cp's OEIS Frontend

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.

%I A108617 #41 Feb 16 2025 08:32:58
%S A108617 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,
%T A108617 32,41,41,32,21,13,21,34,53,73,82,73,53,34,21,34,55,87,126,155,155,
%U A108617 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
%N 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.
%C A108617 Sum of n-th row = 2*A027934(n). - _Reinhard Zumkeller_, Oct 07 2012
%H A108617 Reinhard Zumkeller, <a href="/A108617/b108617.txt">Rows n = 0..120 of triangle, flattened</a>
%H A108617 Hacéne Belbachir and László Szalay, <a href="http://siauliaims.su.lt/pdfai/2014/Belb_Szal_2014.pdf">On the Arithmetic Triangles</a>, Šiauliai Mathematical Seminar, Vol. 9 (17), 2014. See Fig. 1 p. 18.
%H A108617 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>.
%H A108617 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PascalsTriangle.html">Pascal's Triangle</a>.
%H A108617 Wikipedia, <a href="http://www.wikipedia.org/wiki/Fibonacci_number">Fibonacci number</a>.
%H A108617 Wikipedia, <a href="http://en.wikipedia.org/wiki/Pascal&#39;s_triangle">Pascal's triangle</a>.
%H A108617 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A108617 T(n,0) = T(n,n) = A000045(n);
%F A108617 T(n,1) = T(n,n-1) = A000045(n+1) for n>0;
%F A108617 T(n,2) = T(n,n-2) = A000045(n+2) - 2 = A001911(n-1) for n>1;
%F A108617 Sum_{k=0..n} T(n,k) = 2*A027934(n-1) for n>0.
%F A108617 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
%e A108617 Triangle begins:
%e A108617    0;
%e A108617    1,   1;
%e A108617    1,   2,   1;
%e A108617    2,   3,   3,   2;
%e A108617    3,   5,   6,   5,   3;
%e A108617    5,   8,  11,  11,   8,   5;
%e A108617    8,  13,  19,  22,  19,  13,   8;
%e A108617   13,  21,  32,  41,  41,  32,  21,  13;
%e A108617   21,  34,  53,  73,  82,  73,  53,  34,  21;
%e A108617   34,  55,  87, 126, 155, 155, 126,  87,  55,  34;
%e A108617   55,  89, 142, 213, 281, 310, 281, 213, 142,  89,  55;
%p A108617 A108617 := proc(n,k) option remember;
%p A108617     if k = 0 or k=n then
%p A108617         combinat[fibonacci](n) ;
%p A108617     elif k <0 or k > n then
%p A108617         0 ;
%p A108617     else
%p A108617         procname(n-1,k-1)+procname(n-1,k) ;
%p A108617     end if;
%p A108617 end proc: # _R. J. Mathar_, Oct 05 2012
%t A108617 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 *)
%o A108617 (Haskell)
%o A108617 a108617 n k = a108617_tabl !! n !! k
%o A108617 a108617_row n = a108617_tabl !! n
%o A108617 a108617_tabl = [0] : iterate f [1,1] where
%o A108617    f row@(u:v:_) = zipWith (+) ([v - u] ++ row) (row ++ [v - u])
%o A108617 -- _Reinhard Zumkeller_, Oct 07 2012
%o A108617 (Magma)
%o A108617 function T(n,k) // T = A108617
%o A108617   if k eq 0 or k eq n then return Fibonacci(n);
%o A108617   else return T(n-1,k-1) + T(n-1,k);
%o A108617   end if;
%o A108617 end function;
%o A108617 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Oct 20 2023
%o A108617 (SageMath)
%o A108617 def T(n,k): # T = A108617
%o A108617     if (k==0 or k==n): return fibonacci(n)
%o A108617     else: return T(n-1,k-1) + T(n-1,k)
%o A108617 flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Oct 20 2023
%Y A108617 Cf. A007318, A074829, A108037.
%Y A108617 T(2n,n) gives 2*A176085(n).
%K A108617 nonn,easy,tabl
%O A108617 0,5
%A A108617 _Reinhard Zumkeller_, Jun 12 2005