# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a215465 Showing 1-1 of 1 %I A215465 #46 Jan 05 2023 19:09:06 %S A215465 0,1,10,76,540,3751,25840,177451,1217160,8344876,57202750,392089501, %T A215465 2687463360,18420257701,126254611990,865362736876,5931286406640, %U A215465 40653646980451,278644255208560,1909856172864751,13090349042248500 %N A215465 a(n) = (Lucas(4n) - Lucas(2n))/4. %C A215465 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 A215465 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 A215465 Vincenzo Librandi, Table of n, a(n) for n = 0..1000 %H A215465 Peter Bala, Lucas sequences and divisibility sequences %H A215465 E. L. Roettger and H. C. Williams, Appearance of Primes in Fourth-Order Odd Divisibility Sequences, J. Int. Seq., Vol. 24 (2021), Article 21.7.5. %H A215465 Hugh Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory vol. 7 (5) (2011) 1255-1277 %H A215465 Index entries for linear recurrences with constant coefficients, signature (10,-23,10,-1). %F A215465 4*a(n) = A000032(4*n) - A000032(2*n). %F A215465 a(n) = A056854(n)/4 - A005248(n)/4. %F A215465 G.f.: -x*(x-1)*(1+x) / ( (x^2-7*x+1)*(x^2-3*x+1) ). %F A215465 a(n) = 10*a(n-1) - 23*a(n-2) + 10*a(n-3) - a(n-4). - _G. C. Greubel_, Jun 02 2016 %F A215465 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 A215465 a(n) = a(-n) for all n in Z. - _Michael Somos_, Dec 29 2022 %e A215465 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 A215465 := proc(n) %p A215465 (A000032(4*n)-A000032(2*n))/4 ; %p A215465 end proc: %t A215465 Table[(LucasL[4n] - LucasL[2n])/4, {n, 0, 19}] (* _Alonso del Arte_, Aug 11 2012 *) %t A215465 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 A215465 LinearRecurrence[{10,-23,10,-1},{0,1,10,76},50] (* _G. C. Greubel_, Jun 02 2016 *) %o A215465 (Magma) [(Lucas(4*n) - Lucas(2*n))/4: n in [0..20]]; // _Vincenzo Librandi_, Dec 23 2012 %o A215465 (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 A215465 Cf. A085695, A215466, A273622. %K A215465 nonn,easy %O A215465 0,3 %A A215465 _R. J. Mathar_, Aug 11 2012 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE