A120893 a(n) = 3*a(n-1) + 3*a(n-2) - a(n-3); a(0)=1, a(1)=1, a(2)=5.

1, 1, 5, 17, 65, 241, 901, 3361, 12545, 46817, 174725, 652081, 2433601, 9082321, 33895685, 126500417, 472105985, 1761923521, 6575588101, 24540428881, 91586127425, 341804080817, 1275630195845, 4760716702561, 17767236614401

Offset: 0

Author

Lekraj Beedassy, Jul 14 2006

Keywords

• nonn
• easy

Comments

For n>1, hypotenuse of primitive Pythagorean triangles having an angle nearing Pi/3 for larger values of sides. Complete triple (X,Y,Z),XA120892(n),Y=A001353(n),Z=a(n) with recurrence relations X(i+1)=2*{a(i)-(-1)^i}-X(i-1) ; Y(i+1)=2*T(i)-T(i-1)-(-1)^i, where T(i)=Y(i)+a(i)] a(n)=2*A120892(n)-(-1)^n.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• R. C. Alperin, A family of nonlinear recurrences and their linear solutions, Fib. Q., 57:4 (2019), 318-321.
• W. K. Alt, Enumeration of Domino Tilings on the Projective Grid Graph, A Thesis Presented to The Division of Mathematics and Natural Sciences, Reed College, May 2013.
• Index entries for linear recurrences with constant coefficients, signature (3,3,-1).

Programs

Magma

[Floor(((-1)^n+(2-Sqrt(3))^n+(2+Sqrt(3))^n)/3): n in [0..40]]; // Vincenzo Librandi, Jul 09 2012

Maple

a[0]:=1: a[1]:=1: a[2]:=5: for n from 3 to 40 do a[n]:=3*a[n-1]+3*a[n-2]-a[n-3] od: seq(a[n],n=0..30); # Emeric Deutsch, Jul 24 2006

Mathematica

Transpose[NestList[Flatten[{Rest[#],3Last[#]+3#[[2]]- First[#]}]&, {1,1,5},25]][[1]] (* or *)
CoefficientList[Series[(1-2 x-x^2)/(1-3 x-3 x^2+x^3),{x,0,25}],x]  (* Harvey P. Dale, Mar 27 2011 *)

Formula

Union of A103772 and A103974. a(n)=2*{2*a(n-1) + (-1)^n} - a(n-2) ; a(0)=1,a(1)=1.

a(n) = [(-1)^n+(2-sqrt(3))^n+(2+sqrt(3))^n]/3. - Emeric Deutsch, Jul 24 2006

O.g.f: -(-1+2*x+x^2)/((1+x)*(x^2-4*x+1)). - R. J. Mathar, Dec 02 2007

a(n)+a(n+1) = A003699(n+1), n>0. - R. J. Mathar, Oct 15 2013

Extensions

More terms from Emeric Deutsch, Jul 24 2006