

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
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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 (2n1) terms are symmetric about nth term.
The sequence appears to be periodic with periodlength 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. ( 1x3*x^28*x^33*x^4+8*x^59*x^7x^6 ) / ( (x1) *(1+x) *(x^2+1) *(x^4x^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)^21)+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



