A096095 Digital Fibonacci numbers: a(0) = a(1) = 1; a(n+2) = a(n) ++ a(n+1) where ++ stands for digit-wise sum (with no carries).

1, 1, 2, 3, 5, 8, 13, 111, 124, 235, 359, 5814, 511613, 5161427, 567210310, 5612371737, 511795811047, 5161310711827714, 51618211410616387511, 51611337272013271110141225, 5161138888102246817264128736, 5111274911111537425910883742699511, 511127910722491311155271156169109106711171247

Offset

0,3

Comments

In case corresponding digits add to a 2-digit number, the "1" is inserted instead of being added to the next higher significant digit. - M. F. Hasler, Apr 18 2009

A197945(n) = length of longest common prefix of a(n) and a(n+1). [Reinhard Zumkeller, Oct 19 2011]

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..36

Formula

frac( log[10]( a(n) )) tends to 0.7085296011388705685015..., i.e. the first digits 51112791588989116125412156111... eventually remain the same, since on the average a(n+1) has about 1/6 of digits more than a(n). - M. F. Hasler, Apr 18 2009

Example

a(8) = a(6) ++ a(7) = 08 ++ 13 = concatenation of (0+1 = 1 and 8+3 = 11) = 111.
a(9) = a(7) ++ a(8) = 013 ++ 111 =124.

Mathematica

nxt[{a_, b_}]:={b, FromDigits[Flatten[ IntegerDigits/@ (PadLeft[ IntegerDigits[ a], IntegerLength[ b], 0] + IntegerDigits[b])]]}; Transpose[ NestList[ nxt, {1, 1}, 25]][[1]] (* Harvey P. Dale, Dec 16 2012 *)

Prog

(PARI) a=[1, 1]; for(n=2, 30, a=concat(a, a[n]+a[n-1]); my(p=10^#Str(a[n])); while(p\=10, a[n]\p%10+a[n-1]\p%10 > 9 & a[n+1]+=(a[n+1]\p\10-1)*90*p)); a \\ M. F. Hasler, Apr 18 2009
(Haskell)
import Data.List (unfoldr)
a096095 n = a096095_list !! n
a096095_list = 1 : 1 : zipWith dadd a096095_list (tail a096095_list) where
   dadd x y = foldl (\v d -> (if d < 10 then 10 else 100)*v + d)
                    0 $ reverse $ unfoldr f (x, y) where
        f (x, y) | x + y == 0 = Nothing
                | otherwise  = Just (xd + yd, (x', y'))
                where (x', xd) = divMod x 10; (y', yd) = divMod y 10
-- Reinhard Zumkeller, Oct 19 2011

Crossrefs

Cf. A000045, A197945.

Sequence in context: A041957 A329192 A294939 * A041101 A041809 A117566

Adjacent sequences: A096092 A096093 A096094 * A096096 A096097 A096098

Keyword

base,nice,nonn

Author

Amarnath Murthy, Jun 24 2004

Extensions

Edited and extended beyond a(12) by M. F. Hasler, Apr 18 2009

Status

approved