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)
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
KEYWORD
nonn,easy
AUTHOR
Vladimir Kruchinin, Feb 23 2011
EXTENSIONS
More terms from Harvey P. Dale, Mar 14 2011
STATUS
approved