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

Marika Diepenbroek, Monica Maus, and Alex Stoll, Pattern Avoidance in Reverse Double Lists, Preprint 2015. See Table 3.

Hannah Golab, Pattern avoidance in Cayley permutations, Master's Thesis, Northern Arizona Univ. (2024). See p. 41.

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 * A378947 A111281 A018021

Adjacent sequences: A265275 A265276 A265277 * A265279 A265280 A265281

Keyword

nonn,easy

Author

Alois P. Heinz, Apr 06 2016

Status

approved