OFFSET
1,1
COMMENTS
(-93, a(1)) and (A101152(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+569)^2 = y^2.
Lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
Lim_{n -> infinity} a(n)/a(n-1) = (587+102*sqrt(2))/569 for n mod 3 = {0, 2}.
Lim_{n -> infinity} a(n)/a(n-1) = (617139+371510*sqrt(2))/569^2 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)=485, a(2)=569, a(3)=689, a(4)=2221, a(5)=2845, a(6)=3649.
G.f.: (1-x)*(485 +1054*x +1743*x^2 +1054*x^3 +485*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 569*A001653(k) for k >= 1.
EXAMPLE
MATHEMATICA
LinearRecurrence[{0, 0, 6, 0, 0, -1}, {485, 569, 689, 2221, 2845, 3649}, 50] (* G. C. Greubel, Apr 21 2018 *)
PROG
(PARI) {forstep(n=-96, 10000000, [3, 1], if(issquare(2*n^2+1138*n+323761, &k), print1(k, ", ")))}
(PARI) x='x+O('x^30); Vec((1-x)*(485 +1054*x +1743*x^2 +1054*x^3 +485*x^4)/(1-6*x^3+x^6)) \\ G. C. Greubel, Apr 21 2018
(Magma) I:=[485, 569, 689, 2221, 2845, 3649]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, Apr 21 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Klaus Brockhaus, May 04 2009
STATUS
approved