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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 October 15 17:24 EDT 2019. Contains 328037 sequences. (Running on oeis4.)