A051125 Table T(n,k) = max{n,k} read by antidiagonals (n >= 1, k >= 1).

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

Offset

1,2

Comments

Antidiagonal sums = A006578. - Reinhard Zumkeller, Nov 17 2011

The n-th principal submatrix of this array shows the square number n^2 as composed of gnomons of form 2*i+1 with i=0..n-1 (see Deza). - Stefano Spezia, Sep 15 2025

References

Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 3.

Links

Peter Kagey, Antidiagonals n = 1..126 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

From Robert Israel, Jul 22 2016: (Start)

G.f. as table: G(x,y) = x*y*(1-3*x*y+x*y^2+x^2*y)/((1-x*y)*(1-x)^2*(1-y)^2).

G.f. flattened: (1-x)^(-2)*(x^2 + Sum_{j >= 0} x^(2*j^2) *(x+x^2 -2*x^(j+2)-2*x^(-j+2)+2*x^(2*j+2))). (End)

Example

Table begins
  1, 2, 3, 4, 5, ...
  2, 2, 3, 4, 5, ...
  3, 3, 3, 4, 5, ...
  4, 4, 4, 4, 5, ...
  ...

Maple

seq(seq(max(r, d+1-r), r=1..d), d=1..15); # Robert Israel, Jul 22 2016

Mathematica

Flatten[Table[Max[n-k+1, k], {n, 13}, {k, n, 1, -1}]] (* Alonso del Arte, Nov 17 2011 *)

Prog

(PARI) T(n, k) = max(n, k) \\ Charles R Greathouse IV, Feb 07 2017
(Magma) [Max(n-k+1, k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jul 23 2019
(SageMath) [[max(n-k+1, k) for k in (1..n)] for n in (1..15)] # G. C. Greubel, Jul 23 2019
(GAP) Flat(List([1..15], n-> List([1..n], k-> Maximum(n-k+1, k) ))); # G. C. Greubel, Jul 23 2019

Crossrefs

Cf. A003056, A003983, A003984, A004197.

Equals A003984(n) + 1.

Sequence in context: A071647 A034883 A338643 * A321126 A342765 A244580

Adjacent sequences: A051122 A051123 A051124 * A051126 A051127 A051128

Keyword

nonn,tabl,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from Robert Lozyniak

Status

approved