A194508 First coordinate of the (2,3)-Lagrange pair for n.

-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

Offset: 1

Author

Clark Kimberling, Aug 27 2011

Keywords

• sign
• easy

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

Examples

			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
		

References

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

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Crossrefs

Cf. A193509-A194529.

Programs

Maple

A0:= [-1,1,0,2,0]:
f:= n -> A0[(n-1 mod 5)+1]+floor(n/5):
map(f, [$1..100]); # Robert Israel, Jul 29 2019

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}]]

PARI

a(n)=2*n - (3*n+2)\5*3

Formula

From Robert Israel, Jul 29 2019: (Start)

a(n+5) = a(n) + 1.

G.f.: x*(-1+2*x-x^2+2*x^3-x^4)/(1-x-x^5+x^6). (End)

a(n) = 2*n - 3*floor((3*n+2)/5). - Ridouane Oudra, Sep 06 2020

a(n) = n/5 + O(1). - Charles R Greathouse IV, Mar 30 2022