A073145 a(n) = -a(n-1) - a(n-2) + a(n-3), a(0)=3, a(1)=-1, a(2)=-1.

3, -1, -1, 5, -5, -1, 11, -15, 3, 23, -41, 21, 43, -105, 83, 65, -253, 271, 47, -571, 795, -177, -1189, 2161, -1149, -2201, 5511, -4459, -3253, 13223, -14429, -2047, 29699, -42081, 10335, 61445, -113861, 62751, 112555, -289167, 239363, 162359, -690889, 767893, 85355

Offset

0,1

Comments

Previous name was: Sum of the determinants of the principal minors of 2nd order of n-th power of Tribomatrix: first row (1, 1, 0); second row (1, 0, 1); third row (1, 0, 0).

a(n) is related to the generalized Lucas numbers S(n). For instance, 2S(n) = a(n)^2 - a(2n).

a(n) is also the reflected (A074058) sequence of the generalized tribonacci sequence (A001644).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Mario Catalani, Polymatrix and Generalized Polynacci Numbers, arXiv:math/0210201 [math.CO], 2002.

Curtis Cooper, S. Miller, Peter J. C. Moses, M. Sahin, and T. Thanatipanonda, On Identities of Ruggles, Horadam, Howard, and Young, Preprint 2016.

Kai Wang, Identities for generalized enneanacci numbers, Generalized Fibonacci Sequences (2020).

Index entries for linear recurrences with constant coefficients, signature (-1,-1,1).

Formula

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

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

a(n) = -T(n)^2 + 2*T(n-1)^2 + 3*T(n-2)^2 - 2*T(n)*T(n-1) + 2*T(n)*T(n-2) + 4*T(n-1)*T(n-2), where T(n) are tribonacci numbers (A000073).

a(n) = 3*A057597(n+2) + 2*A057597(n+1) + A057597(n). - R. J. Mathar, Jun 06 2011

From Peter Bala, Jun 29 2015: (Start)

a(n) = alpha^n + beta^n + gamma^n, where alpha, beta and gamma are the roots of 1 - x - x^2 - x^3 = 0.

x^2*exp( Sum_{n >= 1} a(n)*x^n/n ) = x^2 - x^3 + 2*x*5 - 3*x^6 + x^7 + ... is the o.g.f. for A057597. (End)

a(n) = A001644(-n) for all n in Z. - Michael Somos, Dec 17 2016

Example

G.f. = 3 - x - x^2 + 5*x^3 - 5*x^4 - x^5 + 11*x^6 - 15*x^7 + 3*x^8 + 23*x^9 + ...

Mathematica

A = Table[0, {3}, {3}]; A[[1, 1]] = 1; A[[1, 2]] = 1; A[[2, 1]] = 1; A[[2, 3]] = 1; A[[3, 1]] = 1; For[i = 1; t = IdentityMatrix[3], i < 50, i++, t = t.A; Print[t[[2, 2]]*t[[3, 3]] - t[[2, 3]]*t[[3, 2]] + t[[1, 1]]*t[[3, 3]] - t[[1, 3]]*t[[3, 1]] + t[[1, 1]]*t[[2, 2]] - t[[1, 2]]*t[[2, 1]]]]
LinearRecurrence[{-1, -1, 1}, {3, -1, -1}, 50] (* Vincenzo Librandi, Aug 17 2013 *)

Prog

(Magma) I:=[3, -1, -1]; [n le 3 select I[n] else -Self(n-1)-Self(n-2)+Self(n-3): n in [1..50]]; // Vincenzo Librandi, Aug 17 2013
(PARI) {a(n) = if( n<0, polsym(1 + x+ x^2 - x^3, -n)[-n+1], polsym(1 - x - x^2 - x^3, n)[n+1])}; /* Michael Somos, Dec 17 2016 */
(PARI) a(n)=([0, 1, 0; 0, 0, 1; 1, -1, -1]^n*[3; -1; -1])[1, 1] \\ Charles R Greathouse IV, Feb 07 2017
(Sage) ((3+2*x+x^2)/(1+x+x^2-x^3)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Apr 22 2019

Crossrefs

Cf. A000073, A001644, A057597.

Sequence in context: A301531 A158418 A124925 * A245368 A354971 A239331

Adjacent sequences: A073142 A073143 A073144 * A073146 A073147 A073148

Keyword

easy,sign

Author

Mario Catalani (mario.catalani(AT)unito.it), Jul 17 2002

Extensions

Better name by Joerg Arndt, Aug 17 2013

More terms from Vincenzo Librandi, Aug 17 2013

Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021

Status

approved