OFFSET
1,1
COMMENTS
(-136, a(1)) and (A129641(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+409)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (473+168*sqrt(2))/409 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (204819+83570*sqrt(2))/409^2 for n mod 3 = 1.
LINKS
Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).
FORMULA
a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=305, a(2)=409, a(3)=641, a(4)=1189, a(5)=2045, a(6)=3541.
G.f.: (1-x)*(305+714*x+1355*x^2+714*x^3+305*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 409*A001653(k) for k >= 1.
EXAMPLE
MATHEMATICA
LinearRecurrence[{0, 0, 6, 0, 0, -1}, {305, 409, 641, 1189, 2045, 3541}, 50] (* or *) Select[Table[Sqrt[x^2+(x+409)^2], {x, -140, 10^6}], IntegerQ] (* The second program generates the first 16 terms of the sequence. To generate more, increase the x constant but the program may take a long time to run. *) (* Harvey P. Dale, Mar 14 2022 *)
PROG
(PARI) {forstep(n=-136, 10000000, [3, 1], if(issquare(2*n^2+818*n+167281, &k), print1(k, ", ")))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Klaus Brockhaus, Jun 08 2009
STATUS
approved