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%I #7 Jun 07 2023 18:42:48
%S 0,9,720,56880,4492809,354875040,28030635360,2214065318409,
%T 174883129518960,13813553166679440,1091095817038156809,
%U 86182755992847708480,6807346627617930813120,537694200825823686528009,42471034518612453304899600,3354674032769557987400540400
%N The list of the k values in the common solutions to the 2 equations 7*k+1=A^2, 11*k+1=B^2.
%C The 2 equations are equivalent to the Pell equation x^2-77*y^2=1,
%C with x=(77*k+9)/2 and y= A*B/2, case C=7 in A160682.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (80,-80,1)
%F k(t+3)=80*(k(t+2)-k(t+1))+k(t).
%F k(t)=((9+w)*((79+9*w)/2)^(t-1)+(9-w)*((79-9*w)/2)^(t-1))/154 where w=sqrt(77).
%F k(t) = floor of ((9+w)*((79+9*w)/2)^(t-1))/154.
%F G.f.: -9*x^2/((x-1)*(x^2-79*x+1)).
%p t:=0: for n from 0 to 1000000 do a:=sqrt(7*n+1): b:=sqrt(11*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[{80,-80,1},{0,9,720},20] (* _Harvey P. Dale_, Jun 07 2023 *)
%Y Cf. A160682, A070998 (sequence of A), A057081 (sequence of B)
%K nonn,easy
%O 1,2
%A _Paul Weisenhorn_, Jun 14 2009
%E Edited, extended by _R. J. Mathar_, Sep 02 2009
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