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The list of the k values in the common solutions to the 2 equations 13*k+1=A^2, 17*k+1=B^2.
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%I #7 Jul 31 2015 22:14:27

%S 0,15,3360,749280,167086095,37259449920,8308690246080,

%T 1852800665425935,413166239699737440,92134218652376023200,

%U 20545517593240153436175,4581558289073901840243840,1021666952945886870220940160

%N 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 The 2 equations are equivalent to the Pell equation x^2-221*y^2=1,

%C with x=(221*k+15)/2 and y= A*B/2, case C=13 of A160682.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (224, -224, 1).

%F k(t+3)=224*(k(t+2)-k(t+1))+k(t).

%F k(t)=((15+w)*((223+15*w)/2)^(t-1)+(15-w)*((223-15*w)/2)^(t-1))/442 where w=sqrt(221).

%F k(t) = floor of ((15+w)*((223+15*w)/2)^(t-1))/442;

%F G.f.: -15*x^2/((x-1)*(x^2-223*x+1)).

%F 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 t:=0: for n from 0 to 1000000 do a:=sqrt(13*n+1): b:=sqrt(17*n+1):

%p if (trunc(a)=a) and (trunc(b)=b) then t:=t+1: print(t,n,a,b): end if: end do:

%t LinearRecurrence[{224,-224,1},{0,15,3360},20] (* _Harvey P. Dale_, Nov 22 2013 *)

%Y Cf. A160682 (sequence of A), A161591 (sequence of B)

%K nonn

%O 1,2

%A _Paul Weisenhorn_, Jun 14 2009

%E Edited, extended by _R. J. Mathar_, Sep 02 2009