213, 281, 310, 281, 213, 142, 89, 55, 89, 144, 231, 355, 494, 591, 591, 494, 355, 231, 144, 89

Offset

0,5

Comments

Sum of n-th row = 2*A027934(n). - Reinhard Zumkeller, Oct 07 2012

Links

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

Hacéne Belbachir and László Szalay, On the Arithmetic Triangles, Šiauliai Mathematical Seminar, Vol. 9 (17), 2014. See Fig. 1 p. 18.

Eric Weisstein's World of Mathematics, Fibonacci Number.

Eric Weisstein's World of Mathematics, Pascal's Triangle.

Wikipedia, Fibonacci number.

Wikipedia, Pascal's triangle.

Index entries for triangles and arrays related to Pascal's triangle

Formula

T(n,0) = T(n,n) = A000045(n);

T(n,1) = T(n,n-1) = A000045(n+1) for n>0;

T(n,2) = T(n,n-2) = A000045(n+2) - 2 = A001911(n-1) for n>1;

Sum_{k=0..n} T(n,k) = 2*A027934(n-1) for n>0.

Sum_{k=0..n} (-1)^k*T(n, k) = 2*((n+1 mod 2)*Fibonacci(n-2) + [n=0]). - G. C. Greubel, Oct 20 2023

Example

Triangle begins:
   0;
   1,   1;
   1,   2,   1;
   2,   3,   3,   2;
   3,   5,   6,   5,   3;
   5,   8,  11,  11,   8,   5;
   8,  13,  19,  22,  19,  13,   8;
  13,  21,  32,  41,  41,  32,  21,  13;
  21,  34,  53,  73,  82,  73,  53,  34,  21;
  34,  55,  87, 126, 155, 155, 126,  87,  55,  34;
  55,  89, 142, 213, 281, 310, 281, 213, 142,  89,  55;

Maple

A108617 := proc(n, k)
    if k = 0 or k=n then
        combinat[fibonacci](n) ;
    elif k <0 or k > n then
        0 ;
    else
        procname(n-1, k-1)+procname(n-1, k) ;
    end if;
end proc: # R. J. Mathar, Oct 05 2012

Mathematica

a[1]:={0}; a[n_]:= a[n]= Join[{Fibonacci[#]}, Map[Total, Partition[a[#], 2, 1]], {Fibonacci[#]}]&[n-1]; Flatten[Map[a, Range[15]]] (* Peter J. C. Moses, Apr 11 2013 *)

Prog

(Haskell)
a108617 n k = a108617_tabl !! n !! k
a108617_row n = a108617_tabl !! n
a108617_tabl = [0] : iterate f [1, 1] where
   f row@(u:v:_) = zipWith (+) ([v - u] ++ row) (row ++ [v - u])
-- Reinhard Zumkeller, Oct 07 2012
(Magma)
function T(n, k) // T = A108617
  if k eq 0 or k eq n then return Fibonacci(n);
  else return T(n-1, k-1) + T(n-1, k);
  end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 20 2023
(SageMath)
def T(n, k): # T = A108617
    if (k==0 or k==n): return fibonacci(n)
    else: return T(n-1, k-1) + T(n-1, k)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 20 2023

Crossrefs

Cf. A007318, A074829, A108037.

Sequence in context: A033775 A033791 A039913 * A092683 A172089 A057475

Adjacent sequences: A108614 A108615 A108616 * A108618 A108619 A108620

Keyword

nonn,easy,tabl

Author

Reinhard Zumkeller, Jun 12 2005

Status

approved