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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

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).

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)

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

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}]]

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 November 18 01:41 EST 2019. Contains 329242 sequences. (Running on oeis4.)