A157456 Expansion of x * (1 - x) / (1 - 16*x + x^2).

1, 15, 239, 3809, 60705, 967471, 15418831, 245733825, 3916322369, 62415424079, 994730462895, 15853271982241, 252657621252961, 4026668668065135, 64174041067789199, 1022757988416562049, 16299953773597203585, 259776502389138695311, 4140124084452621921391

Offset: 1

Author

Paul Weisenhorn, Mar 01 2009

Keywords

• nonn
• easy

Comments

Positive values of x (or y) satisfying x^2 - 16xy + y^2 + 14 = 0. - Colin Barker, Feb 11 2014

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
• Index entries for linear recurrences with constant coefficients, signature (16,-1).

Crossrefs

Cf. A159678.

Cf. similar sequences listed in A238379.

Programs

Magma

I:=[1,15]; [n le 2 select I[n] else 16*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 12 2014

Maple

f:= gfun:-rectoproc({a(n)=16*a(n-1)-a(n-2),a(1)=1,a(2)=15},a(n),remember):
map(f, [$1..30]); # Robert Israel, Jul 07 2015

Mathematica

CoefficientList[Series[(1 - x)/(1 - 16 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 12 2014 *)
LinearRecurrence[{16,-1},{1,15},20] (* Harvey P. Dale, Sep 17 2019 *)

Formula

G.f.: x*(1-x) / ( 1-16*x+x^2 ). - R. J. Mathar, Oct 31 2011

a(n) = 16*a(n-1)-a(n-2). - Colin Barker, Feb 11 2014

a(n) = (1/18)*(9-sqrt(63))*(1+(8+sqrt(63))^(2*n-1))/(8+sqrt(63))^(n-1). [Bruno Berselli, Feb 25 2014]

a(n) = sqrt(2+(8-3*sqrt(7))^(1+2*n)+(8+3*sqrt(7))^(1+2*n))/(3*sqrt(2)). - Gerry Martens, Jun 06 2015

a(n) = A077412(n-1) - A077412(n-2). - R. J. Mathar, Feb 05 2020

Extensions

New name (using the g.f. by R. J. Mathar) from Joerg Arndt, Jun 06 2015