A151162 Number of walks within N^3 (the first octant of Z^3) starting at (0,0,0) and consisting of n steps taken from {(-1, 0, 0), (1, 0, 0), (1, 0, 1), (1, 1, 0)}.
1, 3, 12, 45, 180, 702, 2808, 11097, 44388, 176418, 705672, 2812482, 11249928, 44903484, 179613936, 717517521, 2870070084, 11470898106, 45883592424, 183438670950, 733754683800, 2934026948196, 11736107792784, 46934017407594, 187736069630376, 750833732416212, 3003334929664848
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Paul Barry, A Note on a One-Parameter Family of Catalan-Like Numbers, JIS 12 (2009) 09.5.4
- A. Bostan and M. Kauers, Automatic Classification of Restricted Lattice Walks, arXiv:0811.2899 [math.CO], 2008-2009.
Programs
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Magma
[n le 3 select Factorial(n+1)/2 else (4*n*Self(n-1) + 12*(n-3)*Self(n-2) - 48*(n-3)*Self(n-3))/n: n in [1..41]]; // G. C. Greubel, Nov 09 2022
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Mathematica
aux[i_, j_, k_, n_]:= Which[Min[i, j, k, n]<0 || Max[i, j, k]>n, 0, n==0, KroneckerDelta[i, j, k, n], True, aux[i, j, k, n]= aux[-1+i, -1+j, k, -1+n] + aux[-1+i, j, -1+k, -1+n] + aux[-1+i, j, k, -1+n] + aux[1+i, j, k, -1+n]]; Table[Sum[aux[i,j,k,n], {i,0,n}, {j,0,n}, {k,0,n}], {n,0,30}] a[n_]:= a[n]= If[n<3, (n+2)!/2, (4*(n+1)*a[n-1] +12*(n-2)*a[n-2] -48*(n-2)*a[n- 3])/(n+1)]; Table[a[n], {n,0,40}] (* G. C. Greubel, Nov 09 2022 *) -
SageMath
def a(n): # a = A151162 if (n==0): return 1 elif (n%2==1): return 4*a(n-1) - 3^((n-1)/2)*catalan_number((n-1)/2) else: return 4*a(n-1) [a(n) for n in (0..40)] # G. C. Greubel, Nov 09 2022
Formula
a(n) = Sum_{k=0..n} A120730(n,k)*3^k. - Philippe Deléham, Feb 01 2009
From Philippe Deléham, Feb 02 2009: (Start)
a(2*n+2) = 4*a(2*n+1), a(2*n+1) = 4*a(2*n) - 3^n*A000108(n)
a(2*n+1) = 4*a(2*n) - A005159(n).
G.f.: (sqrt(1 - 12*x^2) + 6*x - 1)/(6*x*(1 - 4*x)). (End)
a(n) ~ 2^(2*n+1)/3. - Vaclav Kotesovec, Oct 31 2017
a(n) = (4*(n+1)*a(n-1) + 12*(n-2)*a(n-2) - 48*(n-2)*a(n-3))/(n+1). - G. C. Greubel, Nov 09 2022
Comments