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

Jorma K. Merikoski, Pentti Haukkanen, Antonio Sasaki, and Timo Tossavainen, On Generalized Eigenvalues of MAX Matrices to MIN Matrices and LCM Matrices to GCD Matrices, J. Int. Seq. (2025) Vol. 28, Art. 25.7.1.

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
(Python)
from math import isqrt
def A003983(n):
    a = (m:=isqrt(k:=n<<1))+(k>m*(m+1))
    x = n-(a*(a-1)>>1)
    return min(x, a-x+1) # Chai Wah Wu, Jun 14 2025

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