A126354 a(n) = 6*a(n-2) - a(n-4) for n > 4, with a(1)=1, a(2)=0, a(3)=3, a(4)=2.

1, 0, 3, 2, 17, 12, 99, 70, 577, 408, 3363, 2378, 19601, 13860, 114243, 80782, 665857, 470832, 3880899, 2744210, 22619537, 15994428, 131836323, 93222358, 768398401, 543339720, 4478554083, 3166815962, 26102926097, 18457556052, 152139002499

Offset: 1

Author

Zak Seidov, Dec 26 2006

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A001109, A001541.

Programs

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

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

Mathematica

LinearRecurrence[{0,6,0,-1}, {1,0,3,2}, 35] (* G. C. Greubel, Mar 16 2019 *)

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

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

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