OFFSET
1,1
COMMENTS
(-29,a(1)) and (A130004(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+449)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (451+30*sqrt(2))/449 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (507363+329222*sqrt(2))/449^2 for n mod 3 = 1.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
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)=421, a(2)=449, a(3)=481, a(4)=2045, a(5)=2245, a(6)=2465.
G.f.: (1-x)*(421+870*x+1351*x^2+870*x^3+421*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 449*A001653(k) for k >= 1.
EXAMPLE
MATHEMATICA
LinearRecurrence[{0, 0, 6, 0, 0, -1}, {421, 449, 481, 2045, 2245, 2465}, 50] (* G. C. Greubel, May 08 2018 *)
PROG
(PARI) {forstep(n=-32, 50000000, [3, 1], if(issquare(2*n^2+898*n+201601, &k), print1(k, ", ")))}
(PARI) x='x+O('x^30); Vec((1-x)*(421+870*x+1351*x^2+870*x^3+421*x^4)/(1- 6*x^3+x^6)) \\ G. C. Greubel, May 08 2018
(Magma) I:=[421, 449, 481, 2045, 2245, 2465]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, May 08 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Klaus Brockhaus, Apr 18 2009
STATUS
approved