login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A030132 Digital root of Fibonacci(n). 17
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Any initial pair (a(0), a(1)) of nonzero single-digit numbers enters a cycle of length 24, except for (3, 3), which enters a cycle of length 8 and (9, 9), which is immediately periodic of period length 1. - Jonathan Vos Post, Dec 29 2005

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

REFERENCES

S. Marivani and others, Problem 10974, Amer. Math. Monthly, 111 (No. 7, 2004), 628.

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

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, 0, 0, 1).

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

EXAMPLE

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

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 *)

PROG

(Haskell)

a030132 n = a030132_list !! n

a030132_list =

   0 : 1 : map a007953 (zipWith (+) a030132_list (tail a030132_list))

-- Reinhard Zumkeller, Aug 20 2011

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

CROSSREFS

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

Cf. A065076.

Sequence in context: A098906 A007887 A105472 * A004090 A104205 A267758

Adjacent sequences:  A030129 A030130 A030131 * A030133 A030134 A030135

KEYWORD

nonn,base,easy,nice

AUTHOR

youngelder(AT)webtv.net (Ana)

EXTENSIONS

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

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified June 28 08:24 EDT 2017. Contains 288813 sequences.