OFFSET
1,2
COMMENTS
The 2 equations are equivalent to the Pell equation x^2-285*y^2 = 1, with x = (285*k+17)/2 and y = A*B/2, case C=15 in A160682.
LINKS
Index entries for linear recurrences with constant coefficients, signature (288,-288,1).
FORMULA
a(n+3) = 288*(a(n+2) - a(n+1)) + a(n).
a(n) = ((17+w)*((287+17*w)/2)^(n-1) + (17-w)*((287-17*w)/2)^(n-1))/570 where w=sqrt(285).
a(n) = floor(((17+w)*((287+17*w)/2)^(n-1))/570).
G.f.: -17*x^2/((x-1)*(x^2-287*x+1)).
Sum_{n>=2} 1/a(n) = (17 - sqrt(285))/2. - Amiram Eldar, Jan 29 2026
MAPLE
t:=0: for n from 0 to 1000000 do a:=sqrt(15*n+1): b:=sqrt(19*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[{288, -288, 1}, {0, 17, 4896}, 15] (* Amiram Eldar, Jan 29 2026 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Weisenhorn, Jun 14 2009
EXTENSIONS
Edited, extended by R. J. Mathar, Sep 02 2009
STATUS
approved