A030132 Digital root of Fibonacci(n).

0, 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9, 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9, 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9, 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8

Offset: 0

Author

youngelder(AT)webtv.net (Ana)

Keywords

• nonn
• base
• easy
• nice

Comments

Any initial pair (a(0), a(1)) of nonzero single-digit numbers enters a cycle of length 24, except for the 8 cases where 3 divides both a(0), a(1) and (a(0), a(1)) != (9, 9), which enter a cycle of length 8 and (9, 9), which is immediately periodic of period length 1. - Jonathan Vos Post, Dec 29 2005 [Corrected by Jianing Song, Apr 17 2021]

First term that differs from A004090 is a(10). In general, all terms of A004090 having one digit are the same in this sequence. - Alonso del Arte, Sep 16 2012

Decimal expansion of 12484270798876404618091 / 1111111111111111111111110 = 0.0[112358437189887641562819] (periodic). - Daniel Forgues, Feb 27 2017

Examples

			a(10) = 1 because F(10) = 55, and since 5 + 5 = 10 and 1 + 0 = 1 is the digital root of 55.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Robert Bruce Gray, Another context for the first 48 terms of this sequence, May 08 2025
• S. Marivani and others, Digital Roots of Fibonacci Numbers: Problem 10974, Amer. Math. Monthly, 111 (No. 7, 2004), 628.
• Colm Mulcahy, Gibonacci Bracelets.
• Marc Renault, The Fibonacci sequence modulo m
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Crossrefs

Cf. A000045 (Fibonacci numbers), A010888 (digital roots), A004090, A007953, A030133.

Cf. A001175, A010077, A017641, A065076.

Programs

Haskell

a030132 n = a030132_list !! n
a030132_list =
   0 : 1 : map a007953 (zipWith (+) a030132_list (tail a030132_list))
-- Reinhard Zumkeller, Aug 20 2011

Mathematica

digitalRoot[n_Integer?Positive] := FixedPoint[Plus@@IntegerDigits[#]&, n]; Table[If[n == 0, 0, digitalRoot[Fibonacci[n]]], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, May 02 2011 *)
Table[NestWhile[Total[IntegerDigits[#]]&, Fibonacci[n], # > 9 &], {n, 0, 90}] (* Harvey P. Dale, May 07 2012 *)
PadRight[{0},120,{9,1,1,2,3,5,8,4,3,7,1,8,9,8,8,7,6,4,1,5,6,2,8,1}] (* Harvey P. Dale, Jul 20 2024 *)

PARI

a(n)=if(n,(fibonacci(n)-1)%9+1,0) \\ Charles R Greathouse IV, Jan 23 2013

Formula

a(n + 1) = sum of digits of (a(n) + a(n - 1)).

Periodic with period 24 = A001175(9) given by {1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9}.

a(n) + a(n + 1) = A010077(n + 4); a(A017641(n)) = 9. - Reinhard Zumkeller, Jul 04 2007

G.f.: x*( -1 -x -2*x^2 -3*x^3 -5*x^4 -8*x^5 -4*x^6 -3*x^7 -7*x^8 -x^9 -8*x^10 -9*x^11 -8*x^12 -8*x^13 -7*x^14 -6*x^15 -4*x^16 -x^17 -5*x^18 -6*x^19 -2*x^20 -8*x^21 -x^22 -9*x^23 ) / ( (x-1) *(1+x+x^2) *(1+x) *(1-x+x^2) *(1+x^2) *(x^4-x^2+1) *(1+x^4) *(x^8-x^4+1) ). - R. J. Mathar, Feb 08 2013

Extensions

Entry revised by N. J. A. Sloane, Aug 29 2004