

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
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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=(c1)*(d1). 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(ncd)+1 and
(4) y(n)=y(ncd)+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. A193509A194529.
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



