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A161584 The list of the k values in the common solutions to the 2 equations 13*k+1=A^2, 17*k+1=B^2.

%I A161584 #7 Jul 31 2015 22:14:27
%S A161584 0,15,3360,749280,167086095,37259449920,8308690246080,
%T A161584 1852800665425935,413166239699737440,92134218652376023200,
%U A161584 20545517593240153436175,4581558289073901840243840,1021666952945886870220940160
%N A161584 The list of the k values in the common solutions to the 2 equations 13*k+1=A^2, 17*k+1=B^2.
%C A161584 The 2 equations are equivalent to the Pell equation x^2-221*y^2=1,
%C A161584 with x=(221*k+15)/2 and y= A*B/2, case C=13 of A160682.
%H A161584 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (224, -224, 1).
%F A161584 k(t+3)=224*(k(t+2)-k(t+1))+k(t).
%F A161584 k(t)=((15+w)*((223+15*w)/2)^(t-1)+(15-w)*((223-15*w)/2)^(t-1))/442 where w=sqrt(221).
%F A161584 k(t) = floor of ((15+w)*((223+15*w)/2)^(t-1))/442;
%F A161584 G.f.: -15*x^2/((x-1)*(x^2-223*x+1)).
%F A161584 a(1)=0, a(2)=15, a(3)=3360, a(n)=224*a(n-1)-224*a(n-2)+a(n-3). - _Harvey P. Dale_, Nov 22 2013
%p A161584 t:=0: for n from 0 to 1000000 do a:=sqrt(13*n+1): b:=sqrt(17*n+1):
%p A161584 if (trunc(a)=a) and (trunc(b)=b) then t:=t+1: print(t,n,a,b): end if: end do:
%t A161584 LinearRecurrence[{224,-224,1},{0,15,3360},20] (* _Harvey P. Dale_, Nov 22 2013 *)
%Y A161584 Cf. A160682 (sequence of A), A161591 (sequence of B)
%K A161584 nonn
%O A161584 1,2
%A A161584 _Paul Weisenhorn_, Jun 14 2009
%E A161584 Edited, extended by _R. J. Mathar_, Sep 02 2009