OFFSET
1,1
COMMENTS
(-15, a(1)) and (A129992(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2 + (x+127)^2 = y^2.
Lim_{n -> infinity} a(n)/a(n-3) = 3 + 2*sqrt(2).
Lim_{n -> infinity} a(n)/a(n-1) = (129 + 16*sqrt(2))/127 for n mod 3 = {0, 2}.
Lim_{n -> infinity} a(n)/a(n-1) = (34947 + 21922*sqrt(2))/127^2 for n mod 3 = 1.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..3900
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)=113, a(2)=127, a(3)=145, a(4)=533, a(5)=635, a(6)=757.
G.f.: (1-x)*(113+240*x+385*x^2+240*x^3+113*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 127*A001653(k) for k >= 1.
EXAMPLE
MATHEMATICA
LinearRecurrence[{0, 0, 6, 0, 0, -1}, {113, 127, 145, 533, 635, 757}, 50] (* Harvey P. Dale, Feb 06 2015 *)
PROG
(PARI) {forstep(n=-16, 500000000, [1, 3], if(issquare(2*n^2+254*n+16129, &k), print1(k, ", ")))}
(Magma) I:=[113, 127, 145, 533, 635, 757]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, Jun 15 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Klaus Brockhaus, Apr 13 2009
STATUS
approved