A121811 a(n) is the floor of the first component of M^n * (0, 1, 2, 3) where M is the matrix [[c, 1/2, 1/2, 1/2], [1/2, c, 1/2, 1/2], [1/2, 1/2, c, 1/2], [1/2, 1/2, 1/2, c]] and c=sqrt(3)/2.
0, 3, 8, 19, 46, 111, 263, 622, 1473, 3485, 8246, 19512, 46166, 109230, 258441, 611480, 1446777, 3423112, 8099170, 19162842, 45339771, 107275050, 253815495, 600533909, 1420878484, 3361834590, 7954186044, 18819806248, 44528139677, 105354709660, 249271919464, 589783693902
Offset: 1
Crossrefs
Cf. A120471.
Programs
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Mathematica
M = N[Abs[MatrixPower[{{0, 1, 1, 1}, {1, 0, 1, 1}, {1, 1, 0, 1}, {1, 1, 1, 0}}, 1/2]]] v[1] = {0, 1, 2, 3} v[n_] := v[n] = M.v[n - 1] Table[Floor[v[n][[1]]], {n, 1, 50}] -
PARI
B(n)={concat([0,0,0],Vec(1/((1 - 6*x + 6*x^2)*(1 + 2*x - 2*x^2)) + O(x^n)))} seq(n)={my(v=B(n)); vector(n, k, (3*(v[k+2]-2*v[k+1]) + sqrtint(108*(v[k+1]-v[k])^2))\2^(k-2))} \\ Andrew Howroyd, Jan 12 2025
Extensions
Name clarified by Sean A. Irvine, Jan 12 2025
a(30) onwards from Andrew Howroyd, Jan 12 2025