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%I #19 May 24 2021 07:40:37
%S 1,6,15,40,97,238,575,1392,3361,8118,19599,47320,114241,275806,665855,
%T 1607520,3880897,9369318,22619535,54608392,131836321,318281038,
%U 768398399,1855077840,4478554081,10812186006,26102926095,63018038200,152139002497,367296043198
%N Expansion of (1+4*x+x^2)/((1-x^2)*(1-2*x-x^2)); a Pellian-related sequence.
%C Elements of odd index give match to A065113: Sum of the squares of the n-th and the (n+1)st triangular numbers (A000217) is a perfect square.
%C Generating floretion: - 1.5'i + 'j + 'k - .5i' + j' + k' + .5'ii' - .5'jj' - .5'kk' - 'ij' + 'ik' - 'ji' + .5'jk' + 2'ki' - .5'kj' + .5e
%H Colin Barker, <a href="/A114696/b114696.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 G.f.: (1 +4*x +x^2)/((1-x)*(1+x)*(1-2*x-x^2)).
%F a(0)=1, a(1)=6, a(2)=15, a(3)=40, a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4). - _Harvey P. Dale_, Jan 23 2014
%F a(n) = (-3 - (-1)^n + (3-2*sqrt(2))*(1-sqrt(2))^n + (1+sqrt(2))^n*(3+2*sqrt(2)))/2. - _Colin Barker_, May 26 2016
%F From _G. C. Greubel_, May 24 2021: (Start)
%F a(n) = 3*A000129(n+1) + A000129(n) - (3 + (-1)^n)/2.
%F a(n) = (1/2)*(A002203(n+2) - 3 - (-1)^n). (End)
%p Q:= proc(n) option remember; # Q=A002203
%p if n<2 then 2
%p else 2*Q(n-1) + Q(n-2)
%p fi; end:
%p seq((Q(n+2) -3 -(-1)^n)/2, n=0..40); # _G. C. Greubel_, May 24 2021
%t CoefficientList[Series[(1+4*x+x^2)/((1-x^2)*(1-2*x-x^2)), {x,0,30}], x] (* or *) LinearRecurrence[{2,2,-2,-1}, {1,6,15,40}, 30] (* _Harvey P. Dale_, Jan 23 2014 *)
%o (PARI) Vec((1+4*x+x^2)/((1-x^2)*(1-2*x-x^2)) + O(x^30)) \\ _Colin Barker_, May 26 2016
%o (Sage) [(lucas_number2(n+2,2,-1) -3 -(-1)^n)/2 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, A114697.
%K easy,nonn
%O 0,2
%A _Creighton Dement_, Feb 18 2006
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