A128424 a(n) = floor(sqrt(a(n-1)^2 + a(n-2)^2 + a(n-1)*a(n-2))), a(1)=1, a(2)=3.

1, 3, 3, 5, 7, 10, 14, 20, 29, 42, 61, 89, 130, 190, 278, 407, 596, 873, 1279, 1874, 2746, 4024, 5897, 8642, 12665, 18561, 27202, 39866, 58426, 85627, 125492, 183917, 269543, 395034, 578950, 848492, 1243525, 1822474, 2670965, 3914489, 5736962

Offset: 1

Author

Zak Seidov, May 04 2007

Keywords

• nonn

Comments

For a triangle with sides a(n-1) and a(n-2) and a 120-degree angle between them, a(n) is the floor of the value of the third side.

a(n) = A020711(n-4) for 4 <= n <= 41. - Georg Fischer, Nov 02 2018

Crossrefs

Cf. A020711, A104803, A104804, A104805.

Programs

Mathematica

a[1]=1;a[2]=3;a[n_]:=a[n]=Floor[Sqrt[a[n-1]^2+a[n-2]^2+a[n-1]*a[n-2]]] Table[a[n],{n,45}]
RecurrenceTable[{a[1]==1,a[2]==3,a[n]==Floor[Sqrt[a[n-1]^2+a[n-2]^2+ a[n-1]*a[n-2]]]},a,{n,50}] (* Harvey P. Dale, Oct 01 2018 *)

Formula

Conjectures from Colin Barker, Nov 03 2018: (Start)

G.f.: x*(1 + x - 2*x^2 + x^3 - 2*x^4 + x^5 - x^6) / ((1 - x)*(1 - x - x^3)).

a(n) = 2*a(n-1) - a(n-2) + a(n-3) - a(n-4) for n>7.

(End)