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A003989 Triangle T from the array A(x, y) = gcd(x,y), for x >= 1, y >= 1, read by antidiagonals. 42
1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 2, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 2, 1, 2, 5, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 7, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 5, 3, 1, 1, 3, 5, 1, 3, 1, 1, 1, 2, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

For m < n, the maximal number of nonattacking queens that can be placed on the n by m rectangular toroidal chessboard is gcd(m,n), except in the case m=3, n=6.

The determinant of the matrix of the first n rows and columns is A001088(n). [Smith, Mansion] - Michael Somos, Jun 25 2012

Imagine a torus having regular polygonal cross-section of m sides. Now, break the torus and twist the free ends, preserving rotational symmetry, then reattach the ends. Let n be the number of faces passed in twisting the torus before reattaching it. For example, if n = m, then the torus has exactly one full twist. Do this for arbitrary m and n (m > 1, n > 0). Now, count the independent, closed paths on the surface of the resulting torus, where a path is "closed" if and only if it returns to its starting point after a finite number of times around the surface of the torus. Conjecture: this number is always gcd(m,n). NOTE: This figure constitutes a group with m and n the binary arguments and gcd(m,n) the resulting value. Twisting in the reverse direction is the inverse operation, and breaking & reattaching in place is the identity operation. - Jason Richardson-White, May 06 2013

Regarded as a triangle, table of gcd(n - k +1, k) for 1 <= k <= n. - Franklin T. Adams-Watters, Oct 09 2014

REFERENCES

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, 2nd ed., 1994, ch. 4.

D. E. Knuth, The Art of Computer Programming, Addison-Wesley, section 4.5.2.

LINKS

T. D. Noe, First 100 antidiagonals of array, flattened

Grant Cairns, Queens on Non-square Tori, El. J. Combinatorics, N6, 2001.

P. Mansion, On an Arithmetical Theorem of Professor Smith's, Messenger of Mathematics, (1878), pp. 81-82.

Kival Ngaokrajang, Pattern of GCD(x,y) > 1 for x and y = 1..60. Non-isolated values larger than 1 (polyomino shapes) are colored.

Marcelo Polezzi, A Geometrical Method for Finding an Explicit Formula for the Greatest Common Divisor, The American Mathematical Monthly, Vol. 104, No. 5 (May, 1997), pp. 445-446.

H. J. S. Smith, On the value of a certain arithmetical determinant, Proc. London Math. Soc. 7 (1875-1876), pp. 208-212.

Index entries for sequences related to lcm's

FORMULA

Multiplicative in both parameters with a(p^e, m) = GCD(p^e, m). - David W. Wilson, Jun 12 2005

T(n, k) = A(n - k + 1, k) = gcd(n - k + 1, k), n >= 1, k = 1..n. See a comment above and the Mathematica program. - Wolfdieter Lang, May 12 2018

EXAMPLE

The array A begins:

x\y 1 2 3 4 5 6 ...

1:  1 1 1 1 1 1 ...

2:  1 2 1 2 1 2 ...

3:  1 1 3 1 1 3 ...

4:  1 2 1 4 1 2 ...

5:  1 1 1 1 5 1 ...

...

The triangle T begins:

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

MATHEMATICA

Table[ GCD[x - y + 1, y], {x, 1, 15}, {y, 1, x}] // Flatten (* Jean-Fran├žois Alcover, Dec 12 2012 *)

PROG

(PARI) {A(n, m) = gcd(n, m)}; /* Michael Somos, Jun 25 2012 */

CROSSREFS

Rows, columns and diagonals: A089128, A109007, A109008, A109009, A109010, A109011, A109012, A109013, A109014, A109015.

A109004 is (0, 0) based.

Cf. A003990, A003991, A050873, A054431, A001088.

Cf. also A091255 for GF(2)[X] polynomial analog.

A(x, y) = A075174(A004198(A075173(x), A075173(y))) = A075176(A004198(A075175(x), A075175(y))).

Antidiagonal sums are in A006579.

Sequence in context: A140194 A159923 A287957 * A091255 A175466 A214403

Adjacent sequences:  A003986 A003987 A003988 * A003990 A003991 A003992

KEYWORD

tabl,nonn,easy,nice,mult

AUTHOR

Marc LeBrun

STATUS

approved

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Last modified August 18 00:28 EDT 2018. Contains 313817 sequences. (Running on oeis4.)