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%I #30 Sep 08 2022 08:45:00
%S 1,5,10,19,33,56,93,153,250,407,661,1072,1737,2813,4554,7371,11929,
%T 19304,31237,50545,81786,132335,214125,346464,560593,907061,1467658,
%U 2374723,3842385,6217112,10059501,16276617,26336122
%N Partial sums of A000285.
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., pp. 189, 194-196.
%D J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 224.
%H G. C. Greubel, <a href="/A053311/b053311.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1).
%F a(n) = a(n-1) + a(n-2) + 4; a(0)=1, a(1)=5; n >= 1.
%F a(n) = 4*F(n+2) + F(n+1) - 4, where F(k) is A000045(k).
%F From _R. J. Mathar_, Apr 29 2013: (Start)
%F G.f.: ( 1+3*x ) / ( (x-1)*(x^2+x-1) ).
%F a(n) = A000071(n+3) + 3*A000071(n+2) = A000285(n+2) - 4. (End)
%t CoefficientList[Series[(1+3*x)/((x-1)*(x^2+x-1)), {x, 0, 50}], x] (* _G. C. Greubel_, May 24 2018 *)
%o (PARI) x='x+O('x^30); Vec((1+3*x)/((x-1)*(x^2+x-1))) \\ _G. C. Greubel_, May 24 2018
%o (Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+3*x)/((x-1)*(x^2+x-1)))); // _G. C. Greubel_, May 24 2018
%Y Cf. A000285.
%Y a(n) = A101220(4, 1, n+1).
%K easy,nonn
%O 0,2
%A _Barry E. Williams_, Mar 06 2000
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