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

0,4

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).

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).

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 *)

Prog

(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
(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

Crossrefs

Cf. A000213, A085697, A107240.

Sequence in context: A225106 A110970 A322939 * A147413 A066137 A269662

Adjacent sequences: A141580 A141581 A141582 * A141584 A141585 A141586

Keyword

easy,nonn

Author

R. J. Mathar, Aug 19 2008

Status

approved