A339828 a(n) = 4*a(n-1) - 2*a(n-2) + a(n-3) - 4*a(n-4) + a(n-5) for n >= 6, where a(1) = 1, a(2) = 3, a(3) = 5, a(4) = 16, a(5) = 53.

1, 2, 5, 16, 53, 179, 610, 2081, 7103, 24250, 82793, 282671, 965098, 3295049, 11249999, 38409898, 131139593, 447738575, 1528675114, 5219223305, 17819542991, 60839725354, 207719815433, 709199811023, 2421359613226, 8267038830857, 28225436096975, 96367666726186

Offset

1,2

Links

Table of n, a(n) for n=1..28.

Index entries for linear recurrences with constant coefficients, signature (4,-2,1,-4,2).

Formula

Let f(n) = floor(r*floor(s*n)) = A184922(n), where r = sqrt(2) and s = r + 1. Let a(1) = 1. Then a(n) = f(a(n-1)) for n >= 2.

Also, a(n) = 4*a(n-1) - 2*a(n-2) + a(n-3) - 4*a(n-4) + a(n-5) for n >= 6, where a(1) = 1, a(2) = 3, a(3) = 5, a(4) = 16, a(5) = 53.

G.f.: x*(-x^4 + x^3 + x^2 + 2*x - 1)/((x - 1)*(x^2 + x + 1)*(2*x^2 - 4*x + 1)). - Chai Wah Wu, Feb 15 2021

Mathematica

z = 40; r = Sqrt[2]; s = 1 + Sqrt[2]; f[x_] := Floor[r*Floor[s*x]];
Table[f[n], {n, 1, z}] m (* A184922 *)
a[1] = 1; a[n_] := f[a[n - 1]]; Table[a[n], {n, 1, z}] (* A339828 *)

Prog

(PARI) a(n)=([0, 1, 0, 0, 0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1; 2, -4, 1, -2, 4]^(n-1)*[1; 2; 5; 16; 53])[1, 1] \\ Charles R Greathouse IV, May 25 2026

Crossrefs

Cf. A184922, A341239, A341240.

Sequence in context: A001428 A055726 A101500 * A047063 A268430 A047006

Adjacent sequences: A339825 A339826 A339827 * A339829 A339830 A339831

Keyword

nonn,easy

Author

Clark Kimberling, Feb 07 2021

Status

approved