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A088132 a(n) equals the square of the n-th partial sum added to twice the n-th partial sum of the squares, divided by a(n-1), for all n>1, with a(0)=1, a(1)=3.

%I A088132 #32 Sep 08 2022 08:45:11
%S A088132 1,3,12,47,185,728,2865,11275,44372,174623,687217,2704496,10643361,
%T A088132 41886227,164840412,648718287,2552986921,10047107272,39539710801,
%U A088132 155605856283,612376317860,2409965560639,9484256386273,37324649227232
%N A088132 a(n) equals the square of the n-th partial sum added to twice the n-th partial sum of the squares, divided by a(n-1), for all n>1, with a(0)=1, a(1)=3.
%H A088132 Vincenzo Librandi, <a href="/A088132/b088132.txt">Table of n, a(n) for n = 0..200</a>
%H A088132 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,0,-1).
%F A088132 a(n) = 4*a(n-1) - a(n-3) for n>3.
%F A088132 G.f.: (1-x)/(1-4*x+x^3).
%F A088132 G.f.: 1/(x - x^2*Sum_{n>=0} A030186(n)*x^n) - 1/x.
%p A088132 seq(coeff(series((1-x)/(1-4*x+x^3), x, n+1), x, n), n = 0..30); # _G. C. Greubel_, Oct 26 2019
%t A088132 LinearRecurrence[{4,0,-1}, {1,3,12}, 30] (* or *) CoefficientList[Series[ (1-x)/(1-4x+x^3), {x,0,30}], x] (* _Harvey P. Dale_, Jun 24 2011 *)
%o A088132 (PARI) {a(n)=if(n==0,1, if(n==1,3, (sum(k=0, n-1, a(k))^2 + 2*sum(k=0, n-1, a(k)^2))/a(n-1)))}
%o A088132 for(n=0,20,print1(a(n),", ")) \\ _Paul D. Hanna_, Feb 20 2014
%o A088132 (PARI) Vec( (1-x)/(1-4*x+x^3) + O(x^66) ) \\ _Joerg Arndt_, Feb 16 2014
%o A088132 (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)/(1-4*x+x^3) )); // _G. C. Greubel_, Oct 26 2019
%o A088132 (Sage)
%o A088132 def A088132_list(prec):
%o A088132     P.<x> = PowerSeriesRing(ZZ, prec)
%o A088132     return P((1-x)/(1-4*x+x^3)).list()
%o A088132 A088132_list(30) # _G. C. Greubel_, Oct 26 2019
%o A088132 (GAP) a:=[1,3,12];; for n in [4..30] do a[n]:=34a[n-1]-a[n-3]; od; a; # _G. C. Greubel_, Oct 26 2019
%Y A088132 Cf. A030186, A088131.
%K A088132 nonn,easy
%O A088132 0,2
%A A088132 _Paul D. Hanna_, Sep 19 2003