OFFSET
1,1
COMMENTS
(-297, a(1)) and (A111258(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+601)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (843+418*sqrt(2))/601 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (361299+5950*sqrt(2))/601^2 for n mod 3 = 1.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..3875
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)=425, a(2)=601, a(3)=1261, a(4)=1289, a(5)=3005, a(6)=7141.
G.f.: (1-x)*(425 +1026*x +2287*x^2 +1026*x^3 +425*x^4)/(1-6*x^3+x^6).
a(3*k-1) = 601*A001653(k) for k >= 1.
EXAMPLE
MATHEMATICA
LinearRecurrence[{0, 0, 6, 0, 0, -1}, {425, 601, 1261, 1289, 3005, 7141}, 50] (* G. C. Greubel, Apr 22 2018 *)
PROG
(PARI) {forstep(n=-300, 10000000, [3, 1], if(issquare(2*n^2+1202*n+361201, &k), print1(k, ", ")))}
(PARI) x='x+O('x^30); Vec((1-x)*(425 +1026*x +2287*x^2 +1026*x^3 +425*x^4 )/(1-6*x^3+x^6)) \\ G. C. Greubel, Apr 22 2018
(Magma) I:=[425, 601, 1261, 1289, 3005, 7141]; [n le 6 select I[n] else 5*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, Apr 22 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Klaus Brockhaus, May 18 2009
STATUS
approved