A168639 - OEIS

%I #15 Dec 26 2023 12:19:51

%S 0,1,2,4,5,7,10,15,21,29,40,56,78,108,149,206,285,394,544,751,1037,

%T 1432,1977,2729,3767,5200,7178,9908,13676,18877,26056,35965,49642,

%U 68520,94577,130543,180186,248707,343285,473829,654016,902724,1246010,1719840,2373857,3276582

%N Expansion of x*(1 + x^2 - x^3) / ( (1-x)*(1-x-x^4) ).

%C Limiting ratio is: 1.38028, see A086106.

%D R. Pallu de la Barriere, Optimal Control Theory,Dover Publications, New York,1967,pages 339-344

%H G. C. Greubel, <a href="/A168639/b168639.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 (2,-1,0,1,-1)

%F a(n)= 2*a(n-1) -a(n-2) +a(n-4) -a(n-5). - _R. J. Mathar_, Dec 02 2009

%F a(n) = A098578(n) - A098578(n-3) + A098578(n-2). - _R. J. Mathar_, May 23 2013

%t Clear[n, m, v, M, a, b, c, K]

%t M = {{0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {1, 0, 0, 1}}

%t v[0] = {0, 0, 0, 1};

%t v[n_] := v[n] = M.v[n - 1] + {1, 1, 1, 1}

%t a = Table[v[n][[1]], {n, 0, 30}]

%t LinearRecurrence[{2,-1,0,1,-1}, {0, 1, 2, 4, 5}, 50] (* _G. C. Greubel_, Jul 28 2016 *)

%o (PARI) a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; -1,1,0,-1,2]^n*[0;1;2;4;5])[1,1] \\ _Charles R Greathouse IV_, Jul 29 2016

%Y Cf. A098578.

%K nonn,easy

%O 0,3

%A _Roger L. Bagula_ and _Gary W. Adamson_, Dec 01 2009