The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A003989 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A003989

Triangle T from the array A(x, y) = gcd(x,y), for x >= 1, y >= 1, read by antidiagonals.
(history; published version)

#139 by Amiram Eldar at Tue Nov 12 02:41:42 EST 2024

STATUS

reviewed

approved

#138 by Stefano Spezia at Tue Nov 12 00:58:11 EST 2024

STATUS

proposed

reviewed

#137 by Michel Marcus at Tue Nov 12 00:04:43 EST 2024

STATUS

editing

proposed

#136 by Michel Marcus at Tue Nov 12 00:04:39 EST 2024

LINKS

Mihai Prunescu and Joseph Shunia, <a href="https://arxiv.org/abs/2411.06430">Arithmetic-term representations for the greatest common divisor</a>, arXiv:2411.06430 [math.NT], 2024.

EXAMPLE

[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

[1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2]

[1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1]

[1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4]

[1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1]

[1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2]

[1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1]

[1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 8]

[1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1]

[1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 2, 1, 2, 5, 2]

[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1]

[1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 12, 1, 2, 3, 4]

...

n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...

1: 1

2: 1 1

3: 1 2 1

4: 1 1 1 1

5: 1 2 3 2 1

6: 1 1 1 1 1 1

7: 1 2 1 4 1 2 1

8: 1 1 3 1 1 3 1 1

9: 1 2 1 2 5 2 1 2 1

10: 1 1 1 1 1 1 1 1 1 1

11: 1 2 3 4 1 6 1 4 3 2 1

12: 1 1 1 1 1 1 1 1 1 1 1 1

13: 1 2 1 2 1 2 7 2 1 2 1 2 1

14: 1 1 3 1 5 3 1 1 3 5 1 3 1 1

15: 1 2 1 4 1 2 1 8 1 2 1 4 1 2 1

... - Wolfdieter Lang, May 12 2018

STATUS

approved

editing

#135 by Michael De Vlieger at Fri Oct 20 06:47:18 EDT 2023

STATUS

reviewed

approved

#134 by Peter Luschny at Fri Oct 20 02:20:47 EDT 2023

STATUS

proposed

reviewed

#133 by Joerg Arndt at Mon Oct 16 10:06:18 EDT 2023

STATUS

editing

proposed

#132 by Joerg Arndt at Mon Oct 16 10:06:14 EDT 2023

LINKS

Marc Chamberland, <a href="https://doi.org/10.1016/j.laa.2011.08.030">Factored matrices can generate combinatorial identities</a>, Linear Algebra and its Applications, Volume 438, Issue 4, 2013, pp. 1667-1677.

STATUS

proposed

editing

#131 by Peter Bala at Mon Oct 16 09:39:15 EDT 2023

STATUS

editing

proposed

#130 by Peter Bala at Mon Oct 16 09:38:17 EDT 2023

LINKS

Marc Chamberland, <a href="https://doi.org/10.1016/j.laa.2011.08.030">Factored matrices can generate combinatorial identities</a>,Linear Algebra and its ApplicationsVolume Applications, Volume 438, Issue 4, 15 February 2013, Pages pp. 1667-1677.

Warren P. Johnson, <a href="https://doi.org/10.1080/0025570X.2003.11953215">An LDU Factorization in Elementary Number Theory</a>, Mathematics Magazine, Vol. 76, No. 5 (Dec., 2003), pp. 392-394.

FORMULA

The LU decomposition of this square array = A051731 * transpose(A054522) (see Johnson (2003) or Chamberland, (2013), p. 1673). - Peter Bala, Oct 15 2023