A003988 Triangle with subscripts (1,1),(2,1),(1,2),(3,1),(2,2),(1,3), etc. in which entry (i,j) is [ i/j ].

1, 2, 0, 3, 1, 0, 4, 1, 0, 0, 5, 2, 1, 0, 0, 6, 2, 1, 0, 0, 0, 7, 3, 1, 1, 0, 0, 0, 8, 3, 2, 1, 0, 0, 0, 0, 9, 4, 2, 1, 1, 0, 0, 0, 0, 10, 4, 2, 1, 1, 0, 0, 0, 0, 0, 11, 5, 3, 2, 1, 1, 0, 0, 0, 0, 0, 12, 5, 3, 2, 1, 1, 0, 0, 0, 0, 0, 0, 13, 6, 3, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 14, 6, 4, 2, 2, 1, 1, 0, 0, 0, 0

Offset: 1

Author

Marc LeBrun

Keywords

• tabl
• nonn
• easy
• nice

Comments

Another version of A010766.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..5050

Crossrefs

Cf. A010766, A003056, A049581, A003991, A004247, A077049.

Row sums are in A006218. Antidiagonal sums are in A002541.

Programs

Haskell

a003988 n k = (n + 1 - k) `div` k
a003988_row n = zipWith div [n,n-1..1] [1..n]
a003988_tabl = map a003988_row [1..]
-- Reinhard Zumkeller, Apr 13 2012

Mathematica

t[n_, k_] := Quotient[n, k]; Table[t[n-k+1, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 21 2013 *)

Formula

From Franklin T. Adams-Watters, Jan 28 2006: (Start)

T(n,k) = Sum_{i=1..k} A077049(n,i).

G.f.: (1/(1-x))*Sum_{k>0} x^k*y^k/(1-x^k) = (1/(1-x))*Sum_{k>0} x^k * y / (1 - x^k y) = (1/(1-x)) * Sum_{k>0} x^k * Sum_{d|k} y^d = A(x,y)/(1-x) where A(x,y) is the g.f. of A077049. (End)

T(n,k) = floor((n + 1 - k) / k). - Reinhard Zumkeller, Apr 13 2012

Extensions

More terms from James Sellers