OFFSET
1,2
COMMENTS
The 2 equations are equivalent to the Pell equation x^2-165*y^2 = 1, with x = (165*k+13)/2 and y = A*B/2, case C=11 in A160682.
LINKS
Index entries for linear recurrences with constant coefficients, signature (168,-168,1).
FORMULA
a(n+3) = 168*(a(n+2)- a(n+1)) + a(n).
a(n) = ((13+w)*((167+13*w)/2)^(n-1) + (13-w)*((167-13*w)/2)^(n-1))/330 where w=sqrt(165).
a(n) = floor(((13+w)*((167+13*w)/2)^(n-1))/330).
G.f.: -13*x^2/((x-1)*(x^2-167*x+1)).
Sum_{n>=2} 1/a(n) = (13 - sqrt(165))/2. - Amiram Eldar, Jan 29 2026
MAPLE
t:=0: for n from 0 to 1000000 do a:=sqrt(11*n+1): b:=sqrt(15*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[{168, -168, 1}, {0, 13, 2184}, 15] (* Amiram Eldar, Jan 29 2026 *)
CoefficientList[Series[-13*x/((x-1)*(x^2-167*x+1)), {x, 0, 20}], x] (* Wesley Ivan Hurt, Feb 12 2026 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Weisenhorn, Jun 14 2009
EXTENSIONS
Edited, extended by R. J. Mathar, Sep 02 2009
STATUS
approved