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Expansion of (1+x+x^2)/((1-x^2)*(1-2*x-x^2)); a Pellian-related sequence.
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%I #31 May 24 2021 07:35:26

%S 1,3,9,22,55,133,323,780,1885,4551,10989,26530,64051,154633,373319,

%T 901272,2175865,5253003,12681873,30616750,73915375,178447501,

%U 430810379,1040068260,2510946901,6061962063,14634871029,35331704122,85298279275,205928262673

%N Expansion of (1+x+x^2)/((1-x^2)*(1-2*x-x^2)); a Pellian-related sequence.

%C Generating floretion: (- .5'j + .5'k - .5j' + .5k' + 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki')*('i + 'j + i').

%H Colin Barker, <a href="/A114697/b114697.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-2,-1).

%F a(n+2) - 2*a(n+1) + a(n) = A111955(n+2).

%F G.f.: (1+x+x^2)/((1-x)*(1+x)*(1-2*x-x^2)).

%F From _Raphie Frank_, Oct 01 2012: (Start)

%F a(2*n) = A216134(2*n+1).

%F a(2*n+1) = A006452(2*n+3)-1.

%F Lim_{n->infinity} a(n+1)/a(n) = A014176. (End)

%F a(n) = (2*A078343(n+2) - A010694(n))/4. - _R. J. Mathar_, Oct 02 2012

%F From _Colin Barker_, May 26 2016: (Start)

%F a(n) = ( 2*(-3 +(-1)^n) + (6-5*sqrt(2))*(1-sqrt(2))^n + (1+sqrt(2))^n*(6+5*sqrt(2)) )/8.

%F a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) for n>3. (End)

%F a(n) = (3*A002203(n) + 10*A000129(n) - 3 + (-1)^n)/4. - _G. C. Greubel_, May 24 2021

%t Table[(3*LucasL[n, 2] +10*Fibonacci[n, 2] -3 +(-1)^n)/4, {n,0,30}] (* _G. C. Greubel_, May 24 2021 *)

%o (PARI) Vec((1+x+x^2)/((1-x^2)*(1-2*x-x^2)) + O(x^40)) \\ _Colin Barker_, Jun 24 2015

%o (Sage) [(4*lucas_number1(n+2,2,-1) -2*lucas_number1(n+1,2,-1) -3 +(-1)^n)/4 for n in (0..30)] # _G. C. Greubel_, May 24 2021

%Y Cf. A000129, A002203, A005409, A100828, A111954, A113224.

%Y Cf. A114647, A114688, A114689, A114695, A116699.

%K easy,nonn

%O 0,2

%A _Creighton Dement_, Feb 18 2006