A107435 Triangle T(n,k), 1<=k<=n, read by rows: T(n,k) = length of Euclidean algorithm starting with n and k.

1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 3, 3, 2, 1, 1, 1, 3, 1, 4, 2, 2, 1, 1, 2, 1, 2, 3, 2, 3, 2, 1, 1, 1, 2, 2, 1, 3, 3, 2, 2, 1, 1, 2, 3, 3, 2, 3, 4, 4, 3, 2, 1, 1, 1, 1, 1, 3, 1, 4, 2, 2, 2, 2, 1, 1, 2, 2, 2, 4, 2, 3, 5, 3, 3, 3, 2, 1, 1, 1, 3, 2, 3, 2, 1, 3, 4, 3, 4, 2, 2, 1

Offset: 1

Author

Philippe Deléham, Jun 09 2005

Keywords

• nonn
• tabl

Comments

Consequence of theorem of Gabriel Lamé (1844): the first value of m in this triangle is T(F(m+2), F(m+1)) where F(n) = A000045(n); example: the first 5 is T(F(7), F(6)) = T(13, 8).

From Bernard Schott, May 01 2022: (Start)

Theorem of Gabriel Lamé (1844): The number of divisions necessary to find the greatest common divisor of two natural numbers n > k by means of the Euclidean algorithm is never greater than five times the number of digits of the smaller number k (see link).

This upper bound 5*length(k) is the best possible; the smallest pairs (n, k) for which T(n, k) = 5 * length(k) when length(k) = 1, 2 or 3 are respectively (F(7), F(6)), (F(12), F(11)) and (F(17), F(16)) where F(n) = A000045(n). This upper bound is not attained when length(k) >= 4. (End)

Examples

			13 = 5*2 + 3, 5 = 3*1 + 2, 3 = 2*1 + 1, 2 = 2*1 + 0 = so that T(13,5) = 4.
Triangle begins:
  1
  1 1
  1 2 1
  1 1 2 1
  1 2 3 2 1
  1 1 1 2 2 1
  1 2 2 3 3 2 1
  1 1 3 1 4 2 2 1
  1 2 1 2 3 2 3 2 1
  1 1 2 2 1 3 3 2 2 1
  1 2 3 3 2 3 4 4 3 2 1
  1 1 1 1 3 1 4 2 2 2 2 1
  1 2 2 2 4 2 3 5 3 3 3 2 1
  1 1 3 2 3 2 1 3 4 3 4 2 2 1
  1 2 1 3 1 2 2 3 3 2 4 2 3 2 1
  1 1 2 1 2 3 3 1 4 4 3 2 3 2 2 1
  1 2 3 2 3 3 3 2 3 4 4 4 3 4 3 2 1
  ..............................
Smallest examples with T(n, k) = 5 * length(k) (Theorem of Gabriel Lamé):
13 = 8*1 + 5, 8 = 5*1 + 3, 5 = 3*1 + 2, 3 = 2*1 + 1, 2 = 2*1 + 0 = so that T(13,8) = 5 = 5 * length(8).
144 = 89*1 + 55, 89 = 55*1 + 34, 55 = 34*1 + 21, 34 = 21*1 + 13, 21 = 13*1 + 8, then 5 steps already seen in the previous example, so that T(144,89) = 10 = 5 * length(89).
1597 = 987*1 + 610, 987 = 610*1 + 377, 610 = 377*1 + 233, 377 = 233*1 + 144, 233 = 144*1 + 89, then 10 steps already seen in the previous examples, so that T(1597,987) = 15 = 5 * length(987).
		

References

• Ross Honsberger, Mathematical Gems II, Dolciani Mathematical Expositions No. 2, Mathematical Association of America, 1976, Chapter 7, A Theorem of Gabriel Lamé, pp. 54-57.
• Wacław Sierpiński, Elementary Theory of Numbers, Theorem 12 (Lamé) p. 21, Warsaw, 1964.

Links

• Peter Kagey, Table of n, a(n) for n = 1..10000
• Gabriel Lamé, Note sur la limite du nombre des divisions dans la recherche du plus grand commun diviseur entre deux nombres entiers, Comptes rendus de l'Académie des sciences XIX, 1844, pages 867-870.
• Wikipedia, Euclidean algorithm.
• Wikipedia, Gabriel Lamé.

Crossrefs

Cf. A000045, A001622, A002390, A034883, A049816, A051010, A055642.

Programs

Maple

F:= proc(n,k) option remember;
   if n mod k = 0 then 1
   else 1 + procname(k, n mod k)
   fi
end proc:
seq(seq(F(n,k),k=1..n), n=1..15); # Robert Israel, Feb 16 2016

Mathematica

T[n_, k_] := T[n, k] = If[Divisible[n, k], 1, 1 + T[k, Mod[n, k]]];
Table[T[n, k], {n, 1, 15}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 12 2019, after Robert Israel *)

PARI

T(n, k) = if ((n % k) == 0, 1, 1 + T(k, n % k)); \\ Michel Marcus, May 02 2022

Formula

T(n, k) = A049816(n, k) + 1.

From Robert Israel, Feb 16 2016: (Start)

T(n, k) = 1 if n == 0 (mod k), otherwise T(n, k) = 1 + T(k, (n mod k)).

G.f. G(x,y) of triangle satisfies G(x,y) = x*y/((1-x)*(1-x*y)) - Sum_{k>=1} (x^2*y)^k/(1-x^k) + Sum_{k>=1} G(x^k*y,x). (End)

From Bernard Schott, Apr 29 2022: (Start)

T(F(m+2), F(m+1)) = m where F(n) = A000045(n) (first comment).

T(n, k) <= 5 * length(k) where length(k) = A055642(k).

T(n, k) <= 1 + floor(log(k)/log(phi)) where log(phi) = A002390; the least numbers for which equality stands are when k and n are consecutive Fibonacci numbers. (End)