A181984 INVERT transform of A028310.

1, 2, 5, 12, 28, 65, 151, 351, 816, 1897, 4410, 10252, 23833, 55405, 128801, 299426, 696081, 1618192, 3761840, 8745217, 20330163, 47261895, 109870576, 255418101, 593775046, 1380359512, 3208946545, 7459895657, 17342153393, 40315615410, 93722435101

Offset: 0

Author

Michael Somos, Apr 04 2012

Keywords

• nonn

Examples

			G.f. = 1 + 2*x + 5*x^2 + 12*x^3 + 28*x^4 + 65*x^5 + 151*x^6 + 351*x^7 + 816*x^8 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Miklos Bona, Rebecca Smith, Pattern avoidance in permutations and their squares, arXiv:1901.00026 [math.CO], 2018. See H(z), Ex. 4.1.
• Index entries for linear recurrences with constant coefficients, signature (3, -2, 1).

Crossrefs

Cf. A000931, A028310, A034943, A185963.

Programs

Magma

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x+x^2)/(1-3*x+2*x^2-x^3))); // G. C. Greubel, Aug 12 2018

Mathematica

CoefficientList[Series[(1-x+x^2)/(1-3*x+2*x^2-x^3), {x, 0, 50}], x] (* G. C. Greubel, Aug 12 2018 *)

PARI

{a(n) = if( n<0, n = -1-n; polcoeff( (1 - x + x^2) / (1 - 2*x + 3*x^2 - x^3) + x * O(x^n), n), polcoeff( (1 - x + x^2) / (1 - 3*x + 2*x^2 - x^3) + x * O(x^n), n))}

PARI

x='x+O('x^50); Vec((1-x+x^2)/(1-3*x+2*x^2-x^3)) \\ G. C. Greubel, Aug 12 2018

Formula

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

G.f.: 1 / (1 - 2*x / (1 - x / (2 + x / (1 - 2*x / (1 + x))))).

a(n) = A034943(n + 2) = A185963(-2 - n).

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

a(n) satisfies 1 = f(a(n-2), a(n-1), a(n)) where f(u, v, w) = u^3 - 5*v^3 + w^3 + u*v * (7*v -4*u) + u*w * (3*u + 2*w) + v*w * (11*v - 6*w) - 9*u*v*w.

a(n) = A000931(3*n + 6). - Michael Somos, Sep 18 2012