OFFSET
1,2
COMMENTS
The Diophantine equation 10*x^2 + k = y^2, for |k| < 10, has integer solutions with the following k values:
k = 1, the nonnegative x values are in A084070;
k = -1, " A097315;
k = 4, " 2*A084070;
k = -4, " 2*A097315;
k = 6, " this sequence;
k = -6, " A221875;
k = 9, " A075836;
k = -9, " A052454.
a(n+1)/a(n) tends alternately to (sqrt(2)+sqrt(5))^2/3 and (2*sqrt(2)+sqrt(5))^2/3; a(n+2)/a(n) tends to A176398^2.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,38,0,-1).
FORMULA
G.f.: x*(1+x)*(1+4*x+x^2)/((1-6*x-x^2)*(1+6*x-x^2)).
a(n) = ((-5*(-1)^n+2*t)*(3+t)^(2*floor(n/2)) - (5*(-1)^n+2*t)*(3-t)^(2*floor(n/2)))/10, where t=sqrt(10).
a(n) = 2*A129556(n) + 1.
a(n)*a(n-3) - a(n-1)*a(n-2) = -36 + 12(-1)^n.
MAPLE
A221874:=proc(q)
local n;
for n from 1 to q do if type(sqrt(10*n^2+6), integer) then print(n);
fi; od; end:
A221874(100000000000000000); # Paolo P. Lava, Feb 11 2013
MATHEMATICA
LinearRecurrence[{0, 38, 0, -1}, {1, 5, 43, 191}, 22]
PROG
(Magma) m:=22; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x)*(1+4*x+x^2)/((1-6*x-x^2)*(1+6*x-x^2))));
(Maxima) makelist(expand(((-5*(-1)^n+2*sqrt(10))*(3+sqrt(10))^(2*floor(n/2))-(5*(-1)^n+2*sqrt(10))*(3-sqrt(10))^(2*floor(n/2)))/10), n, 1, 22);
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Bruno Berselli, Jan 28 2013
STATUS
approved