OFFSET
0,1
COMMENTS
Also a(n) is the trace of A^(-n), where A is the 4 X 4 matrix ((1,1,0,0), (1,0,1,0), (1,0,0,1), (1,0,0,0)).
REFERENCES
R. L. Graham, D. E. Knuth and O. Patashnik, "Concrete Mathematics", Addison-Wesley, Reading, MA, 1998.
LINKS
Indranil Ghosh, Table of n, a(n) for n = 0..8998
A. V. Zarelua, On Matrix Analogs of Fermat's Little Theorem, Mathematical Notes, vol. 79, no. 6, 2006, pp. 783-796. Translated from Matematicheskie Zametki, vol. 79, no. 6, 2006, pp. 840-855.
Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1,1).
FORMULA
a(n) = -a(n-1)-a(n-2)-a(n-3)+a(n-4), a(0)=4, a(1)=-1, a(2)=-1, a(3)=-1.
G.f.: (4+3x+2x^2+x^3)/(1+x+x^2+x^3-x^4).
From Peter Bala, Jan 19 2023: (Start)
a(n) = (-1)^n*A073937(n).
The Gauss congruences hold: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for positive integers n and r and all primes p. See Zarelua. (End)
MATHEMATICA
CoefficientList[Series[(4+3*x+2*x^2+x^3)/(1+x+x^2+x^3-x^4), {x, 0, 1}], x]
PROG
(PARI) polsym(polrecip(1+x+x^2+x^3-x^4), 55) \\ Joerg Arndt, Jan 21 2023
CROSSREFS
KEYWORD
easy,sign
AUTHOR
Mario Catalani (mario.catalani(AT)unito.it), Aug 16 2002
STATUS
approved