OFFSET
1,2
COMMENTS
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.
LINKS
Index entries for linear recurrences with constant coefficients, signature (48,-48,1).
FORMULA
k(t+3) = 48*(k(t+2)-k(t+1))+k(t).
With w = sqrt(5),
k(t) = ((7+3*w)*((47+21*w)/2)^(t-1)+(7-3*w)*((47-21*w)/2)^(t-1))/90.
k(t) = floor((7+3*w)*((47+21*w)/2)^(t-1)/90) = 7*|A156093(t-1)|.
G.f.: -7*x^2/((x-1)*(x^2-47*x+1)).
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
MAPLE
t:=0: for n from 0 to 1000000 do a:=sqrt(5*n+1); b:=sqrt(9*n+1);
if (trunc(a)=a) and (trunc(b)=b) then t:=t+1; print(t, n, a, b): end if: end do:
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Weisenhorn, Jun 14 2009
EXTENSIONS
Edited, extended by R. J. Mathar, Sep 02 2009
STATUS
approved