A121811 - OEIS

%I #23 Jan 12 2025 23:56:38

%S 0,3,8,19,46,111,263,622,1473,3485,8246,19512,46166,109230,258441,

%T 611480,1446777,3423112,8099170,19162842,45339771,107275050,253815495,

%U 600533909,1420878484,3361834590,7954186044,18819806248,44528139677,105354709660,249271919464,589783693902

%N 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.

%t M = N[Abs[MatrixPower[{{0, 1, 1, 1}, {1, 0, 1, 1}, {1, 1, 0, 1}, {1, 1, 1, 0}}, 1/2]]]

%t v[1] = {0, 1, 2, 3}

%t v[n_] := v[n] = M.v[n - 1]

%t Table[Floor[v[n][[1]]], {n, 1, 50}]

%o (PARI) B(n)={concat([0,0,0],Vec(1/((1 - 6*x + 6*x^2)*(1 + 2*x - 2*x^2)) + O(x^n)))}

%o 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

%Y Cf. A120471.

%K nonn,easy,less

%O 1,2

%A _Roger L. Bagula_, Aug 30 2006

%E Name clarified by _Sean A. Irvine_, Jan 12 2025

%E a(30) onwards from _Andrew Howroyd_, Jan 12 2025