%I #39 Dec 30 2023 23:48:27

%S 1,7,29,92,247,591,1300,2683,5270,9955,18228,32551,56967,98086,166681,

%T 280271,467301,773906,1274856,2091266,3419252,5576298,9076280,

%U 14750858,23945893,38839257,62955061,101995694

%N Partial sums of A053739.

%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 189, 194-196.

%H G. C. Greubel, <a href="/A053295/b053295.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (7,-20,29,-20,1,8,-5,1).

%F a(n) = Sum_{i=0..floor(n/2)} C(n+6-i, n-2i), n >= 0.

%F a(n) = a(n-1) + a(n-2) + C(n+5,5); n >= 0; a(-1)=0.

%F G.f.: -1 / ( (x^2 + x - 1)*(x-1)^6 ). - _R. J. Mathar_, Oct 10 2014

%t Table[Sum[Binomial[n+6-j, n-2*j], {j, 0, Floor[n/2]}], {n, 0, 50}] (* _G. C. Greubel_, May 24 2018 *)

%o (PARI) for(n=0, 30, print1(sum(j=0, floor(n/2), binomial(n+6-j, n-2*j)), ", ")) \\ _G. C. Greubel_, May 24 2018

%o (Magma) [(&+[Binomial(n+6-j, n-2*j): j in [0..Floor(n/2)]]): n in [0..30]]; // _G. C. Greubel_, May 24 2018

%Y Cf. A053739, A014166 and A000045.

%Y Right-hand column 12 of triangle A011794.

%Y Cf. A228074.

%K easy,nonn

%O 0,2

%A _Barry E. Williams_, Mar 04 2000