OFFSET
1,2
COMMENTS
Also values x of Pythagorean triples (x, x+89, y).
Corresponding values y of solutions (x, y) are in A160055.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (107+42*sqrt(2))/89 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (8979+2990*sqrt(2))/89^2 for n mod 3 = 0.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).
FORMULA
a(n) = 6*a(n-3) - a(n-6) + 178 with for n > 6; a(1)=0, a(2)=51, a(3)=120, a(4)=267, a(5)=540, a(6)=931.
G.f.: x*(51+69*x+147*x^2-33*x^3-23*x^4-33*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 89*A001652(k), k >= 0. (Zak Seidov, May 28, 2007)
a(1)=0, a(2)=51, a(3)=120, a(4)=267, a(5)=540, a(6)=931, a(7)=1780, a(n) = a(n-1) + 6*a(n-3) - 6*a(n-4) - a(n-6) + a(n-7). - Harvey P. Dale, Sep 21 2013
MATHEMATICA
LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 51, 120, 267, 540, 931, 1780}, 30] (* Harvey P. Dale, Sep 21 2013 *)
PROG
(PARI) {forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+178*n+7921), print1(n, ", ")))};
(PARI) x='x+O('x^30); concat(0, Vec(x*(51+69*x+147*x^2-33*x^3-23*x^4-33*x^5)/((1-x)*(1-6*x^3+x^6)))) \\ G. C. Greubel, Apr 19 2018
(Magma) I:=[0, 51, 120, 267, 540, 931, 1780]; [n le 7 select I[n] else Self(n-1) +6*Self(n-3) -6*Self(n-4) - Self(n-6) + Self(n-7): n in [1..30]]; // G. C. Greubel, Apr 19 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Mohamed Bouhamida, May 26 2007
EXTENSIONS
Edited and three terms added by Klaus Brockhaus, May 04 2009
STATUS
approved