A155881 a(n) is the number of zeros needed to write the integers 1 through Fibonacci(n).

0, 0, 0, 0, 0, 0, 1, 2, 3, 5, 8, 24, 43, 67, 121, 188, 409, 708, 1228, 1946, 4131, 6241, 10525, 17866, 29428, 58369, 87881, 156261, 255242, 412545, 767846, 1280460, 2059307, 3343656, 5510186, 9861418, 16472261, 26422596, 43917688, 73697381, 125281166, 206655249

Offset: 1

Author

Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 29 2009

Keywords

• nonn
• base

Comments

Data suggets a(n) ~= 10 ^ (c * n) where c ~= 0.209. - David A. Corneth, Jan 23 2019

Examples

			F(9)=34, so writing the numbers F(1)..F(9) requires 3 zeros (one each at 10, 20, and 30), thus a(9)=3.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..4770

Crossrefs

Cf. A000045, A055641, A061217.

Programs

Maple

A055641 := proc(n) option remember ; local a,d; if n = 0 then RETURN(a) ; fi; a := 0 ; for d in convert(n,base,10) do if d = 0 then a := a+1 ; fi; od: a ; end: A155881 := proc(n) add(A055641(i),i=1..combinat[fibonacci](n)) ; end: for n from 1 do printf("%d,\n",A155881(n)) ; od; # R. J. Mathar, Feb 19 2009

Mathematica

Block[{n = 32, s}, s = DigitCount[Range@ Fibonacci@ n, 10, 0]; Array[Total@ Take[s, Fibonacci@ #] &, n]] (* Michael De Vlieger, Jan 23 2019 *)

PARI

nb(n) = #Set(select(x->(x==0), digits(n))); \\ A055641
a(n) = sum(k=1, fibonacci(n), nb(k)); \\ Michel Marcus, Jan 23 2019

PARI

a(n) = my(n = fibonacci(n), m=logint(n, 10)); (m+1)*(n+1) - (10^(m+1)-1)/9 + (1/2) * sum(j=1, m+1, (n\10^j * (2*n+2 - (1 + n\10^j) * 10 ^ j) - floor(n/10^j+9/10) * (2*n+2 + ((4/5 - floor(n / 10^j + 9 / 10))*10^j)))) \\ David A. Corneth, Jan 23 2019

Formula

a(n) = A061217(Fibonacci(n)) = A061217(A000045(n)). - David A. Corneth, Jan 23 2019

Extensions

8 more terms from R. J. Mathar, Feb 19 2009

9 more terms from Sean A. Irvine, Dec 10 2009

Edited by Jon E. Schoenfield, Jan 22 2019

More terms from David A. Corneth, Jan 23 2019