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%I #72 Mar 05 2022 01:25:03
%S 1,1,2,5,1,4,1,15,2,11,1,40,1,29,2,105,1,76,1,275,2,199,1,720,1,521,2,
%T 1885,1,1364,1,4935,2,3571,1,12920,1,9349,2,33825,1,24476,1,88555,2,
%U 64079,1,231840,1,167761,2,606965,1,439204,1,1589055,2,1149851,1,4160200,1,3010349,2
%N GCD of all sums of n consecutive Lucas numbers.
%C The sum of two consecutive Lucas number is the sum of four consecutive Fibonacci numbers, which is verified easily enough with the identity L(n) = F(n - 1) + F(n + 1). Therefore a(1) = a(2) = A210209(4).
%H Colin Barker, <a href="/A229339/b229339.txt">Table of n, a(n) for n = 1..1000</a>
%H Dan Guyer and aBa Mbirika, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Mbirika/mb5.pdf">GCD of sums of k consecutive Fibonacci, Lucas, and generalized Fibonacci numbers</a>, Journal of Integer Sequences, 24 No.9, Article 21.9.8 (2021), 25pp; <a href="https://arxiv.org/abs/2104.12262">arXiv preprint</a>, arXiv:2104.12262 [math.NT], 2021.
%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,3,0,1,0,-1,0,-3,0,0,0,1).
%F a(n) = 3*a(n-4) + a(n-6) - a(n-8) - 3*a(n-10) + a(n-14) for n > 14. - _Giovanni Resta_, Oct 04 2013
%F G.f.: x*(x^12 -x^11 +2*x^10 -5*x^9 -2*x^8 -x^7 -6*x^6 +x^5 -2*x^4 +5*x^3 +2*x^2 +x +1) / (-x^14 +3*x^10 +x^8 -x^6 -3*x^4 +1). - _Colin Barker_, Nov 09 2014
%F From _Aba Mbirika_, Jan 04 2022: (Start)
%F a(n) = gcd(L(n+1)-1, L(n+2)-3).
%F a(n) = Lcm_{A106291(m) divides n} m.
%F Proofs of these formulas are given in Theorems 15 and 25 of the Guyer-Mbirika paper. (End)
%e a(3) = 2 because any sum of three consecutive Lucas numbers is an even number.
%e a(4) = 5 because all sums of four consecutive Lucas numbers are divisible by 5.
%e a(5) = 1 because some sums of five consecutive Lucas numbers are coprime.
%t a[n_] := a[n] = If[n <= 14, {1, 1, 2, 5, 1, 4, 1, 15, 2, 11, 1, 40, 1, 29}[[n]], 3*a[n - 4] + a[n - 6] - a[n - 8] - 3*a[n - 10] + a[n - 14]]; Array[a, 64] (* _Giovanni Resta_, Oct 04 2013 *)
%t CoefficientList[Series[(x^12 - x^11 + 2 x^10 - 5 x^9 - 2 x^8 - x^7 - 6 x^6 + x^5 - 2 x^4 + 5 x^3 + 2 x^2 + x + 1) / (-x^14 + 3 x^10 + x^8 - x^6 - 3 x^4 + 1), {x, 0, 40}], x] (* _Vincenzo Librandi_, Nov 09 2014 *)
%t LinearRecurrence[{0,0,0,3,0,1,0,-1,0,-3,0,0,0,1},{1,1,2,5,1,4,1,15,2,11,1,40,1,29},70] (* _Harvey P. Dale_, Jul 21 2021 *)
%t Table[GCD[LucasL[n + 1] - 2, LucasL[n] + 1], {n, 0, 50}] (* _Horst H. Manninger_, Dec 25 2021 *)
%o (PARI) Vec(x*(x^12 -x^11 +2*x^10 -5*x^9 -2*x^8 -x^7 -6*x^6 +x^5 -2*x^4 +5*x^3 +2*x^2 +x +1) / (-x^14 +3*x^10 +x^8 -x^6 -3*x^4 +1) + O(x^100)) \\ _Colin Barker_, Nov 09 2014
%Y Cf. A210209, A022112, A022088, A022098, A106291 (Pisano periods of the Lucas sequence).
%K nonn,easy
%O 1,3
%A _Alonso del Arte_, Sep 23 2013
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