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A161583 The list of the k values in the common solutions to the 2 equations 15*k+1=A^2, 19*k+1=B^2.

%I A161583 #5 Jul 03 2023 11:15:14
%S A161583 0,17,4896,1405152,403273745,115738159680,33216448554432,
%T A161583 9533004996962321,2735939217679631712,785205022469057339040,
%U A161583 225351105509401776672785,64674982076175840847750272,18561494504756956921527655296,5327084247883170460637589319697
%N A161583 The list of the k values in the common solutions to the 2 equations 15*k+1=A^2, 19*k+1=B^2.
%C A161583 The 2 equations are equivalent to the Pell equation x^2-285*y^2=1,
%C A161583 with x=(285*k+17)/2 and y=A*B/2, case C=15 in A160682.
%H A161583 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (288, -288, 1).
%F A161583 k(t+3)=288*(k(t+2)-k(t+1))+k(t).
%F A161583 k(t)=((17+w)*((287+17*w)/2)^(t-1)+(17-w)*((287-17*w)/2)^(t-1))/570 where w=sqrt(285).
%F A161583 k(t) = floor of ((17+w)*((287+17*w)/2)^(t-1))/570;
%F A161583 G.f.: -17*x^2/((x-1)*(x^2-287*x+1)).
%p A161583 t:=0: for n from 0 to 1000000 do a:=sqrt(15*n+1): b:=sqrt(19*n+1):
%p A161583 if (trunc(a)=a) and (trunc(b)=b) then t:=t+1: print(t,n,a,b): end if: end do:
%Y A161583 Cf. A160682, A161595 (sequence of A), A161599 (sequence of B)
%K A161583 nonn
%O A161583 1,2
%A A161583 _Paul Weisenhorn_, Jun 14 2009
%E A161583 Edited, extended by _R. J. Mathar_, Sep 02 2009