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%I #15 Mar 15 2024 12:50:00
%S 1,1,2,7,35,210,1365,9165,62322,425867,2915551,19974626,136884937,
%T 938162617,6430103330,44072167855,302074043195,2070443441970,
%U 14191023001437,97266699113157,666675822475026,4569463931720051
%N a(n) = a(n-1) * (11*a(n-1) - a(n-2)) / (a(n-1) + 4*a(n-2)), with a(0) = a(1) = 1.
%H Harvey P. Dale, <a href="/A172511/b172511.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (11,-33,33,-11,1).
%F a(n) = (4 + A049685(n-1) + 5 * A001519(n)) / 10 = a(1 - n).
%F G.f.: (4 / (1 - x) + (1 - 6*x) / (1 - 7*x + x^2) + (5 - 10*x) / (1 - 3*x + x^2)) / 10.
%F a(0)=1, a(1)=1, a(2)=2, a(3)=7, a(4)=35, a(n)=11*a(n-1)-33*a(n-2)+ 33*a(n-3)- 11*a(n-4)+a(n-5). - _Harvey P. Dale_, Nov 18 2013
%F a(n) = a(1-n) for all n in Z. - _Michael Somos_, Sep 22 2014
%F 0 = a(n)*(+a(n+1) + 4*a(n+2)) + a(n+1)*(-11*a(n+1) + a(n+2)) for all n in Z. - _Michael Somos_, Sep 22 2014
%F a(n) = b(n+1) * b(n) * b(n-1) * b(n-2) / 6 for all n in Z where b = A005247. - _Michael Somos_, Sep 22 2014
%e G.f. = 1 + x + 2*x^2 + 7*x^3 + 35*x^4 + 210*x^5 + 1365*x^6 + 9165*x^7 + ...
%t RecurrenceTable[{a[0]==a[1]==1,a[n]==a[n-1] (11a[n-1]-a[n-2])/(a[n-1]+ 4a[n-2])},a,{n,30}] (* or *) LinearRecurrence[{11,-33,33,-11,1},{1,1,2,7,35},30] (* _Harvey P. Dale_, Nov 18 2013 *)
%o (PARI) {a(n) = (4 + fibonacci(4*n - 1)/3 + fibonacci(4*n - 3)/3 + 5 * fibonacci(2*n - 1)) / 10};
%o (PARI) {a(n) = my(A); if( n<1, n = 1-n); if( n<3, n, A = vector(n, k, k); for(k=3, n, A[k] = A[k-1] * (11*A[k-1] - A[k-2]) / (A[k-1] + 4*A[k-2])); A[n])}; /* _Michael Somos_, Sep 22 2014 */
%Y Cf. A005247, A001519, A049685.
%K nonn,easy
%O 0,3
%A _Michael Somos_, Feb 05 2010
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