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A160999 Row sums of A027052.

%I A160999 #8 Nov 06 2019 02:01:34
%S A160999 1,2,5,12,31,84,233,656,1865,5338,15355,44342,128455,373100,1086087,
%T A160999 3167634,9254009,27074666,79316491,232633206,683026535,2007327660,
%U A160999 5904415195,17381265934,51203990457,150945252394,445252685313
%N A160999 Row sums of A027052.
%H A160999 G. C. Greubel, <a href="/A160999/b160999.txt">Table of n, a(n) for n = 0..1000</a>
%F A160999 a(n) = Sum_{k=0..2*n} A027052(n,k).
%F A160999 Conjecture: (-n+2)*a(n) +(6*n-11)*a(n-1) +(-7*n+1)*a(n-2) +2*(-4*n+27)*a(n-3) +(5*n-28)*a(n-4) +(2*n-3)*a(n-5) +3*(n-5)*a(n-6)=0. - _R. J. Mathar_, May 26 2016
%e A160999 a(2) = 1+0+1+2+1 = 5.
%e A160999 a(3) = 1+0+1+2+3+4+1 = 12.
%p A160999 A027052 := proc(n,k) option remember; if k =0 or k = 2*n then 1; elif k = 1 then 0; elif k =2 then 1; else procname(n-1,k-3)+procname(n-1,k-2)+procname(n-1,k-1) ; fi; end:
%p A160999 A160999 := proc(n) add( A027052(n,k),k=0..2*n) ; end: seq(A160999(n),n=0..30) ;
%t A160999 T[n_, k_]:= T[n, k]= If[k==0 || k==2 || k==2*n, 1, If[k==1, 0, Sum[T[n-1, k-j], {j, 3}]]]; Table[Sum[T[n, k], {k,0,2*n}], {n,0,30}] (* _G. C. Greubel_, Nov 06 2019 *)
%o A160999 (Sage)
%o A160999 @CachedFunction
%o A160999 def T(n, k):
%o A160999     if (k==0 or k==2 or k==2*n): return 1
%o A160999     elif (k==1): return 0
%o A160999     else: return sum(T(n-1, k-j) for j in (1..3))
%o A160999 [sum(T(n, k) for k in (0..2*n)) for n in (0..30)] # _G. C. Greubel_, Nov 06 2019
%K A160999 easy,nonn
%O A160999 0,2
%A A160999 _R. J. Mathar_, Jun 01 2009