A216676 Digital roots of squares of Fibonacci numbers.

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

Offset: 1

Author

Ravi Bhandari, Sep 14 2012

Keywords

• nonn
• base
• easy

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

Examples

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

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).

Programs

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 *)

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

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

Extensions

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

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