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A189735 a(1)=3, a(2)=1, a(n) = 3*a(n-1) + a(n-2).

%I A189735 #57 Jun 23 2024 11:50:47
%S A189735 3,1,6,19,63,208,687,2269,7494,24751,81747,269992,891723,2945161,
%T A189735 9727206,32126779,106107543,350449408,1157455767,3822816709,
%U A189735 12625905894,41700534391,137727509067,454883061592,1502376693843,4962013143121,16388416123206,54127261512739
%N A189735 a(1)=3, a(2)=1, a(n) = 3*a(n-1) + a(n-2).
%C A189735 This is one of two sequences of positive integers a(n) such that 13*a(n)^2 + 68*(-1)^n = b(n)^2. For n >= 2, b(n)^2 = 13*a(n-1)*a(n+1) - 153*(-1)^n and in this sequence with the values of b(n) = a(n-1) + a(n+1) = {9, 20, 69, 227, ...}. The other sequence is {8, 27, 89, 294, ...} with offset 0 and signature (3,1). - _Klaus Purath_, Aug 17 2021
%H A189735 Vincenzo Librandi, <a href="/A189735/b189735.txt">Table of n, a(n) for n = 1..100</a>
%H A189735 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,1).
%F A189735 G.f.: x*(3-8*x)/(1-3*x-x^2). - _Bruno Berselli_, May 24 2011
%F A189735 a(n) = (a(n-1)*a(n-2) + 51*(-1)^n)/a(n-3), n >= 4; a(n) = (a(n-1)^2 - 17*(-1)^n)/a(n-2), n >= 3. - _Klaus Purath_, Aug 17 2021
%F A189735 a(n+3) = 3^(n+1) + Sum_{k=0..n} a(k+1)*3^(n-k). - _Greg Dresden_ and Canran Wang, Jun 13 2024
%t A189735 LinearRecurrence[{3,1},{3,1},40]
%o A189735 (Maxima) a[1]:3$ a[2]:1$ a[n]:=3*a[n-1]+a[n-2]$ makelist(a[n], n, 1, 28); /* _Bruno Berselli_, May 24 2011 */
%o A189735 (PARI) v=vector(99);v[1]=3;v[2]=1;for(i=3,#v,v[i]=3*v[i-1]+v[i-2]);v \\ _Charles R Greathouse IV_, May 24 2011
%K A189735 nonn,easy
%O A189735 1,1
%A A189735 _Harvey P. Dale_, Apr 26 2011