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A161591 The list of the B values in the common solutions to the 2 equations 13*k + 1 = A^2, 17*k + 1 = B^2.

%I A161591 #15 Dec 07 2019 12:18:26
%S A161591 1,16,239,3569,53296,795871,11884769,177475664,2650250191,39576277201,
%T A161591 590993907824,8825332340159,131788991194561,1968009535578256,
%U A161591 29388354042479279,438857301101610929,6553471162481684656,97863210136123658911,1461394680879373199009
%N A161591 The list of the B values in the common solutions to the 2 equations 13*k + 1 = A^2, 17*k + 1 = B^2.
%C A161591 The 2 equations are equivalent to the Pell equation x^2 - 221*y^2 = 1, with x = (221*k+15)/2 and y = A*B/2, case C=13 in A160682.
%H A161591 Andersen, K., Carbone, L. and Penta, D., <a href="https://pdfs.semanticscholar.org/8f0c/c3e68d388185129a56ed73b5d21224659300.pdf">Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields</a>, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.
%H A161591 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (15,-1).
%F A161591 B(t+2) = 15*B(t+1) - B(t).
%F A161591 B(t) = ((221+17*w)*((15+w)/2)^(t-1) + (221-17*w)*((15-w)/2)^(t-1))/442 where w=sqrt(221).
%F A161591 B(t) = floor of ((221+17*w)*((15+w)/2)^(t-1))/442 = A078364(t-2) + A078364(t-1).
%F A161591 G.f.: x*(1+x)/(1-15*x+x^2).
%p A161591 t:=0: for b from 1 to 1000000 do a:=sqrt((13*b^2+4)/17):
%p A161591 if (trunc(a)=a) then t:=t+1: n:=(b^2-1)/17: print(t,a,b,n): end if: end do:
%t A161591 LinearRecurrence[{15,-1},{1,16},30] (* _Harvey P. Dale_, Dec 04 2015 *)
%o A161591 (Sage) [(lucas_number2(n,15,1)-lucas_number2(n-1,15,1))/13 for n in range(1, 20)] # _Zerinvary Lajos_, Nov 10 2009
%Y A161591 Cf. A160682 (sequence of A), A161584 (sequence of k).
%K A161591 nonn
%O A161591 1,2
%A A161591 _Paul Weisenhorn_, Jun 14 2009
%E A161591 Edited, extended by _R. J. Mathar_, Sep 02 2009