A265278 Expansion of (x^4+x^3-x^2+x)/(x^3+x^2-3*x+1).

0, 1, 2, 6, 16, 40, 98, 238, 576, 1392, 3362, 8118, 19600, 47320, 114242, 275806, 665856, 1607520, 3880898, 9369318, 22619536, 54608392, 131836322, 318281038, 768398400, 1855077840, 4478554082, 10812186006, 26102926096, 63018038200, 152139002498, 367296043198

Offset

0,3

Links

Colin Barker, Table of n, a(n) for n = 0..1000

M. Diepenbroek, M. Maus, A. Stoll, Pattern Avoidance in Reverse Double Lists, Preprint 2015. See Table 3.

Index entries for linear recurrences with constant coefficients, signature (3,-1,-1).

Formula

From Colin Barker, Apr 12 2016: (Start)

a(n) = (-2 + (1-sqrt(2))^n + (1+sqrt(2))^n)/2 for n>1.

a(n) = 3*a(n-1)-a(n-2)-a(n-3) for n>4.

(End)

E.g.f.: x + (cosh(sqrt(2)*x) - 1)*exp(x). - Ilya Gutkovskiy, Sep 16 2016

Mathematica

Table[2 Fibonacci[n-1, 2] + LucasL[n-1, 2]/2 + KroneckerDelta[n-1] - 1, {n, 0, 20}] (* Vladimir Reshetnikov, Sep 16 2016 *)
LinearRecurrence[{3, -1, -1}, {0, 1, 2, 6, 16}, 40] (* Harvey P. Dale, Mar 18 2018 *)

Prog

(PARI) concat(0, Vec(x*(1-x+x^2+x^3)/((1-x)*(1-2*x-x^2)) + O(x^50))) \\ Colin Barker, Apr 12 2016

Crossrefs

Agrees with A213667 except for initial terms.

Cf. A002605, A265106, A265107, A270810.

Sequence in context: A302239 A264551 A293004 * A111281 A018021 A215340

Adjacent sequences: A265275 A265276 A265277 * A265279 A265280 A265281

Keyword

nonn,easy

Author

Alois P. Heinz, Apr 06 2016

Status

approved