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A051959 Expansion of (1+6x)/( (1-2x-x^2)(1-x)^2). 1

%I #23 Sep 08 2022 08:44:59

%S 1,10,36,104,273,686,1688,4112,9969,24114,58268,140728,339809,820438,

%T 1980784,4782112,11545121,27872474,67290196,162453000,392196337,

%U 946845822,2285888136

%N Expansion of (1+6x)/( (1-2x-x^2)(1-x)^2).

%H Vincenzo Librandi, <a href="/A051959/b051959.txt">Table of n, a(n) for n = 0..1000</a>

%H A. F. Horadam, <a href="http://www.fq.math.ca/Scanned/5-5/horadam.pdf">Special Properties of the Sequence W(n){a,b; p,q}</a>, Fib. Quart., Vol. 5, No. 5 (1967), pp. 424-434.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,0,1).

%F a(n) = 2a(n-1) + a(n-2) + (7n+1); a(0)=1, a(1)=10.

%F a(n) = ((((25/2)+17*sqrt(2)/2)(1+sqrt(2))^n - ((25/2)-17*sqrt(2)/2)(1-sqrt(2))^n)/2*sqrt(2)) - (7n+15)/2

%F (1/2) (4*Pell(n+2) - 3*Pell(n) - 7n - 15), with Pell(n) = A000129(n). - _Ralf Stephan_, May 15 2007

%F a(n) = 4*a(n-1) - 4*a(n-2) + a(n-4).

%t LinearRecurrence[{4,-4,0,1},{1,10,36,104},40] (* _Vincenzo Librandi_, Jun 22 2012 *)

%o (Magma) I:=[1, 10, 36, 104]; [n le 4 select I[n] else 4*Self(n-1)-4*Self(n-2)+Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, Jun 22 2012

%Y Cf. A048771, A048695.

%K easy,nonn

%O 0,2

%A _Barry E. Williams_, Jan 04 2000

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Last modified March 28 09:04 EDT 2024. Contains 371240 sequences. (Running on oeis4.)