A004197 Triangle read by rows. T(n, k) = n - k if n - k < k, otherwise k.

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 3, 2, 1, 0, 0, 1, 2, 3, 3, 2, 1, 0, 0, 1, 2, 3, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 5, 6, 6, 5, 4, 3, 2, 1, 0, 0, 1, 2

Offset: 0

Author

David W. Wilson

Keywords

• tabl
• nonn
• easy
• nice

Comments

Table of min(x,y), where (x,y) = (0,0),(0,1),(1,0),(0,2),(1,1),(2,0),...

Highest power of 6 that divides A036561. - Fred Daniel Kline, May 29 2012

Triangle T(n,k) read by rows: T(n,k) = min(k,n-k). - Philippe Deléham, Feb 25 2014

Examples

			From _Philippe Deléham_, Feb 25 2014: (Start)
Top left corner of table:
  0 0 0 0
  0 1 1 1
  0 1 2 2
  0 1 2 3
Triangle T(n,k) begins:
  0;
  0, 0;
  0, 1, 0;
  0, 1, 1, 0;
  0, 1, 2, 1, 0;
  0, 1, 2, 2, 1, 0;
  0, 1, 2, 3, 2, 1, 0;
  0, 1, 2, 3, 3, 2, 1, 0;
  0, 1, 2, 3, 4, 3, 2, 1, 0;
  0, 1, 2, 3, 4, 4, 3, 2, 1, 0;
  0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0;
  0, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 0;
  0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0;
  0, 1, 2, 3, 4, 5, 6, 6, 5, 4, 3, 2, 1, 0;
  0, 1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1, 0;
  0, 1, 2, 3, 4, 5, 6, 7, 7, 6, 5, 4, 3, 2, 1, 0;
  ... (End)
		

Links

• Reinhard Zumkeller, Rows n=0..100 of triangle, flattened

Crossrefs

Similar to but strictly different from A087062 and A261684.

Row sums give A002620. - Reinhard Zumkeller, Jul 27 2005

Positions of zero are given in A117142. - Ridouane Oudra, Apr 30 2019

Cf. A144464, A152714, A152716, A152717.

Programs

Haskell

a004197 n k = a004197_tabl !! n !! k
a004197_tabl = map a004197_row [0..]
a004197_row n = hs ++ drop (1 - n `mod` 2) (reverse hs)
   where hs = [0..n `div` 2]
-- Reinhard Zumkeller, Aug 14 2011

Maple

T := (n, k) -> if n - k < k then n - k else k fi:
for n from 0 to 9 do seq(T(n, k), k=0..n) od; # Peter Luschny, May 07 2023

Mathematica

Flatten[Table[IntegerExponent[2^(n - k) 3^k, 6], {n, 0, 20}, {k, 0, n}]] (* Fred Daniel Kline, May 29 2012 *)

PARI

T(x,y)=min(x,y) \\ Charles R Greathouse IV, Feb 07 2017

Formula

a(n) = A003983(n) - 1.

G.f.: x*y/((1-x)*(1-y)*(1-x*y)). - Franklin T. Adams-Watters, Feb 06 2006

2^T(n,k) = A144464(n,k), 3^T(n,k) = A152714(n,k), 4^T(n,k) = A152716(n,k), 5^T(n,k) = A152717(n,k). - Philippe Deléham, Feb 25 2014

a(n) = (1/2)*(t - 1 - abs(t^2 - 2*n - 1)), where t = floor(sqrt(2*n+1)+1/2). - Ridouane Oudra, May 03 2019

Extensions

Mathematica program fixed by Harvey P. Dale, Nov 26 2020

Name edited by Peter Luschny, May 07 2023