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A216676 Digital roots of squares of Fibonacci numbers.

%I A216676 #40 Feb 25 2018 22:50:00
%S A216676 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,
%T A216676 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,
%U A216676 4,1,1,9,1,1,4,9,7,1,7,9,4,1,1,9,1,1,4
%N A216676 Digital roots of squares of Fibonacci numbers.
%C A216676 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.
%C A216676 The sequence appears to be periodic with period-length 12. - _John W. Layman_,  Sep 14 2012
%C A216676 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
%H A216676 Andrew Howroyd, <a href="/A216676/b216676.txt">Table of n, a(n) for n = 1..10000</a>
%H A216676 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,0,0,0,-1,0,1).
%F A216676 a(n) = A010888(A007598(n)).
%F A216676 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
%e A216676 a(7) = 7 because F(7) = 13 and 13^2 = 169, with digits adding up to 16, the digital root is therefore 7.
%t A216676 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 *)
%t A216676 ReplaceAll[Table[Mod[Fibonacci[n]^2, 9], {n, 72}], {0 -> 9}] (* _Alonso del Arte_, Sep 23 2012 *)
%o A216676 (PARI) fibmod(n, m)=((Mod([1, 1; 1, 0], m))^n)[1, 2]
%o A216676 a(n)=lift(fibmod(n,9)^2-1)+1 \\ _Charles R Greathouse IV_, Jun 20 2017
%K A216676 nonn,base,easy
%O A216676 1,3
%A A216676 _Ravi Bhandari_, Sep 14 2012
%E A216676 Terms a(25)-a(72) by _John W. Layman_, Sep 14 2012
%E A216676 Terms a(73) and beyond from _Andrew Howroyd_, Feb 25 2018