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A026522 a(n) = T(n, n-2), where T is given by A026519. Also number of integer strings s(0), ..., s(n), counted by T, such that s(n) = 2.

%I A026522 #17 Dec 20 2021 09:43:35
%S A026522 1,2,5,13,27,76,150,434,845,2470,4797,14085,27377,80584,156900,462620,
%T A026522 902394,2664276,5205950,15387670,30114073,89097932,174609162,
%U A026522 517058502,1014555607,3006637946,5906040623,17514547015,34438443075
%N A026522 a(n) = T(n, n-2), where T is given by A026519. Also number of integer strings s(0), ..., s(n), counted by T, such that s(n) = 2.
%H A026522 G. C. Greubel, <a href="/A026522/b026522.txt">Table of n, a(n) for n = 2..1000</a>
%H A026522 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 A026522 a(n) = A026519(n, n-2).
%t A026522 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 A026522 Table[T[n, n-2], {n,2,40}] (* _G. C. Greubel_, Dec 19 2021 *)
%o A026522 (Sage)
%o A026522 @CachedFunction
%o A026522 def T(n,k): # T = A026552
%o A026522     if (k==0 or k==2*n): return 1
%o A026522     elif (k==1 or k==2*n-1): return (n+1)//2
%o A026522     elif (n%2==0): return T(n-1, k) + T(n-1, k-2)
%o A026522     else: return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
%o A026522 [T(n,n-2) for n in (2..40)] # _G. C. Greubel_, Dec 19 2021
%Y A026522 Cf. A026519, A026520, A026521, A026523, A026524, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026533, A026534, A027262, A027263, A027264, A027265, A027266.
%K A026522 nonn
%O A026522 2,2
%A A026522 _Clark Kimberling_