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A216676 Digital roots of squares of Fibonacci numbers. 5
1, 1, 4, 9, 7, 1, 7, 9, 4, 1, 1, 9, 1, 1, 4, 9, 7, 1, 7, 9, 4, 1, 1, 9, 1, 1, 4, 9, 7, 1, 7, 9, 4, 1, 1, 9, 1, 1, 4, 9, 7, 1, 7, 9, 4, 1, 1, 9, 1, 1, 4, 9, 7, 1, 7, 9, 4, 1, 1, 9, 1, 1, 4, 9, 7, 1, 7, 9, 4, 1, 1, 9, 1, 1, 4, 9, 7, 1, 7, 9, 4, 1, 1, 9, 1, 1, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The first 11 terms are symmetric about 6th term. The first 23 terms are symmetric about 12th term. We can generalize this as follows: the first (2n-1) terms are symmetric about n-th term.

The sequence appears to be periodic with period-length 12. - John W. Layman,  Sep 14 2012

The Fibonacci numbers are periodic modulo any integer. The digital roots of the Fibonacci numbers are given by A030132, a sequence with a period length of 24. Squaring gives {1, 1, 4, 9, 7, 1, 7, 9, 4, 1, 1, 9, 1, 1, 4, 9, 7, 1, 7, 9, 4, 1, 1, 9}, which is a sequence of twelve numbers given twice. Therefore, the previous comment is correct. - Alonso del Arte, Sep 25 2012

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Index entries for linear recurrences with constant coefficients, signature (0,1,0,0,0,-1,0,1).

FORMULA

a(n) = A010888(A007598(n)).

G.f. ( -1-x-3*x^2-8*x^3-3*x^4+8*x^5-9*x^7-x^6 ) / ( (x-1) *(1+x) *(x^2+1) *(x^4-x^2+1) ). - R. J. Mathar, Sep 15 2012

EXAMPLE

a(7) = 7 because F(7) = 13 and 13^2 = 169, with digits adding up to 16, the digital root is therefore 7.

MATHEMATICA

a = {}; For[n = 1, n <= 100, n++, {fn2 = Fibonacci[n]^2; d = IntegerDigits[fn2]; While[Length[d] > 1, d = IntegerDigits[Total[d]]]; AppendTo[a, d[[1]]] }]; a (* John W. Layman,  Sep 14 2012 *)

ReplaceAll[Table[Mod[Fibonacci[n]^2, 9], {n, 72}], {0 -> 9}] (* Alonso del Arte, Sep 23 2012 *)

PROG

(PARI) fibmod(n, m)=((Mod([1, 1; 1, 0], m))^n)[1, 2]

a(n)=lift(fibmod(n, 9)^2-1)+1 \\ Charles R Greathouse IV, Jun 20 2017

CROSSREFS

Sequence in context: A176426 A114720 A053511 * A021909 A018880 A245670

Adjacent sequences:  A216673 A216674 A216675 * A216677 A216678 A216679

KEYWORD

nonn,base,easy

AUTHOR

Ravi Bhandari, Sep 14 2012

EXTENSIONS

Terms a(25)-a(72) by John W. Layman, Sep 14 2012

Terms a(73) and beyond from Andrew Howroyd, Feb 25 2018

STATUS

approved

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Last modified June 15 22:06 EDT 2021. Contains 345053 sequences. (Running on oeis4.)