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A194508 First coordinate of the (2,3)-Lagrange pair for n. 22
-1, 1, 0, 2, 1, 0, 2, 1, 3, 2, 1, 3, 2, 4, 3, 2, 4, 3, 5, 4, 3, 5, 4, 6, 5, 4, 6, 5, 7, 6, 5, 7, 6, 8, 7, 6, 8, 7, 9, 8, 7, 9, 8, 10, 9, 8, 10, 9, 11, 10, 9, 11, 10, 12, 11, 10, 12, 11, 13, 12, 11, 13, 12, 14, 13, 12, 14, 13, 15, 14, 13, 15, 14, 16, 15, 14, 16, 15, 17, 16, 15, 17 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Suppose that c and d are relatively prime integers satisfying 1<c<d.  Every integer n has a representation

(1) n = c*x + d*y

where x and y are integers satisfying

(2) |x - y| < d.

Let h=(c-1)*(d-1).  If n>=h, there is exactly one pair (x,y) satisfying (1) and (2), and, for this pair, x>=0 and y>=0.

For n>=h, write (x,y) as (x(n),y(n)) and call this the (c,d)-Lagrange pair for n.  If n>c*d then

(3) x(n)=x(n-c-d)+1 and

(4) y(n)=y(n-c-d)+1.

If n<h, then n may have more than one representation satisfying (1) and (2); e.g. 1=2*(-3)+7*1=2*4+7*(-1).  To extend the definition of (c,d)-Lagrange pair by stipulating a particular pair (x(n),y(n)) satisfying (1) and (2) for n<h, we reverse (3) and (4): x(n)=x(n+c+d)-1 and y(n)=y(n+c+d)-1 for all integers n.  The initial numbers x(1) and y(1) so determined are also the numbers found by the Euclidean algorithm for 1 as a linear combination c*x+d*y.

Examples:

c..d....x(n)......y(n)

2..3....A194508...A194509

2..5....A194510...A194511

2..7....A194512...A194513

3..4....A194514...A194515

3..5....A194516...A194517

3..7....A194518...A194519

3..8....A194520...A194521

4..5....A194522...A194523

4..7....A194524...A194525

5..6....A194526...A194527

5..8....A194528...A194529

REFERENCES

L. E. Dickson, History of the Theory of Numbers, vol. II:  Diophantine Analysis, Chelsea, 1952, page 47.

LINKS

Table of n, a(n) for n=1..82.

EXAMPLE

This table shows (x(n),y(n)) for 1<=n<=13:

n...... 1..2..3..4..5..6..7..8..9..10..11..12..13

x(n).. -1..1..0..2..1..0..2..1..3..2...1...3...2

y(n)... 1..0..1..0..1..2..1..2..1..2...3...2...3

MATHEMATICA

c = 2; d = 3;

x1 = {-1, 1, 0, 2, 1}; y1 = {1, 0, 1, 0, 1};

x[n_] := If[n <= c + d, x1[[n]], x[n - c - d] + 1]

y[n_] := If[n <= c + d, y1[[n]], y[n - c - d] + 1]

Table[x[n], {n, 1, 100}] (* A194508 *)

Table[y[n], {n, 1, 100}] (* A194509 *)

r[1, n_] := n; r[2, n_] := x[n]; r[3, n_] := y[n]

TableForm[Table[r[m, n], {m, 1, 3}, {n, 1, 30}]]

CROSSREFS

Cf. A193509-A194529.

Sequence in context: A221179 A153247 A071432 * A240808 A263142 A025253

Adjacent sequences:  A194505 A194506 A194507 * A194509 A194510 A194511

KEYWORD

sign

AUTHOR

Clark Kimberling, Aug 27 2011

STATUS

approved

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Last modified March 27 08:41 EDT 2017. Contains 284146 sequences.