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%I #13 Dec 20 2021 09:43:05
%S 1,3,9,28,65,201,430,1316,2721,8259,16793,50680,102102,306958,615024,
%T 1844304,3682545,11024331,21963161,65675764,130648089,390374193,
%U 775797750,2316881892,4601346295,13737041045,27270124455
%N a(n) = T(n, n-4), T given by A026519. Also a(n) = number of integer strings s(0), ..., s(n), counted by T, such that s(n) = 4.
%H G. C. Greubel, <a href="/A026524/b026524.txt">Table of n, a(n) for n = 4..1000</a>
%H Veronika Irvine, Stephen Melczer and Frank Ruskey, <a href="https://arxiv.org/abs/1804.08725">Vertically constrained Motzkin-like paths inspired by bobbin lace</a>, arXiv:1804.08725 [math.CO], 2018.
%F a(n) = A026519(n, n-4).
%t T[n_, k_]:= T[n, k]= If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[(n+1)/2], If[EvenQ[n], T[n-1, k-2] + T[n-1, k], T[n-1, k-1] + T[n-1, k-2] + T[n-1, k]]]]; (* T = A026519 *)
%t Table[T[n, n-4], {n,4,40}] (* _G. C. Greubel_, Dec 19 2021 *)
%o (Sage)
%o @CachedFunction
%o def T(n,k): # T = A026552
%o if (k==0 or k==2*n): return 1
%o elif (k==1 or k==2*n-1): return (n+1)//2
%o elif (n%2==0): return T(n-1, k) + T(n-1, k-2)
%o else: return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
%o [T(n,n-4) for n in (4..40)] # _G. C. Greubel_, Dec 19 2021
%Y Cf. A026519, A026520, A026521, A026522, A026523, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026533, A026534, A027262, A027263, A027264, A027265, A027266.
%K nonn
%O 4,2
%A _Clark Kimberling_
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