A125888 - OEIS

%I #12 Mar 16 2014 21:21:50

%S 0,1,4,11,95,1255,4381,145067,7662223,80819870,3642636055,

%T 790371794974,19684652229146,11040403321665367,3322851650570694710,

%U 219687751171273292755,515022606093026805779903,1946889068386887991263046319,659536996247168335462717076503

%N Numerator of sum of first n ratios of Fibonacci to Lucas numbers.

%C A125889(n) = denominator(SUM[i=1..n]F(i)/L(i)) = SUM[i=1..n] A000045(i)/A000032(i) = n/sqrt(5) + O(1). - _Max Alekseyev_, Dec 07 2006

%C denominator(SUM[i=1..n]A000045(i)/A000032(i)).

%C GCD(F(i),L(i)) <= 2, so the ratio reduces when there is a factor of two in common, every third term. Example as continued fraction: 0 + 1+1/3+2/4+3/7+5/11+8/18+13/29+21/47+34/76+55/123 = 1/1+ 1/19+ 1/4+ 1/1+ 1/13+ 1/2+ 1/4+ 1/1+ 1/6+ 1/6+ 1/1+ 1/85+ 1/1+ 1/4+ 1/2.

%F a(n) = numerator(SUM[i=1..n]F(i)/L(i)) = numerator(SUM[i=1..n]A000045(i)/A000032(i)).

%e The fractions, reduced to lowest terms, begin:

%e 0/1, 1/1, 4/3, 11/6, 95/42, 1255/462, 4381/1386, 145067/40194, 7662223/1889118, 80819870/17946621, 3642636055/735811461, ...

%Y Cf. A000032, A000045.

%K easy,nonn,frac

%O 0,3

%A _Jonathan Vos Post_, Dec 13 2006