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%I #34 Sep 08 2022 08:45:45
%S 1,14,209,3121,46606,695969,10392929,155197966,2317576561,34608450449,
%T 516809180174,7717529252161,115246129602241,1720974414781454,
%U 25699370092119569,383769576967012081,5730844284413061646,85578894689228912609,1277952576054020627489
%N The list of the A values in the common solutions to 13*k+1 = A^2 and 17*k+1 = B^2.
%C This summarizes the case C=13 of common solutions to C*k+1=A^2, (C+4)*k+1=B^2.
%C The 2 equations are equivalent to the Pell equation x^2-C*(C+4)*y^2=1,
%C with x=(C*(C+4)*k+C+2)/2; y=A*B/2 and with smallest values x(1) = (C+2)/2, y(1)=1/2.
%C Generic recurrences are:
%C A(j+2)=(C+2)*A(j+1)-A(j) with A(1)=1; A(2)=C+1.
%C B(j+2)=(C+2)*B(j+1)-B(j) with B(1)=1; B(2)=C+3.
%C k(j+3)=(C+1)*(C+3)*( k(j+2)-k(j+1) )+k(j) with k(1)=0; k(2)=C+2; k(3)=(C+1)*(C+2)*(C+3).
%C x(j+2)=(C^2+4*C+2)*x(j+1)-x(j) with x(1)=(C+2)/2; x(2)=(C^2+4*C+1)*(C+2)/2;
%C Binet-type of solutions of these 2nd order recurrences are:
%C R=C^2+4*C; S=C*sqrt(R); T=(C+2); U=sqrt(R); V=(C+4)*sqrt(R);
%C A(j)=((R+S)*(T+U)^(j-1)+(R-S)*(T-U)^(j-1))/(R*2^j);
%C B(j)=((R+V)*(T+U)^(j-1)+(R-V)*(T-U)^(j-1))/(R*2^j);
%C x(j)+sqrt(R)*y(j)=((T+U)*(C^2*4*C+2+(C+2)*sqrt(R))^(j-1))/2^j;
%C k(j)=(((T+U)*(R+2+T*U)^(j-1)+(T-U)*(R+2-T*U)^(j-1))/2^j-T)/R. [_Paul Weisenhorn_, May 24 2009]
%C .C -A----- -B----- -k-----
%C 01 A001519 A002878 A058038
%C 02 A001653 A002315 A045899/2
%C 03 A004253 A030221 A160695
%C 04 A001653 A002315 A078522/4
%C 05 A049685 A033890 A161582
%C 06 A070997 A057080 A159683/2
%C 07 A070998 A057081 A161585
%C 08 A072256 A054320 A045502/4
%C 09 A078922 A097783 A161586
%C 10 A077417 A077416 A159681/2
%C 11 A085260 A126816 A161588
%C 12 A001570 A028230 A059989/4
%C 13 A160682 A161591 A161584
%C 14 A157456 A159678 A159679/2
%C 15 A161595 A161599 A161583
%C 16 A007805 A049629 A157459/4
%C For n>=2, a(n) equals the permanent of the (2n-2)X(2n-2) tridiagonal matrix with sqrt(13)'s along the main diagonal, and 1's along the superdiagonal and the subdiagonal. [_John M. Campbell_, Jul 08 2011]
%C Positive values of x (or y) satisfying x^2 - 15xy + y^2 + 13 = 0. - _Colin Barker_, Feb 11 2014
%H Vincenzo Librandi, <a href="/A160682/b160682.txt">Table of n, a(n) for n = 1..200</a>
%H J.-C. Novelli, J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
%H <a href="/index/Pea#Pellian">Index entries for the Pell equation</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (15,-1).
%F a(n) = 15*a(n-1)-a(n-2).
%F G.f.: (1-x)*x/(1-15*x+x^2).
%F a(n) = (2^(-1-n)*((15-sqrt(221))^n*(13+sqrt(221))+(-13+sqrt(221))*(15+sqrt(221))^n))/sqrt(221). - _Colin Barker_, Jul 25 2016
%t LinearRecurrence[{15,-1},{1,14},20] (* _Harvey P. Dale_, Oct 08 2012 *)
%t CoefficientList[Series[(1 - x)/(1 - 15 x + x^2), {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 12 2014 *)
%o (Magma) I:=[1,14]; [n le 2 select I[n] else 15*Self(n-1)-Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Feb 12 2014
%o (PARI) a(n) = round((2^(-1-n)*((15-sqrt(221))^n*(13+sqrt(221))+(-13+sqrt(221))*(15+sqrt(221))^n))/sqrt(221)) \\ _Colin Barker_, Jul 25 2016
%Y Cf. similar sequences listed in A238379.
%K nonn,easy
%O 1,2
%A _Paul Weisenhorn_, May 23 2009
%E Edited, extended by _R. J. Mathar_, Sep 02 2009
%E First formula corrected by _Harvey P. Dale_, Oct 08 2012
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