OFFSET
1,3
COMMENTS
Each pair of terms {odd=x, even=y} gives a solution to the Pell equation x^2 - 2y^2 = 1. Note that odd/even terms also have odd/even indices. The ratio a(2k-1)/a(2k) tends to sqrt(2). Interrelations between odd and even terms: a(2k+1) = 3a(2k-1) + 4a(2k); e.g., 99 = 3*17 + 4*12, 577 = 3*99 + 4*70; a(2k) = 3a(2k-2) + 2a(2k-3), e.g., 70 = 3*12 + 2*17, 408 = 3*70 + 2*99. Odd terms = A001541, even terms = 2*A001109.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).
FORMULA
O.g.f.: x*(2*x+1)*(-1+x)^2/((x^2-2*x-1)*(x^2+2*x-1)). - R. J. Mathar, Dec 10 2007
MATHEMATICA
LinearRecurrence[{0, 6, 0, -1}, {1, 0, 3, 2}, 35] (* G. C. Greubel, Mar 16 2019 *)
PROG
(PARI) my(x='x+O('x^35)); Vec(x*(1+2*x)*(1-x)^2/((1-2*x-x^2)*(1+2*x-x^2))) \\ G. C. Greubel, Mar 16 2019
(Magma) I:=[1, 0, 3, 2]; [n le 4 select I[n] else 6*Self(n-2)-Self(n-4): n in [1..35]]; // G. C. Greubel, Mar 16 2019
(Sage) a=(x*(1+2*x)*(1-x)^2/((1-2*x-x^2)*(1+2*x-x^2))).series(x, 35).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Mar 16 2019
(GAP) a:=[1, 0, 3, 2];; for n in [5..35] do a[n]:=6*a[n-2]-a[n-4]; od; a; # G. C. Greubel, Mar 16 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Zak Seidov, Dec 26 2006
STATUS
approved