A153177 - OEIS

%I #24 Sep 08 2022 08:45:39

%S 76,1926,109801,4769326,230701876,10716675201,505618944676,

%T 23714405408926,1114769987764201,52357935173823126,

%U 2459933168462154076,115560463558534156801,5428954301161174383676,255043991670277234750326

%N a(n) = L(9*n)/L(n) where L(n) = Lucas number A000204(n).

%C All numbers in this sequence are:

%C congruent to 1 mod 100 (iff n is congruent to 0 mod 3),

%C congruent to 26 mod 100 (iff n is congruent to 2 or 4 mod 6),

%C congruent to 76 mod 100 (iff n is congruent to 1 or 5 mod 6).

%H G. C. Greubel, <a href="/A153177/b153177.txt">Table of n, a(n) for n = 1..595</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (34,714,-4641,-12376,12376,4641,-714,-34,1).

%F From _R. J. Mathar_, Oct 22 2010: (Start)

%F a(n) = 34*a(n-1) +714*a(n-2) -4641*a(n-3) -12376*a(n-4) +12376*a(n-5) +4641*a(n-6) -714*a(n-7) -34*a(n-8) +a(n-9).

%F G.f.: -x*(76-658*x-9947*x^2+13644*x^3+26020*x^4-5306*x^5-1372*x^6+42*x^7 +x^8) / ((x-1)*(x^2+18*x+1)*(x^2-47*x+1)*(x^2+3*x+1)*(x^2-7*x+1)).

%F a(n) = 1-(-1)^n*A087215(n) -(-1)^n*A005248(n) +A056854(n) +A087265(n). (End)

%t Table[LucasL[9*n]/LucasL[n], {n, 1, 50}]

%t LinearRecurrence[{34,714,-4641,-12376,12376,4641,-714,-34,1},{76,1926,109801,4769326,230701876,10716675201,505618944676,23714405408926,1114769987764201},20] (* _Harvey P. Dale_, Aug 12 2012 *)

%o (PARI) {lucas(n) = fibonacci(n+1) + fibonacci(n-1)};

%o for(n=0,30, print1( lucas(9*n)/lucas(n), ", ")) \\ _G. C. Greubel_, Dec 21 2017

%o (Magma) [Lucas(9*n)/Lucas(n): n in [0..30]]; // _G. C. Greubel_, Dec 21 2017

%Y Cf. A000032, A000204, A110391, A153173, A153175.

%Y Cf. A153179, A153180. - _R. J. Mathar_, Oct 22 2010

%K nonn

%O 1,1

%A _Artur Jasinski_, Dec 20 2008