A141583 Squares of tribonacci numbers A000213.

1, 1, 1, 9, 25, 81, 289, 961, 3249, 11025, 37249, 126025, 426409, 1442401, 4879681, 16507969, 55845729, 188925025, 639128961, 2162157001, 7314525625, 24744863025, 83711270241, 283193201281, 958035736849, 3241011678961

Offset: 0

Author

R. J. Mathar, Aug 19 2008

Keywords

• easy
• nonn

Comments

Partial sums are in A107240.

a(n) is also the number of total dominating sets in the (n-1)-ladder graph. - Eric W. Weisstein, Apr 10 2018

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Ladder Graph
• Eric Weisstein's World of Mathematics, Total Dominating Set
• Index entries for linear recurrences with constant coefficients, signature (2,3,6,-1,0,-1).

Crossrefs

Cf. A000213, A085697, A107240.

Programs

Magma

I:=[1,1,1,9,25,81]; [n le 6 select I[n] else 2*Self(n-1) + 3*Self(n-2) + 6*Self(n-3) - Self(n-4) - Self(n-6): n in [1..30]]; // Vincenzo Librandi, Dec 13 2012

Mathematica

CoefficientList[Series[(1+x)^2*(1-3*x+x^2-x^3)/((1+x+x^2-x^3)*(1-3*x-x^2-x^3)), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 13 2012 *)
Table[RootSum[-1 - # - #^2 + #^3 &, 2 #^n - 4 #^(n + 1) + 3 #^(n + 2) &]^2/121, {n, 0, 20}] (* Eric W. Weisstein, Apr 10 2018 *)
LinearRecurrence[{2,3,6,-1,0,-1}, {1,1,9,25,81,289}, {0, 20}] (* Eric W. Weisstein, Apr 10 2018 *)
LinearRecurrence[{1,1,1},{1,1,1},40]^2 (* Harvey P. Dale, Aug 01 2021 *)

Sage

@CachedFunction
def T(n): # A000213
    if (n<3): return 1
    else: return T(n-1) +T(n-2) +T(n-3)
def A141583(n): return T(n)^2
[A141583(n) for n in (0..40)] # G. C. Greubel, Nov 22 2021

Formula

a(n) = (A000213(n))^2.

O.g.f.: (1+x)^2*(1-3*x+x^2-x^3)/((1+x+x^2-x^3)*(1-3*x-x^2-x^3)).

a(n) = 2*a(n-1) + 3*a(n-2) + 6*a(n-3) - a(n-4) - a(n-6).