A081555 a(n) = 6*a(n-1) - a(n-2) - 4, a(0)=3, a(1)=7.

3, 7, 35, 199, 1155, 6727, 39203, 228487, 1331715, 7761799, 45239075, 263672647, 1536796803, 8957108167, 52205852195, 304278004999, 1773462177795, 10336495061767, 60245508192803, 351136554095047, 2046573816377475, 11928306344169799, 69523264248641315

Offset: 0

Author

Mario Catalani (mario.catalani(AT)unito.it), Mar 24 2003

Keywords

• easy
• nonn

Comments

2*(a(2*n+1) + 1) is a perfect square.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for two-way infinite sequences
• Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Crossrefs

Cf. A003499, A051927.

Programs

GAP

a:=[3,7];; for n in [3..30] do a[n]:=6*a[n-1]-a[n-2]-4; od; a; # G. C. Greubel, Aug 13 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (3-14*x+7*x^2)/((1-x)*(1-6*x+x^2)) )); // G. C. Greubel, Aug 13 2019

Maple

seq(coeff(series((3-14*x+7*x^2)/((1-x)*(1-6*x+x^2)), x, n+1), x, n), n = 0 ..30); # G. C. Greubel, Aug 13 2019

Mathematica

a[n_]:= a[n] = 6*a[n-1] -a[n-2] -4; a[0] = 3; a[1] = 7; Table[a[n], {n, 0, 25}]
LinearRecurrence[{7,-7,1}, {3,7,35}, 30] (* G. C. Greubel, Aug 13 2019 *)

PARI

a(n)=1+2*real((3+quadgen(32))^n)

PARI

a(n)=1+2*subst(poltchebi(abs(n)),x,3)

PARI

a(n)=if(n<0,a(-n),1+polsym(1-6*x+x^2,n)[n+1])

Sage

def A081555_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((3-14*x+7*x^2)/((1-x)*(1-6*x+x^2))).list()
A081555_list(30) # G. C. Greubel, Aug 13 2019

Formula

a(n) = A051927(2n).

a(n) = A003499(n) + 1.

a(2n) + 1 = A003499(n)^2.

a(n) = (3 + 2*sqrt(2))^n + (3 - 2*sqrt(2))^n + 1.

G.f.: (3-14*x+7*x^2)/((1-x)*(1-6*x+x^2)).