A011794 Triangle defined by a(n+1,k)=a(n,k-1)+a(n-1,k), a(n,1)=1, a(1,k)=1, a(2,k)=min(2,k).

1, 1, 2, 1, 2, 3, 1, 3, 4, 5, 1, 3, 6, 7, 8, 1, 4, 7, 11, 12, 13, 1, 4, 10, 14, 19, 20, 21, 1, 5, 11, 21, 26, 32, 33, 34, 1, 5, 15, 25, 40, 46, 53, 54, 55, 1, 6, 16, 36, 51, 72, 79, 87, 88, 89

Offset

1,3

Links

Table of n, a(n) for n=1..55.

D. J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory, arXiv:hep-th/9604128, 1996.

Formula

a(n,n) = Fibonacci(n+1). - Jean-François Alcover, Feb 26 2013

Example

matrix(10,10,n,k,a(n-1,k-1))
[ 0 0 0 0 0 0 0 0 0 0 ]
[ 0 1 1 1 1 1 1 1 1 1 ]
[ 0 1 2 2 2 2 2 2 2 2 ]
[ 0 1 2 3 3 3 3 3 3 3 ]
[ 0 1 3 4 5 5 5 5 5 5 ]
[ 0 1 3 6 7 8 8 8 8 8 ]

Mathematica

a[n_, k_] := a[n, k] = a[n-1, k-1] + a[n-2, k]; a[n_, 1] = 1; a[1, k_] = 1; a[2, k_] := Min[2, k]; Table[a[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 26 2013 *)

Prog

(PARI) a(n, k)=if(n<=0 || k<=0, 0, if(n<=2 || k==1, min(n, k), a(n-1, k-1)+a(n-2, k)))

Crossrefs

Columns include A008619 and (essentially) A055802, A055803, A055804, A055805, A055806. Right-hand columns 1-14 are A000045, A000071, A001911, A001924, A001891, A014162, A053808, A014166, A053809, A053739, A054469, A053295, A054470, A053296.

Essentially a reflected version of A055801.

Sequence in context: A026820 A330661 A091438 * A221640 A073300 A104468

Adjacent sequences: A011791 A011792 A011793 * A011795 A011796 A011797

Keyword

nonn,tabl

Author

N. J. A. Sloane, David Broadhurst

Extensions

Entry improved by comments from Michael Somos

Status

approved