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A161582 The list of the k values in the common solutions to the 2 equations 5*k+1=A^2, 9*k+1=B^2.

%I A161582 #13 Oct 07 2024 01:06:34
%S A161582 0,7,336,15792,741895,34853280,1637362272,76921173511,3613657792752,
%T A161582 169764995085840,7975341111241735,374671267233275712,
%U A161582 17601574218852716736,826899317018844410887,38846666325666834594960,1824966417989322381552240,85734574979172485098360327
%N A161582 The list of the k values in the common solutions to the 2 equations 5*k+1=A^2, 9*k+1=B^2.
%C A161582 The 2 equations are equivalent to the Pell equation x^2-45*y^2=1, with x=(45*k+7)/2 and y= A*B/2, case C=5 in A160682.
%H A161582 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (48,-48,1).
%F A161582 k(t+3) = 48*(k(t+2)-k(t+1))+k(t).
%F A161582 With w = sqrt(5),
%F A161582 k(t) = ((7+3*w)*((47+21*w)/2)^(t-1)+(7-3*w)*((47-21*w)/2)^(t-1))/90.
%F A161582 k(t) = floor((7+3*w)*((47+21*w)/2)^(t-1)/90) = 7*|A156093(t-1)|.
%F A161582 G.f.: -7*x^2/((x-1)*(x^2-47*x+1)).
%F A161582 a(1)=0, a(2)=7, a(3)=336, a(n) = 48*a(n-1)-48*a(n-2)+a(n-3). - _Harvey P. Dale_, Mar 21 2013
%p A161582 t:=0: for n from 0 to 1000000 do a:=sqrt(5*n+1); b:=sqrt(9*n+1);
%p A161582 if (trunc(a)=a) and (trunc(b)=b) then t:=t+1; print(t,n,a,b): end if: end do:
%t A161582 LinearRecurrence[{48,-48,1},{0,7,336},30] (* or *) Rest[CoefficientList[ Series[ -7x^2/((x-1)(x^2-47x+1)),{x,0,30}],x]] (* _Harvey P. Dale_, Mar 21 2013 *)
%Y A161582 Cf. A160682, A049685 (sequence of A), A033890 (sequence of B).
%K A161582 nonn,easy
%O A161582 1,2
%A A161582 _Paul Weisenhorn_, Jun 14 2009
%E A161582 Edited, extended by _R. J. Mathar_, Sep 02 2009