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

Author

Vladimir Kruchinin, Feb 23 2011

Keywords

• nonn
• easy

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)

Examples

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

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).

Crossrefs

Cf. A099837.

Programs

Magma

R:=PowerSeriesRing(Integers(), 35); Coefficients(R!( (1 + 2*x + 6*x^2)/(1 - x - x^2 - 2*x^3))); // Marius A. Burtea, Jan 31 2020

Mathematica

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

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

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)

Extensions

More terms from Harvey P. Dale, Mar 14 2011