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).
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).
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 *)
nxt[{a_, b_, c_}]:={b, c, a-b-c}; NestList[nxt, {3, -1, -1}, 50][[;; , 1]] (* Harvey P. Dale, Jun 16 2024 *)
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
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