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%I #46 Jan 05 2023 19:09:06
%S 0,1,10,76,540,3751,25840,177451,1217160,8344876,57202750,392089501,
%T 2687463360,18420257701,126254611990,865362736876,5931286406640,
%U 40653646980451,278644255208560,1909856172864751,13090349042248500
%N a(n) = (Lucas(4n) - Lucas(2n))/4.
%C This is a divisibility sequence, that is, if n | m then a(n) | a(m). However, it is not a strong divisibility sequence. It is the case k = 3 of a 1-parameter family of 4th-order linear divisibility sequences with o.g.f. x*(1 - x^2)/( (1 - k*x + x^2)*(1 - (k^2 - 2)*x + x^2) ). Compare with A000290 (case k = 2) and A085695 (case k = -3). - _Peter Bala_, Jan 17 2014
%C In general, for distinct integers r and s with r = s (mod 2), the sequence Lucas(r*n) - Lucas(s*n) is a fourth-order divisibility sequence. See A273622 for the case r = 3, s = 1. - _Peter Bala_, May 27 2016
%H Vincenzo Librandi, <a href="/A215465/b215465.txt">Table of n, a(n) for n = 0..1000</a>
%H Peter Bala, <a href="/A273622/a273622.pdf">Lucas sequences and divisibility sequences</a>
%H E. L. Roettger and H. C. Williams, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Roettger/roettger12.html">Appearance of Primes in Fourth-Order Odd Divisibility Sequences</a>, J. Int. Seq., Vol. 24 (2021), Article 21.7.5.
%H Hugh Williams and R. K. Guy, <a href="http://dx.doi.org/10.1142/S1793042111004587">Some fourth-order linear divisibility sequences</a>, Intl. J. Number Theory vol. 7 (5) (2011) 1255-1277
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (10,-23,10,-1).
%F 4*a(n) = A000032(4*n) - A000032(2*n).
%F a(n) = A056854(n)/4 - A005248(n)/4.
%F G.f.: -x*(x-1)*(1+x) / ( (x^2-7*x+1)*(x^2-3*x+1) ).
%F a(n) = 10*a(n-1) - 23*a(n-2) + 10*a(n-3) - a(n-4). - _G. C. Greubel_, Jun 02 2016
%F a(n) = 2^(-2-n)*((7-3*sqrt(5))^n-(3-sqrt(5))^n-(3+sqrt(5))^n+(7+3*sqrt(5))^n). - _Colin Barker_, Jun 02 2016
%F a(n) = a(-n) for all n in Z. - _Michael Somos_, Dec 29 2022
%e a(3) = 76 because the 12th (4 * 3rd) Lucas number is 22, the 6th (2 * 3rd) Lucas number is 18, and (322 - 18)/4 = 304/4 = 76.
%p A215465 := proc(n)
%p (A000032(4*n)-A000032(2*n))/4 ;
%p end proc:
%t Table[(LucasL[4n] - LucasL[2n])/4, {n, 0, 19}] (* _Alonso del Arte_, Aug 11 2012 *)
%t CoefficientList[Series[-x*(x-1)*(1+x)/((x^2 - 7*x + 1)* (x^2 - 3*x + 1)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Dec 23 2012 *)
%t LinearRecurrence[{10,-23,10,-1},{0,1,10,76},50] (* _G. C. Greubel_, Jun 02 2016 *)
%o (Magma) [(Lucas(4*n) - Lucas(2*n))/4: n in [0..20]]; // _Vincenzo Librandi_, Dec 23 2012
%o (PARI) {a(n) = my(w = quadgen(5)^(2*n)); (2*real(w^2-w) + imag(w^2-w))/4}; /* _Michael Somos_, Dec 29 2022 */
%Y Cf. A085695, A215466, A273622.
%K nonn,easy
%O 0,3
%A _R. J. Mathar_, Aug 11 2012
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