%I #32 Mar 11 2019 15:44:33
%S 1,1,1,3,7,31,85,393,1093,5071,14119,65523,182449,846721,2357713,
%T 10941843,30467815,141397231,393723877,1827222153,5087942581,
%U 23612490751,65749529671,305135157603,849655943137,3943144558081
%N a(1)=a(2)=a(3)=1; for n>3, a(n)=(a(n-1)*a(n-2)+a(n-1)+a(n-2))/a(n-3).
%C What accounts for the high proportion of semiprimes in this sequence? Primes: 3, 7, 31, 1093, 846721, 393723877, ... Semiprimes: 85 = 5 * 17 393 = 3 * 131 5071 = 11 * 461 14119 = 7 * 2017 65523 = 3 * 21841 182449 = 43 * 4243 5087942581 = 11113 * 457837 849655943137 = 17 * 49979761361 3943144558081 = 31 * 127198211551 - _Jonathan Vos Post_, Feb 04 2005
%H Seiichi Manyama, <a href="/A072881/b072881.txt">Table of n, a(n) for n = 1..1803</a>
%H P. Heideman and E. Hogan, <a href="http://www.math.wisc.edu/~propp/SSL/heiho.pdf">A new family of Somos-like recurrences</a>.
%H P. Heideman and E. Hogan, <a href="http://www.combinatorics.org/Volume_15/Abstracts/v15i1r54.html">A new family of Somos-like recurrences</a>, El. J. Combin. 15 (2008) #R54. [From _R. J. Mathar_, Dec 04 2008]
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0, 14, 0, -14, 0, 1).
%F Both sequences u=(a(2n-1))_{n>0} and u=(a(2n))_{n>0} satisfy the order 3 linear recursion : u(n)=14u(n-1)-14u(n-2)+u(n-3).
%F a(2*n-1) = ceiling((1/11)*sqrt(1002/5-78*sqrt(33/5))*(sqrt(15)/2+sqrt(11)/ 2)^(2*n-1)).
%F a(2*n) = ceiling((1/11)*(13-sqrt(165))*(sqrt(15)/2+sqrt(11)/2)^(2*n)).
%F G.f.: x*(1+x-13*x^2-11*x^3+7*x^4+3*x^5)/(1-14*x^2+14*x^4-x^6). - _Jaume Oliver Lafont_, Sep 25 2009
%F a(n) = (4-(-1)^n)*a(n-1)-a(n-2)-1. - _Bruno Langlois_, Aug 21 2016
%F Sequences u=(a(2n)) and v=(a(2n-1)) satisfy order 2 linear recursions : u(n)=13*u(n-1)-u(n-2)-5 and v(n)=13*v(n-1)-v(n-2)-7. - _Bruno Langlois_, Aug 21 2016
%t LinearRecurrence[{0, 14, 0, -14, 0, 1},{1, 1, 1, 3, 7, 31},26] (* _Ray Chandler_, Jul 24 2016 *)
%t nxt[{a_,b_,c_}]:={b,c,(c*b+c+b)/a}; NestList[nxt,{1,1,1},30][[All,1]] (* _Harvey P. Dale_, Mar 11 2019 *)
%o (PARI) a(k=3, n) = {K = (k-1)/2; vds = vector(n); for (i=1, 2*K+1, vds[i] = 1;); for (i=2*K+2, n, vds[i] = (vds[i-1]*vds[i-2*K]+vds[i-K]+vds[i-K-1])/vds[i-2*K-1];); for (i=1, n, print1(vds[i], ","););} \\ _Michel Marcus_, Oct 28 2012
%Y A048736 [From _Jaume Oliver Lafont_, Sep 25 2009]
%Y Cf. A092264, A133846, A133847, A133848, A133854.
%K easy,nonn
%O 1,4
%A _Benoit Cloitre_, Jul 28 2002, revised Feb 03 2005
|