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%I #44 Sep 08 2022 08:46:06
%S 1,2,7,28,117,494,2091,8856,37513,158906,673135,2851444,12078909,
%T 51167078,216747219,918155952,3889371025,16475640050,69791931223,
%U 295643364940,1252365390981,5305104928862,22472785106427,95196245354568,403257766524697,1708227311453354,7236167012338111,30652895360805796
%N Expansion of (1-3*x)/(1-5*x+3*x^2+x^3).
%H Vincenzo Librandi, <a href="/A232970/b232970.txt">Table of n, a(n) for n = 0..1000</a>
%H M. Dziemianczuk, <a href="http://paperity.org/p/34654227/counting-lattice-paths-with-four-types-of-steps">Counting Lattice Paths With Four Types of Steps</a>, Graphs and Combinatorics, September 2013, DOI 10.1007/s00373-013-1357-1.
%H Hermann Stamm-Wilbrandt, <a href="/A232970/a232970_2.svg">6 interlaced bisections</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-3,-1).
%F a(n) = 5*a(n-1) - 3*a(n-2) - a(n-3). - _N. J. A. Sloane_, Jun 23 2017
%F a(n) = (Fibonacci(3*n+1) + 1)/2 = Sum_{k=0..n} Fibonacci(3*k-1). - _Ehren Metcalfe_, Apr 15 2019
%F a(2*n) = A294262(2*n); a(2*n+1) = A254627(2*n+2). See "6 interlaced bisections" link. - _Hermann Stamm-Wilbrandt_, Apr 18 2019
%t LinearRecurrence[{5, -3, -1}, {1, 2, 7}, 30] (* _Vincenzo Librandi_, Jun 24 2017 *)
%o (Magma) I:=[1,2,7]; [n le 3 select I[n] else 5*Self(n-1)- 3*Self(n-2)-Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Jun 24 2017
%o (PARI) Vec((1-3*x)/(1-5*x+3*x^2+x^3) + O(x^30)) \\ _Felix Fröhlich_, Apr 15 2019
%o (Sage) [(fibonacci(3*n+1) +1)/2 for n in (0..30)] # _G. C. Greubel_, Apr 19 2019
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Dec 05 2013
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