%I #34 Jan 02 2024 08:53:00

%S 1,6,32,167,868,4508,23409,121554,631180,3277455,17018456,88369736,

%T 458867137,2382705422,12372394248,64244676663,333595777564,

%U 1732223564484,8994713599985,46705791564410,242523671422036

%N a(1)=1, a(n) = ceiling(r(5)*a(n-1)) where r(5) = (1/2)*(5 + sqrt(29)) is the positive root of X^2 = 5*X + 1.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-4,-1).

%F For n > 3, a(n) = 6*a(n-1) - 4*a(n-2) - a(n-3); a(n) = floor(t(5)*r(5)^n) where t(5) = (1/10)*(1 + 7/sqrt(29)) is the positive root of 145*X^2 = 29*X + 1.

%F G.f.: x/((x-1)*(x^2+5*x-1)). - _Colin Barker_, Jan 27 2013

%F G.f.: 1/(1/Q(0) + 3*x^3 - 3*x) where Q(k) = 1 + k*(2*x+1) + 8*x - 2*x*(k+1)*(k+5)/Q(k+1) ; (continued fraction). - _Sergei N. Gladkovskii_, Mar 15 2013

%p a:=n->sum(fibonacci(i,5), i=0..n): seq(a(n), n=1..21); # _Zerinvary Lajos_, Mar 20 2008

%t Join[{a=1,b=6},Table[c=5*b+1*a+1;a=b;b=c,{n,60}]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 06 2011*)

%t LinearRecurrence[{6,-4,-1},{1,6,32},30] (* _Harvey P. Dale_, Jan 06 2012 *)

%Y Cf. A000071, A048739, A049652, A082574.

%K nonn,easy

%O 1,2

%A _Benoit Cloitre_, May 07 2003