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

%I A186575 #68 May 11 2025 23:48:35
%S A186575 1,3,10,15,31,66,127,255,514,1023,2047,4098,8191,16383,32770,65535,
%T A186575 131071,262146,524287,1048575,2097154,4194303,8388607,16777218,
%U A186575 33554431,67108863,134217730,268435455,536870911,1073741826,2147483647,4294967295
%N A186575 Expansion of (1 + 2*x + 6*x^2)/(1 - x - x^2 - 2*x^3) in powers of x.
%C A186575 From _Kai Wang_, May 23 2020: (Start)
%C A186575 Let f(t) = t^3 + u*t^2 + v*t + w and {x,y,z} be the simple roots of f(t).
%C A186575 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.
%C A186575 Then for n >= 0, q(n) = 3*p(n+2) + 2*u*p(n+1) + v*p(n).
%C A186575 In this case, f(t) = t^3 - t^2 - t - 2. q(n) = 3*p(n+2) - 2*p(n+1) - p(n).
%C A186575 p(n) = {0, 0, 1, 1, 2, 5, 9,...}, q(n) = {3, 1, 3, 10, 15, 31,...}.
%C A186575 a(n) = q(n+1), A077939(n) = p(n+2). (End)
%H A186575 Colin Barker, <a href="/A186575/b186575.txt">Table of n, a(n) for n = 0..1000</a>
%H A186575 Gamaliel Cerda-Morales, <a href="https://arxiv.org/abs/1905.00725">A note on Modified Third-order Jacobsthal numbers</a>, arXiv:1905.00725 [math.CO], 2019. See pp. 3-4.
%H A186575 Vladimir Kruchinin, <a href="http://arxiv.org/abs/1009.2565">Composition of ordinary generating functions</a>, arXiv:1009.2565 [math.CO], 2010.
%H A186575 Evren Eyican Polatlı and Yüksel Soykan, <a href="https://doi.org/10.9734/ARJOM/2021/v17i230270">On generalized third-order Jacobsthal numbers</a>, Asian Res. J. of Math. (2021) Vol. 17, No. 2, 1-19, Article No. ARJOM.66022.
%H A186575 Kai Wang, <a href="https://www.researchgate.net/publication/341072691_Closed_Forms_and_Generating_Functions_For_Power_Sums_outline">Closed Forms and Generating Functions For Power Sums</a>, 2020.
%H A186575 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,2).
%F A186575 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.
%F A186575 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).
%F A186575 a(n) = 2^(n+1) + A099837(n+1). - _R. J. Mathar_, Mar 18 2011
%F A186575 a(n) = a(n-1) + a(n-2) + 2*a(n-3) for n>2. - _Colin Barker_, May 03 2019
%F A186575 From _Kai Wang_, May 23 2020: (Start)
%F A186575 a(n) = 3*A077947(n+1) - 2*A077947(n) - A077947(n-1).
%F A186575 A077947(n) = (-8*a(n+3) + 27*a(n+2) - a(n+1))/147. (End)
%e A186575 G.f. = 1 + 3*x + 10*x^2 + 15*x^3 + 31*x^4 + 66*x^5 + 127*x^6 + 255*x^7 + ...
%t A186575 CoefficientList[Series[(1+2x+6x^2)/(1-x-x^2-2x^3),{x,0,40}],x]  (* _Harvey P. Dale_, Mar 14 2011 *)
%o A186575 (PARI) Vec((1 + 2*x + 6*x^2) / ((1 - 2*x)*(1 + x + x^2)) + O(x^40)) \\ _Colin Barker_, May 03 2019
%o A186575 (PARI) polsym(polrecip(1 - x - x^2 - 2*x^3),44)[^1] \\ _Joerg Arndt_, Jun 23 2020
%o A186575 (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
%Y A186575 Cf. A099837.
%K A186575 nonn,easy
%O A186575 0,2
%A A186575 _Vladimir Kruchinin_, Feb 23 2011
%E A186575 More terms from _Harvey P. Dale_, Mar 14 2011