A026521 a(n) = T(n, n-1), T given by A026519. Also a(n) = number of integer strings s(0), ..., s(n), counted by T, such that s(n) = 1.

1, 1, 4, 6, 19, 33, 98, 180, 526, 990, 2887, 5502, 16073, 30863, 90386, 174456, 512128, 992304, 2918954, 5673140, 16716998, 32571858, 96119927, 187675644, 554524660, 1084649644, 3208254571, 6284986554, 18607536319, 36501029265

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Veronika Irvine, Stephen Melczer, and Frank Ruskey, Vertically constrained Motzkin-like paths inspired by bobbin lace, arXiv:1804.08725 [math.CO], 2018.

Crossrefs

Cf. A026519, A026520, A026522, A026523, A026524, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026533, A026534, A027262, A027263, A027264, A027265, A027266.

Cf. A026537.

Programs

Mathematica

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 *)
Table[T[n, n-1], {n,40}] (* G. C. Greubel, Dec 19 2021 *)

Sage

@CachedFunction
def T(n,k): # T = A026552
    if (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n+1)//2
    elif (n%2==0): return T(n-1, k) + T(n-1, k-2)
    else: return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
[T(n,n-1) for n in (1..40)] # G. C. Greubel, Dec 19 2021

Formula

a(n) = A026519(n, n-1).

a(n) = A026537(n+1)/2.