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A049686 a(n) = Fibonacci(8n)/3.

%I A049686 #31 Jul 19 2024 11:30:52
%S A049686 0,7,329,15456,726103,34111385,1602508992,75283811239,3536736619241,
%T A049686 166151337293088,7805576116155895,366695926122033977,
%U A049686 17226902951619441024,809297742799991694151,38019767008647990184073,1786119751663655546957280,83909608561183162716808087,3941965482623944992143022809
%N A049686 a(n) = Fibonacci(8n)/3.
%C A049686 a(n) = (t(i+4n) - t(i))/(t(i+2n+1) - t(i+2n-1)), where (t) is any sequence of the form t(n+2) = 8t(n+1) - 8t(n) + t(n-1) or of the form t(n+1) = 7t(n) - t(n-1) without regard to initial values as long as t(i+2n+1) - t(i+2n-1) != 0. - _Klaus Purath_, Jun 23 2024
%H A049686 Indranil Ghosh, <a href="/A049686/b049686.txt">Table of n, a(n) for n = 0..597</a>
%H A049686 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A049686 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (47,-1).
%F A049686 a(n) = 47*a(n-1) - a(n-2), n>1. a(0)=0, a(1)=7.
%F A049686 G.f.: 7*x/(1-47*x+x^2).
%F A049686 a(n) = A004187(2n).
%F A049686 a(n) = 7*A049668(n). - _R. J. Mathar_, Oct 26 2015
%F A049686 E.g.f.: 2*exp(47*x/2)*sinh(21*sqrt(5)*x/2)/(3*sqrt(5)). - _Stefano Spezia_, Dec 14 2019
%e A049686 a(2) = F(8 * 2) / 3 = F(16) / 3 = 987 / 3 = 329. - _Indranil Ghosh_, Feb 05 2017
%p A049686 with(combinat); seq( fibonacci(8*n)/3, n=0..20); # _G. C. Greubel_, Dec 14 2019
%t A049686 Fibonacci[8(Range[20]-1)]/3 (* _G. C. Greubel_, Dec 14 2019 *)
%t A049686 LinearRecurrence[{47,-1},{0,7},20] (* _Harvey P. Dale_, Dec 27 2019 *)
%o A049686 (PARI) a(n) = fibonacci(8*n)/3; \\ _Michel Marcus_, Feb 05 2017
%o A049686 (Magma) [Fibonacci(8*n)/3: n in [0..20]]; // _G. C. Greubel_, Dec 14 2019
%o A049686 (Sage) [fibonacci(8*n)/3 for n in (0..20)] # _G. C. Greubel_, Dec 14 2019
%o A049686 (GAP) List([0..20], n-> Fibonacci(8*n)/3 ); # _G. C. Greubel_, Dec 14 2019
%Y A049686 Cf. A000045, A004187, A049668.
%K A049686 nonn,easy
%O A049686 0,2
%A A049686 _Clark Kimberling_
%E A049686 Better description and more terms from _Michael Somos_