login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 5

%I

%S 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 32,41,41,32,21,13,21,34,53,73,82,73,53,34,21,34,55,87,126,155,155,

%U 126,87,55,34,55,89,142,213,281,310,281,213,142,89,55,89,144,231,355,494

%N 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 Sum of n-th row = 2*A027934(n). - _Reinhard Zumkeller_, Oct 07 2012

%H Reinhard Zumkeller, <a href="/A108617/b108617.txt">Rows n = 0..120 of triangle, flattened</a>

%H Hacéne Belbachir, 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PascalsTriangle.html">Pascals Triangle</a>

%H Wikipedia, <a href="http://www.wikipedia.org/wiki/Fibonacci_number">Fibonacci number</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Pascal&#39;s_triangle">Pascal's triangle</a>

%H <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>

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

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

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

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

%e 0;

%e 1,1;

%e 1,2,1;

%e 2,3,3,2;

%e 3,5,6,5,3;

%e 5,8,11,11,8,5;

%e 8,13,19,22,19,13,8;

%e 13,21,32,41,41,32,21,13;

%e 21,34,53,73,82,73,53,34,21;

%e 34,55,87,126,155,155,126,87,55,34;

%e 55,89,142,213,281,310,281,213,142,89,55;

%p A108617 := proc(n,k)

%p if k = 0 or k=n then

%p combinat[fibonacci](n) ;

%p elif k <0 or k > n then

%p 0 ;

%p else

%p procname(n-1,k-1)+procname(n-1,k) ;

%p end if;

%p end proc: # _R. J. Mathar_, Oct 05 2012

%t 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 (Haskell)

%o a108617 n k = a108617_tabl !! n !! k

%o a108617_row n = a108617_tabl !! n

%o a108617_tabl = [0] : iterate f [1,1] where

%o f row@(u:v:_) = zipWith (+) ([v - u] ++ row) (row ++ [v - u])

%o -- _Reinhard Zumkeller_, Oct 07 2012

%Y Cf. A108037, A007318.

%K nonn,easy,tabl

%O 0,5

%A _Reinhard Zumkeller_, Jun 12 2005

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 20 20:02 EDT 2020. Contains 337265 sequences. (Running on oeis4.)