A073717 a(n) = T(2n+1), where T(n) are the tribonacci numbers A000073.

0, 1, 4, 13, 44, 149, 504, 1705, 5768, 19513, 66012, 223317, 755476, 2555757, 8646064, 29249425, 98950096, 334745777, 1132436852, 3831006429, 12960201916, 43844049029, 148323355432, 501774317241, 1697490356184, 5742568741225

Offset: 0

Author

Mario Catalani (mario.catalani(AT)unito.it), Aug 05 2002

Keywords

• easy
• nonn

Comments

In general, the bisection of a third-order linear recurrence with signature (x,y,z) will result in a third-order recurrence with signature (x^2 + 2*y, 2*z*x - y^2, z^2). - Gary Detlefs, May 29 2024

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Meng-Han Wu, Henryk A. Witek, Rafał Podeszwa, Clar Covers and Zhang-Zhang Polynomials of Zigzag and Armchair Carbon Nanotubes, MATCH Commun. Math. Comput. Chem. (2025) Vol. 93, 415-462. See p. 437.
• Index entries for linear recurrences with constant coefficients, signature (3,1,1).

Crossrefs

Cf. A000073, A099463, A113300.

Row sums of A216182.

Programs

Magma

[n le 3 select (n-1)^2 else 3*Self(n-1) +Self(n-2) +Self(n-3): n in [1..31]]; // G. C. Greubel, Nov 19 2021

Mathematica

CoefficientList[Series[(x+x^2)/(1-3x-x^2-x^3), {x, 0, 30}], x]
LinearRecurrence[{3,1,1},{0,1,4},30] (* Harvey P. Dale, Sep 07 2015 *)

Sage

def A073717_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1+x)/(1-3*x-x^2-x^3) ).list()
A073717_list(30) # G. C. Greubel, Nov 19 2021

Formula

a(n) = 3*a(n-1) + a(n-2) + a(n-3), a(0)=0, a(1)=1, a(2)=4.

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

a(n+1) = Sum_{k=0..n} A216182(n,k). - Philippe Deléham, Mar 11 2013

a(n) = A113300(n-1) + A113300(n). - R. J. Mathar, Jul 04 2019