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%I #34 Jun 17 2019 03:32:27
%S 3,27,240,2133,18957,168480,1497363,13307787,118272720,1051146693,
%T 9342047517,83027280960,737903481123,6558104049147,58285032961200,
%U 518007192601653,4603779700453677,40916010111481440,363640311302879283
%N a(n+2) = 9*a(n+1) - a(n), a(0) = 3, a(1) = 27.
%C Original definition: a(0) = c, a(1) = p*c^3; a(n+2) = p*c^2*a(n+1) - a(n), for p = 1, c = 3.
%C The sequence could have started with a(0) = 0, then a(1) = 3 etc. Any of the family of sequences x = (0, c, c^3, c^5 - c, ...) with x(n+1) = c^2*x(n) - x(n-1), c >= 1, provides a subset of solutions to A115169 (for n >= 2), their union yields all the solutions. See A052530 for c = 2. - _M. F. Hasler_, Jun 12 2019
%H Harry J. Smith, <a href="/A065100/b065100.txt">Table of n, a(n) for n = 0..100</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H J.-P. Ehrmann et al., <a href="http://forumgeom.fau.edu/POLYA/ProblemCenter/POLYA002.html">Problem POLYA002</a>, Integer pairs (x,y) for which (x^2+y^2)/(1+pxy) is an integer.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (9,-1).
%F G.f.: 3/(1 - 9*x + x^2). - _Floor van Lamoen_, Feb 07 2002
%F a(n) = 3*A018913(n+1). - _R. J. Mathar_, Oct 26 2009
%F a(n) = 9*a(n-1) - a(n-2) (with a(0)=3, a(1)=27). - _Vincenzo Librandi_, Aug 07 2010
%e From _Vincenzo Librandi_, Aug 07 2010: (Start)
%e a(2) = 9*27 - 3 = 240;
%e a(3) = 9*240 - 27 = 2133;
%e a(4) = 9*2133 - 240 = 18957. (End)
%t a[0] = c; a[1] = p*c^3; a[n_] := a[n] = p*c^2*a[n - 1] - a[n - 2]; p = 1; c = 3; Table[ a[n], {n, 0, 20} ]
%t LinearRecurrence[{9,-1},{3,27},30] (* _Harvey P. Dale_, Sep 22 2016 *)
%o (PARI) polya002(1,3,20) \\ See A052530 for definition of function polya002().
%o (PARI) { p=1; c=3; k=p*c^2; for (n=0, 100, if (n>1, a=k*a1 - a2; a2=a1; a1=a, if (n, a=a1=k*c, a=a2=c)); write("b065100.txt", n, " ", a) ) } \\ _Harry J. Smith_, Oct 07 2009
%Y Cf. A052530 (analog for c = 2).
%K easy,nonn
%O 0,1
%A _N. J. A. Sloane_, Nov 12 2001
%E Definition simplified by _M. F. Hasler_, Jun 12 2019
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