A186575 Expansion of (1 + 2*x + 6*x^2)/(1 - x - x^2 - 2*x^3) in powers of x.

1, 3, 10, 15, 31, 66, 127, 255, 514, 1023, 2047, 4098, 8191, 16383, 32770, 65535, 131071, 262146, 524287, 1048575, 2097154, 4194303, 8388607, 16777218, 33554431, 67108863, 134217730, 268435455, 536870911, 1073741826, 2147483647, 4294967295

Offset

0,2

Comments

From Kai Wang, May 23 2020: (Start)

Let f(t) = t^3 + u*t^2 + v*t + w and {x,y,z} be the simple roots of f(t).

For n>=0, let p(n) = x^n/((x-y)(x-z)) + y^n/((y-x)(y-z)) + z^n/((z-x)(z-y)) and q(n) = x^n + y^n + z^n.

Then for n >= 0, q(n) = 3*p(n+2) +2*u*p(n+1) + v*p(n).

In this case, f(t) = t^3 - t^2 - t - 2. q(n) = 3*p(n+2} - 2*p(n+1) - p(n).

p(n) = {0, 0, 1, 1, 2, 5, 9,...}, q(n) = {3, 1, 3, 10, 15, 31,...}.

a(n) = q(n+1), A077939(n) = p(n+2). (End)

Links

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

Gamaliel Cerda-Morales, A note on Modified Third-order Jacobsthal numbers, arXiv:1905.00725 [math.CO], 2019. See pp. 3-4.

Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

Evren Eyican Polatlı and Yüksel Soykan, On generalized third-order Jacobsthal numbers, Asian Res. J. of Math. (2021) Vol. 17, No. 2, 1-19, Article No. ARJOM.66022.

Kai Wang, Closed Forms and Generating Functions For Power Sums, 2020.

Index entries for linear recurrences with constant coefficients, signature (1,1,2).

Formula

a(n+1) = n*Sum_{k=1..n} Sum_{j=n-3*k..k} 2^(k-j)*binomial(j,n-3*k+2*j)*binomial(k,j)/k.

G.f.: [log(1/(1 - x - x^2 - 2*x^3))]', (x + x^2 + 2*x^3)^k = Sum_{n>=k} Sum_{j=n-3*k..k} 2^(k-j)*binomial(j,n-3*k+2*j)*binomial(k,j)*x^n (see link).

a(n) = 2^(n+1) + A099837(n+1). - R. J. Mathar, Mar 18 2011

a(n) = a(n-1) + a(n-2) + 2*a(n-3) for n>2. - Colin Barker, May 03 2019

From Kai Wang, May 23 2020: (Start)

a(n) = 3*A077947(n+1) - 2*A077947(n) - A077947(n-1).

A077947(n) = (-8*a(n+3) + 27*a(n+2) - a(n+1))/147. (End)

Example

G.f. = 1 + 3*x + 10*x^2 + 15*x^3 + 31*x^4 + 66*x^5 + 127*x^6 + 255*x^7 + ...

Mathematica

CoefficientList[Series[(1+2x+6x^2)/(1-x-x^2-2x^3), {x, 0, 40}], x]  (* Harvey P. Dale, Mar 14 2011 *)

Prog

(PARI) Vec((1 + 2*x + 6*x^2) / ((1 - 2*x)*(1 + x + x^2)) + O(x^40)) \\ Colin Barker, May 03 2019
(PARI) polsym(polrecip(1 - x - x^2 - 2*x^3), 44)[^1] \\ Joerg Arndt, Jun 23 2020
(Magma) R<x>:=PowerSeriesRing(Integers(), 35); Coefficients(R!( (1 + 2*x + 6*x^2)/(1 - x - x^2 - 2*x^3))); // Marius A. Burtea, Jan 31 2020

Crossrefs

Cf. A099837.

Sequence in context: A030005 A192163 A129307 * A356318 A233312 A330940

Adjacent sequences: A186572 A186573 A186574 * A186576 A186577 A186578

Keyword

nonn,easy

Author

Vladimir Kruchinin, Feb 23 2011

Extensions

More terms from Harvey P. Dale, Mar 14 2011

Status

approved