A194517 -id:A194517 - cp's OEIS frontend

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

Clark Kimberling, Aug 27 2011

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

			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

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

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).

Cf. A193509-A194529.

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

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
```

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

A194516 First coordinate of (3,5)-Lagrange pair for n.

Original entry on oeis.org

2, -1, 1, 3, 0, 2, -1, 1, 3, 0, 2, 4, 1, 3, 0, 2, 4, 1, 3, 5, 2, 4, 1, 3, 5, 2, 4, 6, 3, 5, 2, 4, 6, 3, 5, 7, 4, 6, 3, 5, 7, 4, 6, 8, 5, 7, 4, 6, 8, 5, 7, 9, 6, 8, 5, 7, 9, 6, 8, 10, 7, 9, 6, 8, 10, 7, 9, 11, 8, 10, 7, 9, 11, 8, 10, 12, 9, 11, 8, 10, 12, 9, 11, 13, 10, 12, 9, 11, 13, 10, 12, 14, 11, 13, 10

Offset: 1

Clark Kimberling, Aug 28 2011

sign

See A194508.

			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)... 2.-1..1..3..0..2.-1..1..3..0...2...4...1
y(n).. -1..1..0.-1..1..0..2..1..0..2...1...0...2

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,1,-1).

Cf. A194508, A194517.

c = 3; d = 5;
x1 = {2, -1, 1, 3, 0, 2, -1, 1}; y1 = {-1, 1, 0, -1, 1, 0, 2, 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}]  (* A194516 *)
Table[y[n], {n, 1, 100}]  (* A194517 *)
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}]]
LinearRecurrence[{1,0,0,0,0,0,0,1,-1},{2,-1,1,3,0,2,-1,1,3},100] (* Harvey P. Dale, Nov 29 2024 *)

From Chai Wah Wu, Jan 21 2020: (Start)
a(n) = a(n-1) + a(n-8) - a(n-9) for n > 9.
G.f.: x*(2*x^7 - 3*x^6 + 2*x^5 - 3*x^4 + 2*x^3 + 2*x^2 - 3*x + 2)/(x^9 - x^8 - x + 1). (End)
a(n) = 2*n - 5*floor((3*n + 3)/8). - Ridouane Oudra, Dec 29 2020

cp's OEIS Frontend

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

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