A003983 Array read by antidiagonals with T(n,k) = min(n,k).

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 3, 2, 1, 1, 2, 3, 4, 3, 2, 1, 1, 2, 3, 4, 4, 3, 2, 1, 1, 2, 3, 4, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 7, 7, 6, 5, 4, 3, 2, 1

Offset

1,5

Comments

Also, "correlation triangle" for the constant sequence 1. - Paul Barry, Jan 16 2006

Antidiagonal sums are in A002620.

As a triangle, row sums are A002620. T(2n,n)=n+1. Diagonal sums are A001399. Construction: Take antidiagonal triangle of MM^T where M is the sequence array for the constant sequence 1 (lower triangular matrix with all 1's). - Paul Barry, Jan 16 2006

From Franklin T. Adams-Watters, Sep 25 2011: (Start)

As a triangle, count up to ceiling(n/2) and back down again (repeating the central term when n is even).

When the first two instances of each number are removed from the sequence, the original sequence is recovered.

(End)

Links

Reinhard Zumkeller, Rows n=1..100 of triangle, flattened

Formula

Number triangle T(n, k) = Sum_{j=0..n} [j<=k][j<=n-k]. - Paul Barry, Jan 16 2006

G.f.: 1/((1-x)*(1-x*y)*(1-x^2*y)). - Christian G. Bower, Jan 17 2006

a(n) = min(floor( 1/2 + sqrt(2*n)) - (2*n + round(sqrt(2*n)) - round(sqrt(2*n))^2)/2+1, (2*n + round(sqrt(2*n)) - round(sqrt(2*n))^2)/2). - Leonid Bedratyuk, Dec 13 2009

Example

Triangle version begins
  1;
  1, 1;
  1, 2, 1;
  1, 2, 2, 1;
  1, 2, 3, 2, 1;
  1, 2, 3, 3, 2, 1;
  1, 2, 3, 4, 3, 2, 1;
  1, 2, 3, 4, 4, 3, 2, 1;
  1, 2, 3, 4, 5, 4, 3, 2, 1;
  ...

Maple

a(n) = min(floor(1/2 + sqrt(2*n)) - (2*n + round(sqrt(2*n)) - round(sqrt(2*n))^2)/2+1, (2*n + round(sqrt(2*n)) - round(sqrt(2*n))^2)/2) # Leonid Bedratyuk, Dec 13 2009

Mathematica

Flatten[Table[Min[n-k+1, k], {n, 1, 14}, {k, 1, n}]] (* Jean-François Alcover, Feb 23 2012 *)

Prog

(Haskell)
a003983 n k = a003983_tabl !! (n-1) !! (k-1)
a003983_tabl = map a003983_row [1..]
a003983_row n = hs ++ drop m (reverse hs)
   where hs = [1..n' + m]
         (n', m) = divMod n 2
-- Reinhard Zumkeller, Aug 14 2011
(PARI) T(n, k) = min(n, k) \\ Charles R Greathouse IV, Feb 06 2017

Crossrefs

Cf. A002620, A001399, A087062, A115236, A115237, A124258, A006752, A120268, A173945, A173947, A173948, A173949, A173953, A173954, A173955, A173973, A173982-A173986, A004197.

Sequence in context: A330190 A356300 A348041 * A087062 A204026 A300119

Adjacent sequences: A003980 A003981 A003982 * A003984 A003985 A003986

Keyword

tabl,nonn,easy,nice

Author

Marc LeBrun

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Nov 08 2000

Entry revised by N. J. A. Sloane, Dec 05 2006

Status

approved