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 A108617 Triangle read by rows: T(n,k) = T(n-1,k-1)+T(n-1,k) for 0
 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 (list; table; graph; refs; listen; history; text; internal format)
 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, 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, Pascals Triangle Wikipedia, Fibonacci number Wikipedia, 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. EXAMPLE 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 CROSSREFS Cf. A108037, A007318. 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

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Last modified April 20 20:26 EDT 2021. Contains 343137 sequences. (Running on oeis4.)