0,2

It appears that fractions of this kind with a(0)=1 exist only for a(1) equal to 4 (this sequence) and 13 (A320336).

Next term has 98 digits. - Giovanni Resta, Oct 11 2018

1/(4-1) = 0.3333...

At the beginning instead of 4 we have 3 as first decimal digit. Adding the second term this is fixed.

1/(4-1) + 4/(41 - 4) = 0.441441...

1/(4-1) + 4/(41 - 4) + 41/(89814 - 41) = 0.44189814891 ...

The sum is 0.4 41 89814 98285430640360 ...

P:=proc(q, h) local a, b, d, t, x, n; x:=1; a:=1/(h-1); b:=ilog10(h-1)+1; d:=h; print(d); t:=h; for n from h+1 to q do if trunc(evalf(a+t/(n-t), 100)*10^(b+ilog10(n)+1))=d*10^(ilog10(n)+1)+n then b:=b+ilog10(n)+1; d:=d*10^(ilog10(n)+1)+n; a:=a+t/(n-t); t:=n; x:=n+1; print(n); fi; od; end: P(10^10, 4);

Cf. A302932, A302933, A303388, A304285, A304286, A304287, A304288, A304289, A305661, A305662, A305663, A305664, A305665, A305666, A320306, A320307, A320308, A320309, A320336.

Sequence in context: A333731 A326899 A022500 * A076652 A078288 A349455

Adjacent sequences: A320332 A320333 A320334 * A320336 A320337 A320338

nonn,base

Paolo P. Lava, Oct 11 2018

a(4)-a(5) from Giovanni Resta, Oct 11 2018

approved