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%I #19 Mar 27 2024 08:59:54
%S 1,1,0,3,0,4,2,5,5,8,9,14,16,24,29,41,52,71,92,124,162,217,285,380,
%T 501,666,880,1168,1545,2049,2712,3595,4760,6308,8354,11069,14661,
%U 19424,25729,34086,45152,59816,79237,104969,139052,184207,244020,323260
%N Expansion of (1+2*x)/((1+x)*(1-x^2-x^3)).
%C Diagonal sums of A098599.
%C The signed sequence 1,-1,0,-3,0,-4,... gives the diagonal sums of A100218. - _Paul Barry_, Nov 09 2004
%H G. C. Greubel, <a href="/A098601/b098601.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-1,1,2,1).
%F G.f.: x/((1+x)*(1-x^2-x^3)) + 1/(1-x^2-x^3).
%F a(n) = Sum_{k=0..floor(n/2)} (binomial(k, n-2*k) + binomial(k-1, n-2*k-1)).
%F a(n) = -a(n-1) + a(n-2) + 2*a(n-3) + a(n-4).
%F Inverse binomial transform of A135364. - _Paul Curtz_, Apr 25 2008
%t CoefficientList[Series[(1+2x)/((1+x)(1-x^2-x^3)),{x,0,50}],x] (* or *) LinearRecurrence[{-1,1,2,1},{1,1,0,3},50] (* _Harvey P. Dale_, Dec 14 2011 *)
%o (Magma) I:=[1,1,0,3]; [n le 4 select I[n] else -Self(n-1) +Self(n-2) +2*Self(n-3) +Self(n-4): n in [1..55]]; // _G. C. Greubel_, Mar 27 2024
%o (SageMath)
%o def A098601(n): return sum( binomial(k, n-2*k) + binomial(k-1, n-2*k-1) for k in range(1+n//2))
%o [A098601(n) for n in range(56)] # _G. C. Greubel_, Mar 27 2024
%Y Cf. A000931, A077883, A098599, A100218, A135364.
%K easy,nonn
%O 0,4
%A _Paul Barry_, Sep 17 2004
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