A049581 Table T(n,k) = |n-k| read by antidiagonals (n >= 0, k >= 0).

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

Offset

0,4

Comments

Commutative non-associative operator with identity 0. T(nx,kx) = x T(n,k). A multiplicative analog is A089913. - Marc LeBrun, Nov 14 2003

For the characteristic polynomial of the n X n matrix M_n with entries M_n(i, j) = |i-j| see A203993. - Wolfdieter Lang, Feb 04 2018

For the determinant of the n X n matrix M_n with entries M_n(i, j) = |i-j| see A085750. - Bernard Schott, May 13 2020

a(n) = 0 iff n = 4 times triangular number (A046092). - Bernard Schott, May 13 2020

Links

Peter Kagey, Rows n = 0..125 of triangle, flattened

Formula

G.f.: (x + y - 4xy + x^2y + xy^2)/((1-x)^2 (1-y)^2) (1-xy)) = (x/(1-x)^2 + y/(1-y)^2)/(1-xy). T(n,0) = T(0,n) = n; T(n+1,k+1) = T(n,k). - Franklin T. Adams-Watters, Feb 06 2006

a(n) = |A002260(n+1)-A004736(n+1)| or a(n) = |((n+1)-t(t+1)/2) - (t*t+3*t+4)/2-(n+1))| where t=floor[(-1+sqrt(8*(n+1)-7))/2]. - Boris Putievskiy, Dec 24 2012; corrected by Altug Alkan, Sep 30 2015

From Robert Israel, Sep 30 2015: (Start)

If b(n) = a(n+1) - 2*a(n) + a(n-1), then for n >= 3 we have

b(n) = -1 if n = (j^2+5j+4)/2 for some integer j >= 1

b(n) = -3 if n = (j^2+5j+6)/2 for some integer j >= 0

b(n) = 4 if n = 2j^2 + 6j + 4 for some integer j >= 0

b(n) = 2 if n = 2j^2 + 8j + 7 or 2j^2 + 8j + 8 for some integer j >= 0

b(n) = 0 otherwise. (End)

Triangle t(n,k) = max(k, n-k) - min(k, n-k). - Peter Luschny, Jan 26 2018

Triangle t(n, k) = |n - 2*k| for n >= 0, k = 0..n. See the Maple and Mathematica programs. Hence t(n, k)= t(n, n-k). - Wolfdieter Lang, Feb 04 2018

a(n) = |t^2 - 2*n - 1|, where t = floor(sqrt(2*n+1) + 1/2). - Ridouane Oudra, Jun 07 2019; Dec 11 2020

As a rectangle, T(n,k) = |n-k| = max(n,k) - min(n,k). - Clark Kimberling, May 11 2020

Example

Displayed as a triangle t(n, k):
n\k   0 1 2 3 4 5 6 7 8 9 10 ...
0:    0
1:    1 1
2:    2 0 2
3:    3 1 1 3
4:    4 2 0 2 4
5:    5 3 1 1 3 5
6:    6 4 2 0 2 4 6
7:    7 5 3 1 1 3 5 7
8:    8 6 4 2 0 2 4 6 8
9:    9 7 5 3 1 1 3 5 7 9
10:  10 8 6 4 2 0 2 4 6 8 10
... reformatted by Wolfdieter Lang, Feb 04 2018
Displayed as a table:
  0 1 2 3 4 5 6 ...
  1 0 1 2 3 4 5 ...
  2 1 0 1 2 3 4 ...
  3 2 1 0 1 2 3 ...
  4 3 2 1 0 1 2 ...
  5 4 3 2 1 0 1 ...
  6 5 4 3 2 1 0 ...
  ...

Maple

seq(seq(abs(n-2*k), k=0..n), n=0..12); # Robert Israel, Sep 30 2015

Mathematica

Table[Abs[(n-k) -k], {n, 0, 12}, {k, 0, n}]//Flatten (* Michael De Vlieger, Sep 29 2015 *)
Table[Join[Range[n, 0, -2], Range[If[EvenQ[n], 2, 1], n, 2]], {n, 0, 12}]//Flatten (* Harvey P. Dale, Sep 18 2023 *)

Prog

(PARI) a(n) = abs(2*(n+1)-binomial((sqrtint(8*(n+1))+1)\2, 2)-(binomial(1+floor(1/2 + sqrt(2*(n+1))), 2))-1);
vector(100, n , a(n-1)) \\ Altug Alkan, Sep 29 2015
(PARI) {t(n, k) = abs(n-2*k)}; \\ G. C. Greubel, Jun 07 2019
(GAP) a := Flat(List([0..12], n->List([0..n], k->Maximum(k, n-k)-Minimum(k, n-k)))); # Muniru A Asiru, Jan 26 2018
(Magma) [[Abs(n-2*k): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Jun 07 2019
(Sage) [[abs(n-2*k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jun 07 2019
(Python)
from math import isqrt
def A049581(n): return abs((k:=n+1<<1)-((m:=isqrt(k))+(k>m*(m+1)))**2-1) # Chai Wah Wu, Nov 09 2024

Crossrefs

Cf. A003989, A003990, A003991, A003056, A004247, A002260, A004736.

Cf. A089913. Apart from signs, same as A114327. A203993.

Sequence in context: A194547 A257570 A220417 * A114327 A330240 A330237

Adjacent sequences: A049578 A049579 A049580 * A049582 A049583 A049584

Keyword

nonn,tabl,easy,nice

Author

N. J. A. Sloane

Status

approved