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%I #61 Apr 14 2023 13:56:35
%S 1,0,0,1,1,0,1,1,1,1,1,1,2,1,1,2,2,1,2,2,2,2,2,2,3,2,2,3,3,2,3,3,3,3,
%T 3,3,4,3,3,4,4,3,4,4,4,4,4,4,5,4,4,5,5,4,5,5,5,5,5,5,6,5,5,6,6,5,6,6,
%U 6,6,6,6,7,6,6,7,7,6,7,7,7,7,7,7,8,7
%N Expansion of 1/((1-x^3)*(1-x^4)).
%C Number of partitions of n into parts 3 and 4. - _Reinhard Zumkeller_, Feb 09 2009
%C Convolution of A112689 (shifted left once) by A033999. - _R. J. Mathar_, Feb 13 2009
%C With four 0's prepended and offset 0, a(n) is the number of partitions of n into four parts whose largest three parts are equal. - _Wesley Ivan Hurt_, Jan 06 2021
%H Vincenzo Librandi, <a href="/A008679/b008679.txt">Table of n, a(n) for n = 0..1000</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=216">Encyclopedia of Combinatorial Structures 216</a>
%H A. V. Kitaev and A. Vartanian, <a href="https://arxiv.org/abs/2304.05671">Algebroid Solutions of the Degenerate Third Painlevé Equation for Vanishing Formal Monodromy Parameter</a>, arXiv:2304.05671 [math.CA], 2023. See p. 20.
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,1,1,0,0,-1).
%F a(n+12) = a(n) + 1. - _Reinhard Zumkeller_, Feb 09 2009
%F G.f.: 1/((1-x)^2*(1+x)*(1+x+x^2)*(1+x^2)). - _R. J. Mathar_, Feb 13 2009
%F a(n) = 1 + floor(n/3) + floor(-n/4). - _Tani Akinari_, Sep 02 2013
%F E.g.f.: (1/72)*(9*exp(-x)+21*exp(x)+6*exp(x)*x+18*cos(x)+24*exp(-x/2)*cos(sqrt(3)*x/2)-18*sin(x)+8*sqrt(3)*exp(-x/2)*sin(sqrt(3)*x/2)). - _Stefano Spezia_, Sep 09 2019
%F a(n) = A005044(n+3) - A005044(n+1). - _Yuchun Ji_, Oct 10 2020
%F From _Wesley Ivan Hurt_, Jan 17 2021: (Start)
%F a(n) = a(n-3) + a(n-4) - a(n-7).
%F a(n) = Sum_{k=1..floor((n+4)/4)} Sum_{j=k..floor((n+4-k)/3)} Sum_{i=j..floor((n+4-j-k)/2)} [j = i = n+4-i-k-j], where [ ] is the Iverson bracket. (End)
%p seq(coeff(series(1/((1-x^3)*(1-x^4)), x, n+1), x, n), n = 0..90); # _G. C. Greubel_, Sep 09 2019
%t LinearRecurrence[{0,0,1,1,0,0,-1}, {1,0,0,1,1,0,1}, 90] (* _Vladimir Joseph Stephan Orlovsky_, Feb 23 2012 *)
%t CoefficientList[Series[1/((1-x)^2(1+x)(1+x+x^2)(1+x^2)), {x,0,90}], x] (* _Vincenzo Librandi_, Jun 11 2013 *)
%o (PARI) my(x='x+O('x^90)); Vec(1/((1-x^3)*(1-x^4))) \\ _G. C. Greubel_, Sep 09 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 90); Coefficients(R!( 1/((1-x^3)*(1-x^4)) )); // _G. C. Greubel_, Sep 09 2019
%o (Sage)
%o def A008679_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P(1/((1-x^3)*(1-x^4))).list()
%o A008679_list(90) # _G. C. Greubel_, Sep 09 2019
%o (GAP) a:=[1,0,0,1,1,0,1,1];; for n in [8..90] do a[n]:=a[n-3]+a[n-4]-a[n-7]; od; a; # _G. C. Greubel_, Sep 09 2019
%Y Cf. A005044, A033999, A112689.
%K nonn,easy
%O 0,13
%A _N. J. A. Sloane_
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