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%I
%S 0,1,1,2,1,1,4,1,3,2,11,1,8,1,29,2,21,1,76,1,55,2,199,1,144,1,521,2,
%T 377,1,1364,1,987,2,3571,1,2584,1,9349,2,6765,1,24476,1,17711,2,64079,
%U 1,46368,1,167761,2,121393,1,439204,1,317811,2,1149851,1,832040
%N GCD of all sums of n consecutive Fibonacci numbers.
%C Early on in the Posamentier & Lehmann (2007) book, the fact that the sum of any ten consecutive Fibonacci numbers is a multiple of 11 is presented as an interesting property of the Fibonacci numbers. Much later in the book a proof of this fact is given, using arithmetic modulo 11. An alternative proof could demonstrate that 11F(n + 6) = sum_{i = n .. n + 9} F(i).
%D Alfred S. Posamentier & Ingmar Lehmann, The (Fabulous) Fibonacci Numbers, Prometheus Books, New York (2007) p. 33.
%H Alois P. Heinz, <a href="/A210209/b210209.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rea#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (0,0,0,3,0,1,0,-1,0,-3,0,0,0,1).
%F G.f.: -x*(x^12-x^11+2*x^10-x^9-2*x^8-x^7-6*x^6+x^5-2*x^4+x^3+2*x^2+x+1) / (x^14-3*x^10-x^8+x^6+3*x^4-1) = -1/(x^4+x^2-1) +(x^2+1)/(x^4-x^2-1) +(x+2)/(6*(x^2+x+1)) +(x-2)/(6*(x^2-x+1)) -2/(3*(x+1)) -2/(3*(x-1)). - _Alois P. Heinz_, Mar 18 2012
%e a(3) = 2 because all sums of three consecutive Fibonacci numbers are divisible by 2 (F_n + F_(n-1) + F_(n-2) = 2F_n), but since the GCD of 3+5+8=16 and 5+8+13=26 is 2, no number larger than 2 divides all sums of three consecutive Fibonacci numbers.
%e a(4) = 1 because the GCD of 1+1+2+3=7 and 1+2+3+5=12 is 1, so the sums of four consecutive Fibonacci numbers have no factors in common.
%p a:= n-> (Matrix(7, (i, j)-> `if` (i=j-1, 1, `if`(i=7, [1, 0, -3, -1, 1, 3, 0][j], 0)))^iquo(n, 2, 'r'). `if`(r=0, <<0, 1, 1, 4, 3, 11, 8>>, <<1, 2, 1, 1, 2, 1, 1>>))[1, 1]: seq (a(n), n=0..80); # _Alois P. Heinz_, Mar 18 2012
%Y Cf. A000071, sum of the first n Fibonacci numbers.
%Y Bisections give: A005013 (even part), A131534 (odd part). - _Alois P. Heinz_, Mar 18 2012
%K nonn,easy
%O 0,4
%A _Alonso del Arte_, Mar 18 2012
%E More terms from _Alois P. Heinz_, Mar 18 2012
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